Approximation operators constructed by means of Sheffer sequences

Maria Crăciun
(original), Approximation Theory, html, paper, PDF, works

Abstract

In this paper we introduce a class of positive linear operators by using the “umbral calculus”, and we study some approximation properties of it.

Let \(Q\) be a delta operator, and \(S\) an invertible shift invariant operator.

For \(f\in C[0,1]\) we define \((L_{n}^{Q,S}f)(x)=\frac{1}{sn_{(1)}}\sum \limits_{k=0}^{n}\binomial{n}{k} p_{k}(x)s_{n-k}(1-x)f(\frac{k}{n})\), where \((p_{n})_{n\geq0}\) is a binomial sequence which is the basic sequence for \(Q\), and \((s_{n})_{n\geq0}\) is a Sheffer set, \(s_{n}=S^{-1}p_{n}\).

These operators generalize the binomial operators of T. Popoviciu.

Authors

Maria Craciun

Maria Crăciun

Keywords

approximation operators; Sheffer sequences; basic sequences; delta operators

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https://ictp.acad.ro/craciun/papers/craciun-2001-ANTA.pdf

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M. Crăciun, Approximation operators constructed by means of Sheffer sequences, Rev. Anal. Numér. Théor. Approx., vol. 30 (2001), no. 2, 135-150, https://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no2-art3

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Revue d’Analyse Numérique et de Théorie de l’Approximation

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Editura Academiei Române

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https://doi.org/10.33993/jnaat302-692

 

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1222-9024

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jnaat,+Journal+manager,+2001-2-Craciun-15-10-29
Rev. Anal. Numér. Théor. Approx., vol. 30 (2001) no. 2, pp. 135-150
ictp.acad.ro/jnaat

APPROXIMATION OPERATORS
CONSTRUCTED BY MEANS OF SHEFFER SEQUENCES

MARIA CRĂCIUN

Abstract

In this paper we introduce a class of positive linear operators by using the "umbral calculus", and we study some approximation properties of it. Let Q Q QQQ be a delta operator, and S S SSS an invertible shift invariant operator. For f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] we define ( L n Q , S f ) ( x ) = 1 s n ( 1 ) ∑ k = 0 n ( n k ) p k ( x ) s n − k ( 1 − x ) f ( k n ) , L n Q , S f ( x ) = 1 s n ( 1 ) ∑ k = 0 n   ( n k ) p k ( x ) s n − k ( 1 − x ) f k n , (L_(n)^(Q,S)f)(x)=(1)/(s_(n)(1))sum_(k=0)^(n)((n)/(k))p_(k)(x)s_(n-k)(1-x)f((k)/(n)),\left(L_{n}^{Q, S} f\right)(x)=\frac{1}{s_{n}(1)} \sum_{k=0}^{n}\binom{n}{k} p_{k}(x) s_{n-k}(1-x) f\left(\frac{k}{n}\right),(LnQ,Sf)(x)=1sn(1)∑k=0n(nk)pk(x)sn−k(1−x)f(kn), where ( p n ) n ≥ 0 p n n ≥ 0 (p_(n))_(n >= 0)\left(p_{n}\right)_{n \geq 0}(pn)n≥0 is a binomial sequence which is the basic sequence for Q Q QQQ, and ( s n ) n ≥ 0 s n n ≥ 0 (s_(n))_(n >= 0)\left(s_{n}\right)_{n \geq 0}(sn)n≥0 is a Sheffer set, s n = S − 1 p n s n = S − 1 p n s_(n)=S^(-1)p_(n)s_{n}=S^{-1} p_{n}sn=S−1pn. These operators generalize the binomial operators of T. Popoviciu.

MSC 2000. 41A36, 05A40.

1. INTRODUCTION

Let P P PPP be the linear space of all polynomials with real coefficients, and P n P n P_(n)P_{n}Pn the linear space of all polynomials of degree at most n n nnn.
We will consider some linear operators defined on P P PPP. We will denote by I I III the identity and by D D DDD the derivative. The shift operator E a : P → P E a : P → P E^(a):P rarr PE^{a}: P \rightarrow PEa:P→P is defined by E a p ( x ) = p ( x + a ) E a p ( x ) = p ( x + a ) E^(a)p(x)=p(x+a)E^{a} p(x)=p(x+a)Eap(x)=p(x+a).
A linear operator T T TTT which commutes with all shift operators is called a shift invariant operator. In symbols, E a T = T E a E a T = T E a E^(a)T=TE^(a)E^{a} T=T E^{a}EaT=TEa, for all real a a aaa.
Let us remind that if T 1 T 1 T_(1)T_{1}T1 and T 2 T 2 T_(2)T_{2}T2 are shift invariant operators, then T 1 T 2 = T 2 T 1 T 1 T 2 = T 2 T 1 T_(1)T_(2)=T_(2)T_(1)T_{1} T_{2}= T_{2} T_{1}T1T2=T2T1.
Definition 1. A shift invariant operator for which Q x = Q x = Qx=Q x=Qx= const ≠ 0 ≠ 0 !=0\neq 0≠0 is called a delta operator.
By a polynomial sequence we shall denote a sequence of polynomials p n ( x ) p n ( x ) p_(n)(x)p_{n}(x)pn(x), n = 0 , 1 , 2 , … n = 0 , 1 , 2 , … n=0,1,2,dotsn=0,1,2, \ldotsn=0,1,2,… where p n ( x ) p n ( x ) p_(n)(x)p_{n}(x)pn(x) is of degree exactly n n nnn for all n n nnn.
A sequence of binomial type is a polynomial sequence ( p n ) n ≥ 0 p n n ≥ 0 (p_(n))_(n >= 0)\left(p_{n}\right)_{n \geq 0}(pn)n≥0 with p 0 ( x ) = 1 p 0 ( x ) = 1 p_(0)(x)=1p_{0}(x)=1p0(x)=1 and satisfying the identities
p n ( x + y ) = ∑ k = 0 n ( n k ) p k ( x ) p n − k ( y ) , p n ( x + y ) = ∑ k = 0 n   ( n k ) p k ( x ) p n − k ( y ) , p_(n)(x+y)=sum_(k=0)^(n)((n)/(k))p_(k)(x)p_(n-k)(y),p_{n}(x+y)=\sum_{k=0}^{n}\binom{n}{k} p_{k}(x) p_{n-k}(y),pn(x+y)=∑k=0n(nk)pk(x)pn−k(y),
for all x , y x , y x,yx, yx,y and n = 0 , 1 , 2 , … n = 0 , 1 , 2 , … n=0,1,2,dotsn=0,1,2, \ldotsn=0,1,2,….
Definition 2. Let Q Q QQQ be a delta operator and ( p n ( x ) ) n ≥ 0 p n ( x ) n ≥ 0 (p_(n)(x))_(n >= 0)\left(p_{n}(x)\right)_{n \geq 0}(pn(x))n≥0 a polynomial sequence. If
i) p 0 ( x ) = 1 p 0 ( x ) = 1 quadp_(0)(x)=1\quad p_{0}(x)=1p0(x)=1,
ii) p n ( 0 ) = 0 , n = 1 , 2 , … p n ( 0 ) = 0 , n = 1 , 2 , … quadp_(n)(0)=0,n=1,2,dots\quad p_{n}(0)=0, n=1,2, \ldotspn(0)=0,n=1,2,…,
iii) Q p n = n p n − 1 , n = 1 , 2 , … Q p n = n p n − 1 , n = 1 , 2 , … quad Qp_(n)=np_(n-1),n=1,2,dots\quad Q p_{n}=n p_{n-1}, n=1,2, \ldotsQpn=npn−1,n=1,2,…,
then ( p n ) p n (p_(n))\left(p_{n}\right)(pn) is called the sequence of basic polynomials for Q Q QQQ.
Proposition 1. 8].
i) Every delta operator has a unique sequence of basic polynomials.
ii) If p n ( x ) p n ( x ) p_(n)(x)p_{n}(x)pn(x) is a basic sequence for some delta operator Q Q QQQ, then it is binomial.
iii) If p n ( x ) p n ( x ) p_(n)(x)p_{n}(x)pn(x) is a binomial sequence, then it is a basic sequence for some delta operator Q Q QQQ.
Let X X XXX be the multiplication operator defined as ( X p ) ( x ) = x p ( x ) ( X p ) ( x ) = x p ( x ) (Xp)(x)=xp(x)(X p)(x)=x p(x)(Xp)(x)=xp(x) for every polynomial p p ppp.
For any operator T T TTT defined on P P PPP, the operator T ′ = T X − X T T ′ = T X − X T T^(')=TX-XTT^{\prime}=T X-X TT′=TX−XT is called the Pincherle derivative of the operator T T TTT.
Proposition 2. 8].
i) If T T TTT is a shift invariant operator, then its Pincherle derivative is also a shift invariant operator.
ii) If Q Q QQQ is a delta operator, then its Pincherle derivative Q ′ Q ′ Q^(')Q^{\prime}Q′ is an invertible operator.
Proposition 3. [8], [11]. If ( p n ( x ) ) n ≥ 0 p n ( x ) n ≥ 0 (p_(n)(x))_(n >= 0)\left(p_{n}(x)\right)_{n \geq 0}(pn(x))n≥0 is a sequence of basic polynomials for the delta operator Q Q QQQ then
i) p n ( x ) = X ( Q ′ ) − 1 p n − 1 ( x ) , n = 1 , 2 , … p n ( x ) = X Q ′ − 1 p n − 1 ( x ) , n = 1 , 2 , … quadp_(n)(x)=X(Q^('))^(-1)p_(n-1)(x),n=1,2,dots\quad p_{n}(x)=X\left(Q^{\prime}\right)^{-1} p_{n-1}(x), n=1,2, \ldotspn(x)=X(Q′)−1pn−1(x),n=1,2,…,
ii) p n ( x ) = x ∑ k = 0 n − 1 ( n − 1 k ) p n − 1 − k ( x ) p k + 1 ′ ( 0 ) , n = 1 , 2 , … p n ( x ) = x ∑ k = 0 n − 1   ( n − 1 k ) p n − 1 − k ( x ) p k + 1 ′ ( 0 ) , n = 1 , 2 , … quadp_(n)(x)=xsum_(k=0)^(n-1)((n-1)/(k))p_(n-1-k)(x)p_(k+1)^(')(0),n=1,2,dots\quad p_{n}(x)=x \sum_{k=0}^{n-1}\binom{n-1}{k} p_{n-1-k}(x) p_{k+1}^{\prime}(0), n=1,2, \ldotspn(x)=x∑k=0n−1(n−1k)pn−1−k(x)pk+1′(0),n=1,2,…
Definition 3. A polynomial sequence ( s n ( x ) ) n ≥ 0 s n ( x ) n ≥ 0 (s_(n)(x))_(n >= 0)\left(s_{n}(x)\right)_{n \geq 0}(sn(x))n≥0 is called a Sheffer set relative to the delta operator Q Q QQQ if:
i) s 0 ( x ) = s 0 ( x ) = quads_(0)(x)=\quad s_{0}(x)=s0(x)= const ≠ 0 ≠ 0 !=0\neq 0≠0
ii) Q s n = n s n − 1 , n = 1 , 2 , … Q s n = n s n − 1 , n = 1 , 2 , … Qs_(n)=ns_(n-1),n=1,2,dotsQ s_{n}=n s_{n-1}, n=1,2, \ldotsQsn=nsn−1,n=1,2,…
An Appel set is a Sheffer set relative to the derivative D D DDD.
Proposition 4. 11. Let Q Q QQQ be a delta operator with basic polynomial set ( p n ( x ) ) n ≥ 0 p n ( x ) n ≥ 0 (p_(n)(x))_(n >= 0)\left(p_{n}(x)\right)_{n \geq 0}(pn(x))n≥0 and ( s n ( x ) ) n ≥ 0 s n ( x ) n ≥ 0 (s_(n)(x))_(n >= 0)\left(s_{n}(x)\right)_{n \geq 0}(sn(x))n≥0 a polynomial sequence. The next statements are equivalent:
i) s n ( x ) s n ( x ) s_(n)(x)s_{n}(x)sn(x) is a Sheffer set relative to Q Q QQQ.
ii) There exists an invertible shift invariant operator S S SSS such that s n ( x ) = S − 1 p n ( x ) s n ( x ) = S − 1 p n ( x ) s_(n)(x)=S^(-1)p_(n)(x)s_{n}(x)= S^{-1} p_{n}(x)sn(x)=S−1pn(x).
iii) For all x , y ∈ R x , y ∈ R x,y inRx, y \in \mathbb{R}x,y∈R and n = 0 , 1 , 2 , … n = 0 , 1 , 2 , … n=0,1,2,dotsn=0,1,2, \ldotsn=0,1,2,…, the following identity holds:
s n ( x + y ) = ∑ k = 0 n ( n k ) p k ( x ) s n − k ( y ) s n ( x + y ) = ∑ k = 0 n   ( n k ) p k ( x ) s n − k ( y ) s_(n)(x+y)=sum_(k=0)^(n)((n)/(k))p_(k)(x)s_(n-k)(y)s_{n}(x+y)=\sum_{k=0}^{n}\binom{n}{k} p_{k}(x) s_{n-k}(y)sn(x+y)=∑k=0n(nk)pk(x)sn−k(y)
From the previous Proposition it results that the pair ( Q , S Q , S Q,SQ, SQ,S ) gives us a unique Sheffer set.

2. THE OPERATORS CONSTRUCTED BY MEANS OF SHEFFER POLYNOMIALS AND THEIR CONVERGENCE

In 1931 in [9] Tiberiu Popoviciu has used binomial sequences in order to construct some operators of the form
(1) ( L n f ) ( x ) = 1 p n ( 1 ) ∑ k = 0 n ( n k ) p k ( x ) p n − k ( 1 − x ) f ( k n ) (1) L n f ( x ) = 1 p n ( 1 ) ∑ k = 0 n   ( n k ) p k ( x ) p n − k ( 1 − x ) f k n {:(1)(L_(n)f)(x)=(1)/(p_(n)(1))sum_(k=0)^(n)((n)/(k))p_(k)(x)p_(n-k)(1-x)f((k)/(n)):}\begin{equation*} \left(L_{n} f\right)(x)=\frac{1}{p_{n}(1)} \sum_{k=0}^{n}\binom{n}{k} p_{k}(x) p_{n-k}(1-x) f\left(\frac{k}{n}\right) \tag{1} \end{equation*}(1)(Lnf)(x)=1pn(1)∑k=0n(nk)pk(x)pn−k(1−x)f(kn)
where f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] and x ∈ [ 0 , 1 ] x ∈ [ 0 , 1 ] x in[0,1]x \in[0,1]x∈[0,1]. These operators are called binomial operators.
Such operators and their generalizations have been studied by the Romanian mathematicians as: D. D. Stancu, A. Lupaş, L. Lupaş, G. Moldovan, C. Manole, O. Agratini, A. Vernescu, and others.
Let Q Q QQQ be a delta operator and S S SSS an invertible shift invariant operator. Let ( p n ( x ) ) n ≥ 0 p n ( x ) n ≥ 0 (p_(n)(x))_(n >= 0)\left(p_{n}(x)\right)_{n \geq 0}(pn(x))n≥0 be the sequence of basic polynomials for Q Q QQQ, and ( s n ( x ) ) n ≥ 0 s n ( x ) n ≥ 0 (s_(n)(x))_(n >= 0)\left(s_{n}(x)\right)_{n \geq 0}(sn(x))n≥0 a Sheffer set relative to Q , s n = S − 1 p n Q , s n = S − 1 p n Q,s_(n)=S^(-1)p_(n)Q, s_{n}=S^{-1} p_{n}Q,sn=S−1pn with s n ( 1 ) ≠ 0 s n ( 1 ) ≠ 0 s_(n)(1)!=0s_{n}(1) \neq 0sn(1)≠0 for any positive integer n n nnn.
In this note we want to study the operators L n Q , S : C [ 0 , 1 ] → C [ 0 , 1 ] L n Q , S : C [ 0 , 1 ] → C [ 0 , 1 ] L_(n)^(Q,S):C[0,1]rarr C[0,1]L_{n}^{Q, S}: C[0,1] \rightarrow C[0,1]LnQ,S:C[0,1]→C[0,1],
(2) ( L n Q , S f ) ( x ) = 1 s n ( 1 ) ∑ k = 0 n ( n k ) p k ( x ) s n − k ( 1 − x ) f ( k n ) (2) L n Q , S f ( x ) = 1 s n ( 1 ) ∑ k = 0 n   ( n k ) p k ( x ) s n − k ( 1 − x ) f k n {:(2)(L_(n)^(Q,S)f)(x)=(1)/(s_(n)(1))sum_(k=0)^(n)((n)/(k))p_(k)(x)s_(n-k)(1-x)f((k)/(n)):}\begin{equation*} \left(L_{n}^{Q, S} f\right)(x)=\frac{1}{s_{n}(1)} \sum_{k=0}^{n}\binom{n}{k} p_{k}(x) s_{n-k}(1-x) f\left(\frac{k}{n}\right) \tag{2} \end{equation*}(2)(LnQ,Sf)(x)=1sn(1)∑k=0n(nk)pk(x)sn−k(1−x)f(kn)
Because p k ( 0 ) = δ k , 0 p k ( 0 ) = δ k , 0 p_(k)(0)=delta_(k,0)p_{k}(0)=\delta_{k, 0}pk(0)=δk,0 (from the definition of basic polynomials), we have ( L n Q , S f ) ( 0 ) = f ( 0 ) L n Q , S f ( 0 ) = f ( 0 ) (L_(n)^(Q,S)f)(0)=f(0)\left(L_{n}^{Q, S} f\right)(0)=f(0)(LnQ,Sf)(0)=f(0).
In order to evaluate expression ( L n Q , S e m ) ( x ) L n Q , S e m ( x ) (L_(n)^(Q,S)e_(m))(x)\left(L_{n}^{Q, S} e_{m}\right)(x)(LnQ,Sem)(x), where e m ( x ) = x m e m ( x ) = x m e_(m)(x)=x^(m)e_{m}(x)=x^{m}em(x)=xm we shall make use of C. Manole's method for binomial operators (see [5]) which we have adapted to our purposes.
Let us introduce the polynomials
(3) S m ( x , y , n ) = ∑ k = 0 n ( n k ) p k ( x ) s n − k ( y ) ( k n ) m (3) S m ( x , y , n ) = ∑ k = 0 n   ( n k ) p k ( x ) s n − k ( y ) k n m {:(3)S_(m)(x","y","n)=sum_(k=0)^(n)((n)/(k))p_(k)(x)s_(n-k)(y)((k)/(n))^(m):}\begin{equation*} S_{m}(x, y, n)=\sum_{k=0}^{n}\binom{n}{k} p_{k}(x) s_{n-k}(y)\left(\frac{k}{n}\right)^{m} \tag{3} \end{equation*}(3)Sm(x,y,n)=∑k=0n(nk)pk(x)sn−k(y)(kn)m
From Proposition 4 iii) we have S 0 ( x , y , n ) = s n ( x + y ) S 0 ( x , y , n ) = s n ( x + y ) S_(0)(x,y,n)=s_(n)(x+y)S_{0}(x, y, n)=s_{n}(x+y)S0(x,y,n)=sn(x+y).
In the following we consider that x x xxx is the variable. Let us denote θ = X ( Q ′ ) − 1 θ = X Q ′ − 1 theta=X(Q^('))^(-1)\theta= X\left(Q^{\prime}\right)^{-1}θ=X(Q′)−1.
From Proposition 3 i) it results that θ p k ( x ) = p k + 1 ( x ) θ p k ( x ) = p k + 1 ( x ) thetap_(k)(x)=p_(k+1)(x)\theta p_{k}(x)=p_{k+1}(x)θpk(x)=pk+1(x) and consequently the linear operator θ θ theta\thetaθ is called the shift operator for the sequence ( p n ) n ≥ 0 p n n ≥ 0 (p_(n))_(n >= 0)\left(p_{n}\right)_{n \geq 0}(pn)n≥0 (see [10]). Therefore θ Q p k ( x ) = θ ( k p k − 1 ( x ) ) = k p k ( x ) θ Q p k ( x ) = θ k p k − 1 ( x ) = k p k ( x ) theta Qp_(k)(x)=theta(kp_(k-1)(x))=kp_(k)(x)\theta Q p_{k}(x)=\theta\left(k p_{k-1}(x)\right)=k p_{k}(x)θQpk(x)=θ(kpk−1(x))=kpk(x); consequently k k kkk is an eigenvalue for the operator θ Q θ Q theta Q\theta QθQ, with its eigenvector p k ( x ) p k ( x ) p_(k)(x)p_{k}(x)pk(x). We have
(4) ( θ Q ) m = k m p k ( x ) (4) ( θ Q ) m = k m p k ( x ) {:(4)(theta Q)^(m)=k^(m)p_(k)(x):}\begin{equation*} (\theta Q)^{m}=k^{m} p_{k}(x) \tag{4} \end{equation*}(4)(θQ)m=kmpk(x)
for every positive integer m m mmm, and then
S m ( x , y , n ) = 1 n m ( θ Q ) m ∑ k = 0 n ( n k ) p k ( x ) s n − k ( y ) = 1 n m ( θ Q ) m S 0 ( x , y , n ) = 1 n m ( θ Q ) m s n ( x + y ) S m ( x , y , n ) = 1 n m ( θ Q ) m ∑ k = 0 n   ( n k ) p k ( x ) s n − k ( y ) = 1 n m ( θ Q ) m S 0 ( x , y , n ) = 1 n m ( θ Q ) m s n ( x + y ) {:[S_(m)(x","y","n)=(1)/(n^(m))(theta Q)^(m)sum_(k=0)^(n)((n)/(k))p_(k)(x)s_(n-k)(y)],[=(1)/(n^(m))(theta Q)^(m)S_(0)(x","y","n)=(1)/(n^(m))(theta Q)^(m)s_(n)(x+y)]:}\begin{aligned} S_{m}(x, y, n) & =\frac{1}{n^{m}}(\theta Q)^{m} \sum_{k=0}^{n}\binom{n}{k} p_{k}(x) s_{n-k}(y) \\ & =\frac{1}{n^{m}}(\theta Q)^{m} S_{0}(x, y, n)=\frac{1}{n^{m}}(\theta Q)^{m} s_{n}(x+y) \end{aligned}Sm(x,y,n)=1nm(θQ)m∑k=0n(nk)pk(x)sn−k(y)=1nm(θQ)mS0(x,y,n)=1nm(θQ)msn(x+y)
In this way we obtain
(5) S m ( x , y , n ) = 1 n m ( θ Q ) m E y s n ( x ) (5) S m ( x , y , n ) = 1 n m ( θ Q ) m E y s n ( x ) {:(5)S_(m)(x","y","n)=(1)/(n^(m))(theta Q)^(m)E^(y)s_(n)(x):}\begin{equation*} S_{m}(x, y, n)=\frac{1}{n^{m}}(\theta Q)^{m} E^{y} s_{n}(x) \tag{5} \end{equation*}(5)Sm(x,y,n)=1nm(θQ)mEysn(x)
Using the operational formula (see for instance [10])
( θ Q ) m = ∑ k = 0 n S ( m , k ) θ k Q k ( θ Q ) m = ∑ k = 0 n   S ( m , k ) θ k Q k (theta Q)^(m)=sum_(k=0)^(n)S(m,k)theta^(k)Q^(k)(\theta Q)^{m}=\sum_{k=0}^{n} S(m, k) \theta^{k} Q^{k}(θQ)m=∑k=0nS(m,k)θkQk
where S ( m , k ) = [ 0 , 1 , … , k ; e m ] S ( m , k ) = 0 , 1 , … , k ; e m S(m,k)=[0,1,dots,k;e_(m)]S(m, k)=\left[0,1, \ldots, k ; e_{m}\right]S(m,k)=[0,1,…,k;em] are the Stirling numbers of the second kind, relation (5) becomes:
(6) S m ( x , y , n ) = 1 n m ∑ k = 0 n S ( m , k ) θ k Q k E y s n ( x ) (6) S m ( x , y , n ) = 1 n m ∑ k = 0 n   S ( m , k ) θ k Q k E y s n ( x ) {:(6)S_(m)(x","y","n)=(1)/(n^(m))sum_(k=0)^(n)S(m","k)theta^(k)Q^(k)E^(y)s_(n)(x):}\begin{equation*} S_{m}(x, y, n)=\frac{1}{n^{m}} \sum_{k=0}^{n} S(m, k) \theta^{k} Q^{k} E^{y} s_{n}(x) \tag{6} \end{equation*}(6)Sm(x,y,n)=1nm∑k=0nS(m,k)θkQkEysn(x)
Because Q Q QQQ is shift invariant and Q k s n ( x ) = n ( n − 1 ) … ( n − k + 1 ) s n − k ( x ) = n [ k ] s n − k ( x ) Q k s n ( x ) = n ( n − 1 ) … ( n − k + 1 ) s n − k ( x ) = n [ k ] s n − k ( x ) Q^(k)s_(n)(x)=n(n-1)dots(n-k+1)s_(n-k)(x)=n^([k])s_(n-k)(x)Q^{k} s_{n}(x)=n(n-1) \ldots(n-k+1) s_{n-k}(x)= n^{[k]} s_{n-k}(x)Qksn(x)=n(n−1)…(n−k+1)sn−k(x)=n[k]sn−k(x) we obtain
(7) S m ( x , y , n ) = 1 n m ∑ k = 0 n ( n k ) k ! S ( m , k ) θ k E y s n − k ( x ) , ∀ m ∈ N ∗ (7) S m ( x , y , n ) = 1 n m ∑ k = 0 n   ( n k ) k ! S ( m , k ) θ k E y s n − k ( x ) , ∀ m ∈ N ∗ {:(7)S_(m)(x","y","n)=(1)/(n^(m))sum_(k=0)^(n)((n)/(k))k!S(m","k)theta^(k)E^(y)s_(n-k)(x)","AA m inN^(**):}\begin{equation*} S_{m}(x, y, n)=\frac{1}{n^{m}} \sum_{k=0}^{n}\binom{n}{k} k!S(m, k) \theta^{k} E^{y} s_{n-k}(x), \forall m \in \mathbb{N}^{*} \tag{7} \end{equation*}(7)Sm(x,y,n)=1nm∑k=0n(nk)k!S(m,k)θkEysn−k(x),∀m∈N∗
Theorem 1. If L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S is the linear operator defined by (2) then
( L n Q , S e 0 ) ( x ) = e 0 ( x ) (8) ( L n Q , S e 1 ) ( x ) = a n e 1 ( x ) ( L n Q , S e 2 ) ( x ) = b n x 2 + x ( a n − b n − c n ) , L n Q , S e 0 ( x ) = e 0 ( x ) (8) L n Q , S e 1 ( x ) = a n e 1 ( x ) L n Q , S e 2 ( x ) = b n x 2 + x a n − b n − c n , {:[(L_(n)^(Q,S)e_(0))(x)=e_(0)(x)],[(8)(L_(n)^(Q,S)e_(1))(x)=a_(n)e_(1)(x)],[(L_(n)^(Q,S)e_(2))(x)=b_(n)x^(2)+x(a_(n)-b_(n)-c_(n))","]:}\begin{align*} & \left(L_{n}^{Q, S} e_{0}\right)(x)=e_{0}(x) \\ & \left(L_{n}^{Q, S} e_{1}\right)(x)=a_{n} e_{1}(x) \tag{8}\\ & \left(L_{n}^{Q, S} e_{2}\right)(x)=b_{n} x^{2}+x\left(a_{n}-b_{n}-c_{n}\right), \end{align*}(LnQ,Se0)(x)=e0(x)(8)(LnQ,Se1)(x)=ane1(x)(LnQ,Se2)(x)=bnx2+x(an−bn−cn),
where
a n = [ ( Q ′ ) − 1 s n − 1 ] ( 1 ) s n ( 1 ) (9) b n = n − 1 n [ ( Q ′ ) − 2 s n − 2 ] ( 1 ) s n ( 1 ) c n = n − 1 n [ ( Q ′ ) − 2 ( S − 1 ) ′ S s n − 2 ] ( 1 ) s n ( 1 ) a n = Q ′ − 1 s n − 1 ( 1 ) s n ( 1 ) (9) b n = n − 1 n Q ′ − 2 s n − 2 ( 1 ) s n ( 1 ) c n = n − 1 n Q ′ − 2 S − 1 ′ S s n − 2 ( 1 ) s n ( 1 ) {:[a_(n)=([(Q^('))^(-1)s_(n-1)](1))/(s_(n)(1))],[(9)b_(n)=(n-1)/(n)([(Q^('))^(-2)s_(n-2)](1))/(s_(n)(1))],[c_(n)=(n-1)/(n)([(Q^('))^(-2)(S^(-1))^(')Ss_(n-2)](1))/(s_(n)(1))]:}\begin{align*} & a_{n}=\frac{\left[\left(Q^{\prime}\right)^{-1} s_{n-1}\right](1)}{s_{n}(1)} \\ & b_{n}=\frac{n-1}{n} \frac{\left[\left(Q^{\prime}\right)^{-2} s_{n-2}\right](1)}{s_{n}(1)} \tag{9}\\ & c_{n}=\frac{n-1}{n} \frac{\left[\left(Q^{\prime}\right)^{-2}\left(S^{-1}\right)^{\prime} S s_{n-2}\right](1)}{s_{n}(1)} \end{align*}an=[(Q′)−1sn−1](1)sn(1)(9)bn=n−1n[(Q′)−2sn−2](1)sn(1)cn=n−1n[(Q′)−2(S−1)′Ssn−2](1)sn(1)
Proof. Using the notation (3) we can write
(10) ( L n Q , S e m ) ( x ) = S m ( x , 1 − x , n ) / s n ( 1 ) (10) L n Q , S e m ( x ) = S m ( x , 1 − x , n ) / s n ( 1 ) {:(10)(L_(n)^(Q,S)e_(m))(x)=S_(m)(x","1-x","n)//s_(n)(1):}\begin{equation*} \left(L_{n}^{Q, S} e_{m}\right)(x)=S_{m}(x, 1-x, n) / s_{n}(1) \tag{10} \end{equation*}(10)(LnQ,Sem)(x)=Sm(x,1−x,n)/sn(1)
Because S 0 ( x , 1 − x , n ) = s n ( 1 ) S 0 ( x , 1 − x , n ) = s n ( 1 ) S_(0)(x,1-x,n)=s_(n)(1)S_{0}(x, 1-x, n)=s_{n}(1)S0(x,1−x,n)=sn(1) we have ( L n Q , S e 0 ) ( x ) = e 0 ( x ) L n Q , S e 0 ( x ) = e 0 ( x ) (L_(n)^(Q,S)e_(0))(x)=e_(0)(x)\left(L_{n}^{Q, S} e_{0}\right)(x)=e_{0}(x)(LnQ,Se0)(x)=e0(x).
As we have
(11) θ E y s n − 1 ( x ) = X ( Q ′ ) − 1 E y s n − 1 ( x ) = X E y ( Q ′ ) − 1 s n − 1 ( x ) (11) θ E y s n − 1 ( x ) = X Q ′ − 1 E y s n − 1 ( x ) = X E y Q ′ − 1 s n − 1 ( x ) {:(11)thetaE^(y)s_(n-1)(x)=X(Q^('))^(-1)E^(y)s_(n-1)(x)=XE^(y)(Q^('))^(-1)s_(n-1)(x):}\begin{equation*} \theta E^{y} s_{n-1}(x)=X\left(Q^{\prime}\right)^{-1} E^{y} s_{n-1}(x)=X E^{y}\left(Q^{\prime}\right)^{-1} s_{n-1}(x) \tag{11} \end{equation*}(11)θEysn−1(x)=X(Q′)−1Eysn−1(x)=XEy(Q′)−1sn−1(x)
we obtain from (7): S 1 ( x , 1 − x , n ) = x [ ( Q ′ ) − 1 s n − 1 ] ( 1 ) S 1 ( x , 1 − x , n ) = x Q ′ − 1 s n − 1 ( 1 ) S_(1)(x,1-x,n)=x[(Q^('))^(-1)s_(n-1)](1)S_{1}(x, 1-x, n)=x\left[\left(Q^{\prime}\right)^{-1} s_{n-1}\right](1)S1(x,1−x,n)=x[(Q′)−1sn−1](1); consequently we get:
( L n Q , S e 1 ) ( x ) = [ ( Q ′ ) − 1 s n − 1 ] ( 1 ) s n ( 1 ) x . L n Q , S e 1 ( x ) = Q ′ − 1 s n − 1 ( 1 ) s n ( 1 ) x . (L_(n)^(Q,S)e_(1))(x)=([(Q^('))^(-1)s_(n-1)](1))/(s_(n)(1))x.\left(L_{n}^{Q, S} e_{1}\right)(x)=\frac{\left[\left(Q^{\prime}\right)^{-1} s_{n-1}\right](1)}{s_{n}(1)} x .(LnQ,Se1)(x)=[(Q′)−1sn−1](1)sn(1)x.
Using the Pincherle derivative of the shift operator E y E y E^(y)E^{y}Ey
(12) ( E y ) ′ = y E y = E y X − X E y (12) E y ′ = y E y = E y X − X E y {:(12)(E^(y))^(')=yE^(y)=E^(y)X-XE^(y):}\begin{equation*} \left(E^{y}\right)^{\prime}=y E^{y}=E^{y} X-X E^{y} \tag{12} \end{equation*}(12)(Ey)′=yEy=EyX−XEy
we can write
θ E y s n − k ( x ) = E y X ( Q ′ ) − 1 s n − k ( x ) − y E y ( Q ′ ) − 1 s n − k ( x ) θ E y s n − k ( x ) = E y X Q ′ − 1 s n − k ( x ) − y E y Q ′ − 1 s n − k ( x ) thetaE^(y)s_(n-k)(x)=E^(y)X(Q^('))^(-1)s_(n-k)(x)-yE^(y)(Q^('))^(-1)s_(n-k)(x)\theta E^{y} s_{n-k}(x)=E^{y} X\left(Q^{\prime}\right)^{-1} s_{n-k}(x)-y E^{y}\left(Q^{\prime}\right)^{-1} s_{n-k}(x)θEysn−k(x)=EyX(Q′)−1sn−k(x)−yEy(Q′)−1sn−k(x)
Then
(13) θ 2 E y s n − k ( x ) = X E y ( Q ′ ) − 1 E y X ( Q ′ ) − 1 s n − k ( x ) − y X E y ( Q ′ ) − 2 s n − k ( x ) (13) θ 2 E y s n − k ( x ) = X E y Q ′ − 1 E y X Q ′ − 1 s n − k ( x ) − y X E y Q ′ − 2 s n − k ( x ) {:(13)theta^(2)E^(y)s_(n-k)(x)=XE^(y)(Q^('))^(-1)E^(y)X(Q^('))^(-1)s_(n-k)(x)-yXE^(y)(Q^('))^(-2)s_(n-k)(x):}\begin{equation*} \theta^{2} E^{y} s_{n-k}(x)=X E^{y}\left(Q^{\prime}\right)^{-1} E^{y} X\left(Q^{\prime}\right)^{-1} s_{n-k}(x)-y X E^{y}\left(Q^{\prime}\right)^{-2} s_{n-k}(x) \tag{13} \end{equation*}(13)θ2Eysn−k(x)=XEy(Q′)−1EyX(Q′)−1sn−k(x)−yXEy(Q′)−2sn−k(x)
Because s n − k = S − 1 p n − k , X S − 1 = S − 1 X − ( S − 1 ) ′ s n − k = S − 1 p n − k , X S − 1 = S − 1 X − S − 1 ′ s_(n-k)=S^(-1)p_(n-k),XS^(-1)=S^(-1)X-(S^(-1))^(')s_{n-k}=S^{-1} p_{n-k}, X S^{-1}=S^{-1} X-\left(S^{-1}\right)^{\prime}sn−k=S−1pn−k,XS−1=S−1X−(S−1)′ (from the definition of Pincherle derivative) and ( Q ′ ) − 1 p n − k ( x ) = p n − k + 1 ( x ) / x Q ′ − 1 p n − k ( x ) = p n − k + 1 ( x ) / x (Q^('))^(-1)p_(n-k)(x)=p_(n-k+1)(x)//x\left(Q^{\prime}\right)^{-1} p_{n-k}(x)=p_{n-k+1}(x) / x(Q′)−1pn−k(x)=pn−k+1(x)/x (from Proposition 3 i), we obtain
(14) θ 2 E y s n − k ( x ) = X E y ( Q ′ ) − 1 s n − k + 1 ( x ) − X E y ( Q ′ ) − 2 ( S − 1 ) ′ S s n − k ( x ) − − y X E y ( Q ′ ) − 2 s n − k ( x ) (14) θ 2 E y s n − k ( x ) = X E y Q ′ − 1 s n − k + 1 ( x ) − X E y Q ′ − 2 S − 1 ′ S s n − k ( x ) − − y X E y Q ′ − 2 s n − k ( x ) {:[(14)theta^(2)E^(y)s_(n-k)(x)=XE^(y)(Q^('))^(-1)s_(n-k+1)(x)-XE^(y)(Q^('))^(-2)(S^(-1))^(')Ss_(n-k)(x)-],[-yXE^(y)(Q^('))^(-2)s_(n-k)(x)]:}\begin{align*} \theta^{2} E^{y} s_{n-k}(x)= & X E^{y}\left(Q^{\prime}\right)^{-1} s_{n-k+1}(x)-X E^{y}\left(Q^{\prime}\right)^{-2}\left(S^{-1}\right)^{\prime} S s_{n-k}(x)- \tag{14}\\ & -y X E^{y}\left(Q^{\prime}\right)^{-2} s_{n-k}(x) \end{align*}(14)θ2Eysn−k(x)=XEy(Q′)−1sn−k+1(x)−XEy(Q′)−2(S−1)′Ssn−k(x)−−yXEy(Q′)−2sn−k(x)
Replacing (11) and (14) in (7) we can write
S 2 ( x , y , n ) = x E y ( Q ′ ) − 1 s n − 1 ( x ) − n − 1 n [ x E y ( Q ′ ) − 2 ( S − 1 ) ′ S s n − 2 ( x ) − − y x E y ( Q ′ ) − 2 s n − 2 ( x ) ] S 2 ( x , y , n ) = x E y Q ′ − 1 s n − 1 ( x ) − n − 1 n x E y Q ′ − 2 S − 1 ′ S s n − 2 ( x ) − − y x E y Q ′ − 2 s n − 2 ( x ) {:[S_(2)(x","y","n)=xE^(y)(Q^('))^(-1)s_(n-1)(x)-(n-1)/(n)[xE^(y)(Q^('))^(-2)(S^(-1))^(')Ss_(n-2)(x)-:}],[{:-yxE^(y)(Q^('))^(-2)s_(n-2)(x)]]:}\begin{aligned} S_{2}(x, y, n)= & x E^{y}\left(Q^{\prime}\right)^{-1} s_{n-1}(x)-\frac{n-1}{n}\left[x E^{y}\left(Q^{\prime}\right)^{-2}\left(S^{-1}\right)^{\prime} S s_{n-2}(x)-\right. \\ & \left.-y x E^{y}\left(Q^{\prime}\right)^{-2} s_{n-2}(x)\right] \end{aligned}S2(x,y,n)=xEy(Q′)−1sn−1(x)−n−1n[xEy(Q′)−2(S−1)′Ssn−2(x)−−yxEy(Q′)−2sn−2(x)]
From (10) and the previous relation one obtains expression L n Q , S e 2 L n Q , S e 2 L_(n)^(Q,S)e_(2)L_{n}^{Q, S} e_{2}LnQ,Se2 from theorem's conclusion.
Lemma 1. Let Q Q QQQ be a delta operator and S S SSS an invertible shift invariant operator. Let ( p n ( x ) ) n ≥ 0 p n ( x ) n ≥ 0 (p_(n)(x))_(n >= 0)\left(p_{n}(x)\right)_{n \geq 0}(pn(x))n≥0 be the sequence of basic polynomials for Q Q QQQ and ( s n ( x ) ) n ≥ 0 s n ( x ) n ≥ 0 (s_(n)(x))_(n >= 0)\left(s_{n}(x)\right)_{n \geq 0}(sn(x))n≥0 a Sheffer set relative to Q , s n = S − 1 p n Q , s n = S − 1 p n Q,s_(n)=S^(-1)p_(n)Q, s_{n}=S^{-1} p_{n}Q,sn=S−1pn with s n ( 1 ) ≠ 0 s n ( 1 ) ≠ 0 s_(n)(1)!=0s_{n}(1) \neq 0sn(1)≠0 for any positive integer n n nnn. If p k ′ ( 0 ) ≥ 0 p k ′ ( 0 ) ≥ 0 p_(k)^(')(0) >= 0p_{k}^{\prime}(0) \geq 0pk′(0)≥0 and s k ( 0 ) ≥ 0 s k ( 0 ) ≥ 0 s_(k)(0) >= 0s_{k}(0) \geq 0sk(0)≥0 for n = 0 , 1 , 2 , … n = 0 , 1 , 2 , … n=0,1,2,dotsn=0,1,2, \ldotsn=0,1,2,… then the operator L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S defined by (2) is positive.
Proof. If p k ′ ( 0 ) ≥ 0 p k ′ ( 0 ) ≥ 0 p_(k)^(')(0) >= 0p_{k}^{\prime}(0) \geq 0pk′(0)≥0 using Proposition 3 ii), it is easy to prove by induction that p k ( x ) ≥ 0 , ∀ k ∈ N p k ( x ) ≥ 0 , ∀ k ∈ N p_(k)(x) >= 0,AA k inNp_{k}(x) \geq 0, \forall k \in \mathbb{N}pk(x)≥0,∀k∈N and ∀ x ∈ [ 0 , 1 ] ∀ x ∈ [ 0 , 1 ] AA x in[0,1]\forall x \in[0,1]∀x∈[0,1].
If we consider x = 0 x = 0 x=0x=0x=0 in Proposition 4 iii) we obtain
s n ( x ) = ∑ k = 0 n ( n k ) p k ( x ) s n − k ( 0 ) ; s n ( x ) = ∑ k = 0 n   ( n k ) p k ( x ) s n − k ( 0 ) ; s_(n)(x)=sum_(k=0)^(n)((n)/(k))p_(k)(x)s_(n-k)(0);s_{n}(x)=\sum_{k=0}^{n}\binom{n}{k} p_{k}(x) s_{n-k}(0) ;sn(x)=∑k=0n(nk)pk(x)sn−k(0);
accordingly, for s k ( 0 ) ≥ 0 s k ( 0 ) ≥ 0 s_(k)(0) >= 0s_{k}(0) \geq 0sk(0)≥0 and p k ( x ) ≥ 0 , ∀ k ∈ N , ∀ x ∈ [ 0 , 1 ] p k ( x ) ≥ 0 , ∀ k ∈ N , ∀ x ∈ [ 0 , 1 ] p_(k)(x) >= 0,AA k inN,AA x in[0,1]p_{k}(x) \geq 0, \forall k \in \mathbb{N}, \forall x \in[0,1]pk(x)≥0,∀k∈N,∀x∈[0,1], we have s k ( x ) ≥ 0 , ∀ k ∈ N s k ( x ) ≥ 0 , ∀ k ∈ N s_(k)(x) >= 0,AA k inNs_{k}(x) \geq 0, \forall k \in \mathbb{N}sk(x)≥0,∀k∈N and ∀ x ∈ [ 0 , 1 ] ∀ x ∈ [ 0 , 1 ] AA x in[0,1]\forall x \in[0,1]∀x∈[0,1]. Therefore the operator L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S is positive.
Lemma 2. If the operator L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S is positive, then a n ∈ [ 0 , 1 ] , b n ≤ 1 a n ∈ [ 0 , 1 ] , b n ≤ 1 a_(n)in[0,1],b_(n) <= 1a_{n} \in[0,1], b_{n} \leq 1an∈[0,1],bn≤1 and 0 ≤ c n ≤ min { 1 − b n 2 , a n − a n 2 } , ∀ n ∈ N 0 ≤ c n ≤ min 1 − b n 2 , a n − a n 2 , ∀ n ∈ N 0 <= c_(n) <= min{(1-b_(n))/(2),a_(n)-a_(n)^(2)},AA n inN0 \leq c_{n} \leq \min \left\{\frac{1-b_{n}}{2}, a_{n}-a_{n}^{2}\right\}, \forall n \in \mathbb{N}0≤cn≤min{1−bn2,an−an2},∀n∈N, where a n , b n a n , b n a_(n),b_(n)a_{n}, b_{n}an,bn and c n c n c_(n)c_{n}cn are defined by (9).
Proof. Since 0 ≤ e 1 ( t ) ≤ 1 , ∀ t ∈ [ 0 , 1 ] 0 ≤ e 1 ( t ) ≤ 1 , ∀ t ∈ [ 0 , 1 ] 0 <= e_(1)(t) <= 1,AA t in[0,1]0 \leq e_{1}(t) \leq 1, \forall t \in[0,1]0≤e1(t)≤1,∀t∈[0,1] and the operator L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S is positive, we have 0 ≤ ( L n Q , S e 1 ) ( x ) ≤ 1 , ∀ x ∈ [ 0 , 1 ] 0 ≤ L n Q , S e 1 ( x ) ≤ 1 , ∀ x ∈ [ 0 , 1 ] 0 <= (L_(n)^(Q,S)e_(1))(x) <= 1,AA x in[0,1]0 \leq\left(L_{n}^{Q, S} e_{1}\right)(x) \leq 1, \forall x \in[0,1]0≤(LnQ,Se1)(x)≤1,∀x∈[0,1], and as ( L n Q , S e 1 ) ( x ) = a n x L n Q , S e 1 ( x ) = a n x (L_(n)^(Q,S)e_(1))(x)=a_(n)x\left(L_{n}^{Q, S} e_{1}\right)(x)=a_{n} x(LnQ,Se1)(x)=anx, we get a n ∈ [ 0 , 1 ] a n ∈ [ 0 , 1 ] a_(n)in[0,1]a_{n} \in[0,1]an∈[0,1].
From t ( 1 − t ) ≥ 0 t ( 1 − t ) ≥ 0 t(1-t) >= 0t(1-t) \geq 0t(1−t)≥0 it results that ( L n Q , S e 1 ) ( x ) − ( L n Q , S e 2 ) ( x ) ≥ 0 L n Q , S e 1 ( x ) − L n Q , S e 2 ( x ) ≥ 0 (L_(n)^(Q,S)e_(1))(x)-(L_(n)^(Q,S)e_(2))(x) >= 0\left(L_{n}^{Q, S} e_{1}\right)(x)-\left(L_{n}^{Q, S} e_{2}\right)(x) \geq 0(LnQ,Se1)(x)−(LnQ,Se2)(x)≥0, which leads to x ( 1 − x ) b n + x c n ≥ 0 , ∀ x ∈ [ 0 , 1 ] x ( 1 − x ) b n + x c n ≥ 0 , ∀ x ∈ [ 0 , 1 ] x(1-x)b_(n)+xc_(n) >= 0,AA x in[0,1]x(1-x) b_{n}+x c_{n} \geq 0, \forall x \in[0,1]x(1−x)bn+xcn≥0,∀x∈[0,1] and choosing x = 1 x = 1 x=1x=1x=1, we get c n ≥ 0 c n ≥ 0 c_(n) >= 0c_{n} \geq 0cn≥0.
Since t 2 − t + 1 / 4 ≥ 0 t 2 − t + 1 / 4 ≥ 0 t^(2)-t+1//4 >= 0t^{2}-t+1 / 4 \geq 0t2−t+1/4≥0, we obtain ( L n Q , S e 2 ) ( x ) − ( L n Q , S e 1 ) ( x ) + ( L n Q , S e 0 ) ( x ) / 4 ≥ 0 , ∀ x ∈ [ 0 , 1 ] L n Q , S e 2 ( x ) − L n Q , S e 1 ( x ) + L n Q , S e 0 ( x ) / 4 ≥ 0 , ∀ x ∈ [ 0 , 1 ] (L_(n)^(Q,S)e_(2))(x)-(L_(n)^(Q,S)e_(1))(x)+(L_(n)^(Q,S)e_(0))(x)//4 >= 0,AA x in[0,1]\left(L_{n}^{Q, S} e_{2}\right)(x)-\left(L_{n}^{Q, S} e_{1}\right)(x)+\left(L_{n}^{Q, S} e_{0}\right)(x) / 4 \geq 0, \forall x \in[0,1](LnQ,Se2)(x)−(LnQ,Se1)(x)+(LnQ,Se0)(x)/4≥0,∀x∈[0,1], relation equivalent to x 2 b n − x b n − x c n + 1 / 4 ≥ 0 , ∀ x ∈ [ 0 , 1 ] x 2 b n − x b n − x c n + 1 / 4 ≥ 0 , ∀ x ∈ [ 0 , 1 ] x^(2)b_(n)-xb_(n)-xc_(n)+1//4 >= 0,AA x in[0,1]x^{2} b_{n}-x b_{n}-x c_{n}+1 / 4 \geq 0, \forall x \in[0,1]x2bn−xbn−xcn+1/4≥0,∀x∈[0,1]. If we consider x = 1 / 2 x = 1 / 2 x=1//2x=1 / 2x=1/2, it results that c n ≤ ( 1 − b n ) / 2 c n ≤ 1 − b n / 2 c_(n) <= (1-b_(n))//2c_{n} \leq\left(1-b_{n}\right) / 2cn≤(1−bn)/2 and because c n ≥ 0 c n ≥ 0 c_(n) >= 0c_{n} \geq 0cn≥0 we get b n ≤ 1 b n ≤ 1 b_(n) <= 1b_{n} \leq 1bn≤1.
Finally, from the Schwarz's inequality,
[ ( L n Q , S e 1 ) ( x ) ] 2 ≤ ( L n Q , S e 2 ) ( x ) ( L n Q , S e 0 ) ( x ) , L n Q , S e 1 ( x ) 2 ≤ L n Q , S e 2 ( x ) L n Q , S e 0 ( x ) , [(L_(n)^(Q,S)e_(1))(x)]^(2) <= (L_(n)^(Q,S)e_(2))(x)(L_(n)^(Q,S)e_(0))(x),\left[\left(L_{n}^{Q, S} e_{1}\right)(x)\right]^{2} \leq\left(L_{n}^{Q, S} e_{2}\right)(x)\left(L_{n}^{Q, S} e_{0}\right)(x),[(LnQ,Se1)(x)]2≤(LnQ,Se2)(x)(LnQ,Se0)(x),
we have a n 2 x 2 ≤ b n x 2 + x ( a n − b n − c n ) , ∀ x ∈ [ 0 , 1 ] a n 2 x 2 ≤ b n x 2 + x a n − b n − c n , ∀ x ∈ [ 0 , 1 ] a_(n)^(2)x^(2) <= b_(n)x^(2)+x(a_(n)-b_(n)-c_(n)),AA x in[0,1]a_{n}^{2} x^{2} \leq b_{n} x^{2}+x\left(a_{n}-b_{n}-c_{n}\right), \forall x \in[0,1]an2x2≤bnx2+x(an−bn−cn),∀x∈[0,1]. For x = 1 x = 1 x=1x=1x=1 that implies c n ≤ a n − a n 2 c n ≤ a n − a n 2 c_(n) <= a_(n)-a_(n)^(2)c_{n} \leq a_{n}-a_{n}^{2}cn≤an−an2.
Theorem 2. Let Q Q QQQ be a delta operator and S S SSS an invertible shift invariant operator. Let ( p n ( x ) ) n ≥ 0 p n ( x ) n ≥ 0 (p_(n)(x))_(n >= 0)\left(p_{n}(x)\right)_{n \geq 0}(pn(x))n≥0 be the sequence of basic polynomials for Q Q QQQ, with p n ′ ( 0 ) ≥ 0 , ∀ n ∈ N p n ′ ( 0 ) ≥ 0 , ∀ n ∈ N p_(n)^(')(0) >= 0,AA n inNp_{n}^{\prime}(0) \geq 0, \forall n \in \mathbb{N}pn′(0)≥0,∀n∈N, and ( s n ( x ) ) n ≥ 0 s n ( x ) n ≥ 0 (s_(n)(x))_(n >= 0)\left(s_{n}(x)\right)_{n \geq 0}(sn(x))n≥0 a Sheffer set relative to Q , s n = S − 1 p n Q , s n = S − 1 p n Q,s_(n)=S^(-1)p_(n)Q, s_{n}=S^{-1} p_{n}Q,sn=S−1pn with s n ( 1 ) ≠ 0 s n ( 1 ) ≠ 0 s_(n)(1)!=0s_{n}(1) \neq 0sn(1)≠0 and s n ( 0 ) ≥ 0 , ∀ n ∈ N s n ( 0 ) ≥ 0 , ∀ n ∈ N s_(n)(0) >= 0,AA n inNs_{n}(0) \geq 0, \forall n \in \mathbb{N}sn(0)≥0,∀n∈N. If f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] and lim n → ∞ a n = lim n → ∞ b n = 1 lim n → ∞   a n = lim n → ∞   b n = 1 lim_(n rarr oo)a_(n)=lim_(n rarr oo)b_(n)=1\lim _{n \rightarrow \infty} a_{n}= \lim _{n \rightarrow \infty} b_{n}=1limn→∞an=limn→∞bn=1, where a n a n a_(n)a_{n}an and b n b n b_(n)b_{n}bn are defined by (9), then the operator L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S converges to the function f f fff, uniformly on the interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1].
Proof. If lim n → ∞ a n = 1 lim n → ∞   a n = 1 lim_(n rarr oo)a_(n)=1\lim _{n \rightarrow \infty} a_{n}=1limn→∞an=1 then lim n → ∞ ( L n Q , S e 1 ) ( x ) = e 1 ( x ) lim n → ∞   L n Q , S e 1 ( x ) = e 1 ( x ) lim_(n rarr oo)(L_(n)^(Q,S)e_(1))(x)=e_(1)(x)\lim _{n \rightarrow \infty}\left(L_{n}^{Q, S} e_{1}\right)(x)=e_{1}(x)limn→∞(LnQ,Se1)(x)=e1(x). From Lemma 2 , c n ≤ a n − a n 2 2 , c n ≤ a n − a n 2 2,c_(n) <= a_(n)-a_(n)^(2)2, c_{n} \leq a_{n}-a_{n}^{2}2,cn≤an−an2 so we have lim n → ∞ c n = 0 lim n → ∞   c n = 0 lim_(n rarr oo)c_(n)=0\lim _{n \rightarrow \infty} c_{n}=0limn→∞cn=0, and as lim n → ∞ b n = 1 lim n → ∞   b n = 1 lim_(n rarr oo)b_(n)=1\lim _{n \rightarrow \infty} b_{n}=1limn→∞bn=1, we get lim n → ∞ ( L n Q , S e 2 ) ( x ) = e 2 ( x ) lim n → ∞   L n Q , S e 2 ( x ) = e 2 ( x ) lim_(n rarr oo)(L_(n)^(Q,S)e_(2))(x)=e_(2)(x)\lim _{n \rightarrow \infty}\left(L_{n}^{Q, S} e_{2}\right)(x)=e_{2}(x)limn→∞(LnQ,Se2)(x)=e2(x). Therefore lim n → ∞ ( L n Q , S e i ) ( x ) = e i ( x ) lim n → ∞   L n Q , S e i ( x ) = e i ( x ) lim_(n rarr oo)(L_(n)^(Q,S)e_(i))(x)=e_(i)(x)\lim _{n \rightarrow \infty}\left(L_{n}^{Q, S} e_{i}\right)(x)=e_{i}(x)limn→∞(LnQ,Sei)(x)=ei(x) for i = 0 , 1 , 2 i = 0 , 1 , 2 i=0,1,2i= 0,1,2i=0,1,2 so we can use the convergence criterion of Bohman-Korokvin.

3. REPRESENTATIONS OF THE OPERATOR L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S

Theorem 3. The operator L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S can be represented in the form
(15) ( L n Q , S f ) ( x ) = ∑ k = 0 n k ! n k ( n k ) [ 0 , 1 n , … , k n ; f ] d k , n ( x ) (15) L n Q , S f ( x ) = ∑ k = 0 n   k ! n k ( n k ) 0 , 1 n , … , k n ; f d k , n ( x ) {:(15)(L_(n)^(Q,S)f)(x)=sum_(k=0)^(n)(k!)/(n^(k))((n)/(k))[0,(1)/(n),dots,(k)/(n);f]d_(k,n)(x):}\begin{equation*} \left(L_{n}^{Q, S} f\right)(x)=\sum_{k=0}^{n} \frac{k!}{n^{k}}\binom{n}{k}\left[0, \frac{1}{n}, \ldots, \frac{k}{n} ; f\right] d_{k, n}(x) \tag{15} \end{equation*}(15)(LnQ,Sf)(x)=∑k=0nk!nk(nk)[0,1n,…,kn;f]dk,n(x)
where
d k , n ( x ) = 1 s n ( 1 ) ( θ k E 1 − x s n − k ) ( x ) d k , n ( x ) = 1 s n ( 1 ) θ k E 1 − x s n − k ( x ) d_(k,n)(x)=(1)/(s_(n)(1))(theta^(k)E^(1-x)s_(n-k))(x)d_{k, n}(x)=\frac{1}{s_{n}(1)}\left(\theta^{k} E^{1-x} s_{n-k}\right)(x)dk,n(x)=1sn(1)(θkE1−xsn−k)(x)
Moreover L n Q , S ( P m ) ⊆ P m , ∀ m ∈ N L n Q , S P m ⊆ P m , ∀ m ∈ N L_(n)^(Q,S)(P_(m))subeP_(m),AA m inNL_{n}^{Q, S}\left(P_{m}\right) \subseteq P_{m}, \forall m \in \mathbb{N}LnQ,S(Pm)⊆Pm,∀m∈N.
Proof. From the Newton interpolation formula we have
f ( k n ) = ∑ j = 0 k j ! n j ( k j ) [ 0 , 1 n , … , j n ; f ] f k n = ∑ j = 0 k   j ! n j ( k j ) 0 , 1 n , … , j n ; f f((k)/(n))=sum_(j=0)^(k)(j!)/(n^(j))((k)/(j))[0,(1)/(n),dots,(j)/(n);f]f\left(\frac{k}{n}\right)=\sum_{j=0}^{k} \frac{j!}{n^{j}}\binom{k}{j}\left[0, \frac{1}{n}, \ldots, \frac{j}{n} ; f\right]f(kn)=∑j=0kj!nj(kj)[0,1n,…,jn;f]
If we denote w k , n ( x , y ) = ( n k ) p k ( x ) s n − k ( y ) w k , n ( x , y ) = ( n k ) p k ( x ) s n − k ( y ) w_(k,n)(x,y)=((n)/(k))p_(k)(x)s_(n-k)(y)w_{k, n}(x, y)=\binom{n}{k} p_{k}(x) s_{n-k}(y)wk,n(x,y)=(nk)pk(x)sn−k(y) then
∑ k = 0 n w k , n ( x , y ) f ( k n ) = ∑ k = 0 n k ! n k [ 0 , 1 n , … , k n ; f ] ∑ j = k n ( j k ) w j , n ( x , y ) ∑ k = 0 n   w k , n ( x , y ) f k n = ∑ k = 0 n   k ! n k 0 , 1 n , … , k n ; f ∑ j = k n   ( j k ) w j , n ( x , y ) sum_(k=0)^(n)w_(k,n)(x,y)f((k)/(n))=sum_(k=0)^(n)(k!)/(n^(k))[0,(1)/(n),dots,(k)/(n);f]sum_(j=k)^(n)((j)/(k))w_(j,n)(x,y)\sum_{k=0}^{n} w_{k, n}(x, y) f\left(\frac{k}{n}\right)=\sum_{k=0}^{n} \frac{k!}{n^{k}}\left[0, \frac{1}{n}, \ldots, \frac{k}{n} ; f\right] \sum_{j=k}^{n}\binom{j}{k} w_{j, n}(x, y)∑k=0nwk,n(x,y)f(kn)=∑k=0nk!nk[0,1n,…,kn;f]∑j=kn(jk)wj,n(x,y)
But
∑ j = k n ( j k ) w j , n ( x , y ) = ( n k ) ∑ j = k n ( n − k j − k ) p j ( x ) s n − j ( y ) = ( n k ) ∑ j = 0 n − k ( n − k j ) p j + k ( x ) s n − k − j ( y ) = ( n k ) θ k ∑ j = 0 n − k ( n − k j ) p j ( x ) s n − k − j ( y ) = ( n k ) θ k E y s n − k ( x ) ∑ j = k n   ( j k ) w j , n ( x , y ) = ( n k ) ∑ j = k n   ( n − k j − k ) p j ( x ) s n − j ( y ) = ( n k ) ∑ j = 0 n − k   ( n − k j ) p j + k ( x ) s n − k − j ( y ) = ( n k ) θ k ∑ j = 0 n − k   ( n − k j ) p j ( x ) s n − k − j ( y ) = ( n k ) θ k E y s n − k ( x ) {:[sum_(j=k)^(n)((j)/(k))w_(j,n)(x","y)=((n)/(k))sum_(j=k)^(n)((n-k)/(j-k))p_(j)(x)s_(n-j)(y)],[=((n)/(k))sum_(j=0)^(n-k)((n-k)/(j))p_(j+k)(x)s_(n-k-j)(y)],[=((n)/(k))theta^(k)sum_(j=0)^(n-k)((n-k)/(j))p_(j)(x)s_(n-k-j)(y)],[=((n)/(k))theta^(k)E^(y)s_(n-k)(x)]:}\begin{aligned} \sum_{j=k}^{n}\binom{j}{k} w_{j, n}(x, y) & =\binom{n}{k} \sum_{j=k}^{n}\binom{n-k}{j-k} p_{j}(x) s_{n-j}(y) \\ & =\binom{n}{k} \sum_{j=0}^{n-k}\binom{n-k}{j} p_{j+k}(x) s_{n-k-j}(y) \\ & =\binom{n}{k} \theta^{k} \sum_{j=0}^{n-k}\binom{n-k}{j} p_{j}(x) s_{n-k-j}(y) \\ & =\binom{n}{k} \theta^{k} E^{y} s_{n-k}(x) \end{aligned}∑j=kn(jk)wj,n(x,y)=(nk)∑j=kn(n−kj−k)pj(x)sn−j(y)=(nk)∑j=0n−k(n−kj)pj+k(x)sn−k−j(y)=(nk)θk∑j=0n−k(n−kj)pj(x)sn−k−j(y)=(nk)θkEysn−k(x)
Since ( L n Q , S f ) ( x ) = 1 s n ( 1 ) ∑ k = 0 n w k , n ( x , 1 − x ) f ( k n ) L n Q , S f ( x ) = 1 s n ( 1 ) ∑ k = 0 n   w k , n ( x , 1 − x ) f k n (L_(n)^(Q,S)f)(x)=(1)/(s_(n)(1))sum_(k=0)^(n)w_(k,n)(x,1-x)f((k)/(n))\left(L_{n}^{Q, S} f\right)(x)=\frac{1}{s_{n}(1)} \sum_{k=0}^{n} w_{k, n}(x, 1-x) f\left(\frac{k}{n}\right)(LnQ,Sf)(x)=1sn(1)∑k=0nwk,n(x,1−x)f(kn) we obtain (15).
In order to show that L n Q , S ( P m ) ⊆ P m L n Q , S P m ⊆ P m L_(n)^(Q,S)(P_(m))subeP_(m)L_{n}^{Q, S}\left(P_{m}\right) \subseteq P_{m}LnQ,S(Pm)⊆Pm we shall prove that deg ( d k , n ( x ) ) = k deg d k , n ( x ) = k deg(d_(k,n)(x))=k\operatorname{deg}\left(d_{k, n}(x)\right)=kdeg(dk,n(x))=k.
We remind that if ( p n p n p_(n)p_{n}pn ) is a basic sequence for Q = q ( D ) Q = q ( D ) Q=q(D)Q=q(D)Q=q(D) and h ( t ) h ( t ) h(t)h(t)h(t) is the compositional inverse of q ( t ) q ( t ) q(t)q(t)q(t), then the generating function for ( p n ) p n (p_(n))\left(p_{n}\right)(pn) is
(16) ∑ k = 0 ∞ p k ( x ) t k k ! = e x h ( t ) (16) ∑ k = 0 ∞   p k ( x ) t k k ! = e x h ( t ) {:(16)sum_(k=0)^(oo)p_(k)(x)(t^(k))/(k!)=e^(xh(t)):}\begin{equation*} \sum_{k=0}^{\infty} p_{k}(x) \frac{t^{k}}{k!}=e^{x h(t)} \tag{16} \end{equation*}(16)∑k=0∞pk(x)tkk!=exh(t)
and if s n = S − 1 p n s n = S − 1 p n s_(n)=S^(-1)p_(n)s_{n}=S^{-1} p_{n}sn=S−1pn, with S = s ( D ) S = s ( D ) S=s(D)S=s(D)S=s(D) then
(17) ∑ k = 0 ∞ s k ( x ) t k k ! = 1 s ( h ( t ) ) e x h ( t ) (17) ∑ k = 0 ∞   s k ( x ) t k k ! = 1 s ( h ( t ) ) e x h ( t ) {:(17)sum_(k=0)^(oo)s_(k)(x)(t^(k))/(k!)=(1)/(s(h(t)))e^(xh(t)):}\begin{equation*} \sum_{k=0}^{\infty} s_{k}(x) \frac{t^{k}}{k!}=\frac{1}{s(h(t))} e^{x h(t)} \tag{17} \end{equation*}(17)∑k=0∞sk(x)tkk!=1s(h(t))exh(t)
If we differentiate the relation (16) m m mmm times with respect to t t ttt, we get
(18) ∑ k = 0 ∞ p k + m ( x ) t k k ! = d m d x m ( e x h ( t ) ) = ( x h 1 ( t ) + x 2 h 1 ( t ) + ⋯ + x m h m ( t ) ) e x h ( t ) ∑ k = 0 ∞   p k + m ( x ) t k k ! = d m d x m e x h ( t ) = x h 1 ( t ) + x 2 h 1 ( t ) + ⋯ + x m h m ( t ) e x h ( t ) sum_(k=0)^(oo)p_(k+m)(x)(t^(k))/(k!)=(d^(m))/(dx^(m))(e^(xh(t)))=(xh_(1)(t)+x^(2)h_(1)(t)+cdots+x^(m)h_(m)(t))e^(xh(t))\sum_{k=0}^{\infty} p_{k+m}(x) \frac{t^{k}}{k!}=\frac{d^{m}}{d x^{m}}\left(e^{x h(t)}\right)=\left(x h_{1}(t)+x^{2} h_{1}(t)+\cdots+x^{m} h_{m}(t)\right) e^{x h(t)}∑k=0∞pk+m(x)tkk!=dmdxm(exh(t))=(xh1(t)+x2h1(t)+⋯+xmhm(t))exh(t),
where every h i ( t ) h i ( t ) h_(i)(t)h_{i}(t)hi(t) is a product of derivatives of h ( t ) h ( t ) h(t)h(t)h(t).
Let us denote r ( k , m , x ) = ∑ j = 0 k ( k j ) p j + m ( x ) s k − j ( 1 − x ) r ( k , m , x ) = ∑ j = 0 k   ( k j ) p j + m ( x ) s k − j ( 1 − x ) r(k,m,x)=sum_(j=0)^(k)((k)/(j))p_(j+m)(x)s_(k-j)(1-x)r(k, m, x)=\sum_{j=0}^{k}\binom{k}{j} p_{j+m}(x) s_{k-j}(1-x)r(k,m,x)=∑j=0k(kj)pj+m(x)sk−j(1−x). Expanding 1 s ( h ( t ) ) h i ( t ) e h ( t ) = ∑ k ≥ 0 α i k t k k ! 1 s ( h ( t ) ) h i ( t ) e h ( t ) = ∑ k ≥ 0   α i k t k k ! (1)/(s(h(t)))h_(i)(t)e^(h(t))=sum_(k >= 0)alpha_(ik)(t^(k))/(k!)\frac{1}{s(h(t))} h_{i}(t) e^{h(t)}=\sum_{k \geq 0} \alpha_{i k} \frac{t^{k}}{k!}1s(h(t))hi(t)eh(t)=∑k≥0αiktkk!, from (17) and (18) we get r ( k , m , x ) = x α 1 k + x 2 α 2 k + ⋯ + x m α m k r ( k , m , x ) = x α 1 k + x 2 α 2 k + ⋯ + x m α m k r(k,m,x)=xalpha_(1k)+x^(2)alpha_(2k)+cdots+x^(m)alpha_(mk)r(k, m, x)=x \alpha_{1 k}+ x^{2} \alpha_{2 k}+\cdots+x^{m} \alpha_{m k}r(k,m,x)=xα1k+x2α2k+⋯+xmαmk.
Because d k , n ( x ) = r ( n − k , k , x ) / s n ( 1 ) d k , n ( x ) = r ( n − k , k , x ) / s n ( 1 ) d_(k,n)(x)=r(n-k,k,x)//s_(n)(1)d_{k, n}(x)=r(n-k, k, x) / s_{n}(1)dk,n(x)=r(n−k,k,x)/sn(1) we obtain deg ( d k , n ( x ) ) = k deg d k , n ( x ) = k deg(d_(k,n)(x))=k\operatorname{deg}\left(d_{k, n}(x)\right)=kdeg(dk,n(x))=k.
Suppose that p ∈ P m p ∈ P m p inP_(m)p \in P_{m}p∈Pm. Then [ 0 , 1 n , … , k n ; p ] = 0 0 , 1 n , … , k n ; p = 0 [0,(1)/(n),dots,(k)/(n);p]=0\left[0, \frac{1}{n}, \ldots, \frac{k}{n} ; p\right]=0[0,1n,…,kn;p]=0 for k ≥ m + 1 k ≥ m + 1 k >= m+1k \geq m+1k≥m+1, and using (15) we get L n Q , S ( P m ) ⊆ P m L n Q , S P m ⊆ P m L_(n)^(Q,S)(P_(m))subeP_(m)L_{n}^{Q, S}\left(P_{m}\right) \subseteq P_{m}LnQ,S(Pm)⊆Pm.
Remark 1. For Q = D Q = D Q=DQ=DQ=D (it means that s n s n s_(n)s_{n}sn is an Appell set A n A n A_(n)A_{n}An ) we have θ = X θ = X theta=X\theta= Xθ=X, therefore in this case d k , n = A n − k ( 1 ) A n ( 1 ) x k d k , n = A n − k ( 1 ) A n ( 1 ) x k d_(k,n)=(A_(n-k)(1))/(A_(n)(1))x^(k)d_{k, n}=\frac{A_{n-k}(1)}{A_{n}(1)} x^{k}dk,n=An−k(1)An(1)xk. This representation for operators constructed with Appell sequences was given by C. Manole in [5].
Theorem 4. Suppose that all the assumptions of Theorem 2 are true, then there exists θ 1 n , θ 2 n , θ 3 n ∈ [ 0 , 1 ] θ 1 n , θ 2 n , θ 3 n ∈ [ 0 , 1 ] theta_(1n),theta_(2n),theta_(3n)in[0,1]\theta_{1 n}, \theta_{2 n}, \theta_{3 n} \in[0,1]θ1n,θ2n,θ3n∈[0,1] such that ∀ x ∈ [ 0 , 1 ] ∀ x ∈ [ 0 , 1 ] AA x in[0,1]\forall x \in[0,1]∀x∈[0,1] and ∀ f ∈ C [ 0 , 1 ] ∀ f ∈ C [ 0 , 1 ] AA f in C[0,1]\forall f \in C[0,1]∀f∈C[0,1] we have
( L n Q , S f ) ( x ) = f ( a n x ) + α ( x , n ) [ θ 1 n , θ 2 n , θ 3 n ; f ] L n Q , S f ( x ) = f a n x + α ( x , n ) θ 1 n , θ 2 n , θ 3 n ; f (L_(n)^(Q,S)f)(x)=f(a_(n)x)+alpha(x,n)[theta_(1n),theta_(2n),theta_(3n);f]\left(L_{n}^{Q, S} f\right)(x)=f\left(a_{n} x\right)+\alpha(x, n)\left[\theta_{1 n}, \theta_{2 n}, \theta_{3 n} ; f\right](LnQ,Sf)(x)=f(anx)+α(x,n)[θ1n,θ2n,θ3n;f]
where α ( x , n ) = x 2 ( b n − a n 2 ) + x ( a n − b n − c n ) α ( x , n ) = x 2 b n − a n 2 + x a n − b n − c n alpha(x,n)=x^(2)(b_(n)-a_(n)^(2))+x(a_(n)-b_(n)-c_(n))\alpha(x, n)=x^{2}\left(b_{n}-a_{n}^{2}\right)+x\left(a_{n}-b_{n}-c_{n}\right)α(x,n)=x2(bn−an2)+x(an−bn−cn).
Proof. First we shall prove that f ( a n x ) ≤ ( L n Q , S f ) ( x ) f a n x ≤ L n Q , S f ( x ) f(a_(n)x) <= (L_(n)^(Q,S)f)(x)f\left(a_{n} x\right) \leq\left(L_{n}^{Q, S} f\right)(x)f(anx)≤(LnQ,Sf)(x) for every convex function f f fff.
Let us denote c k = 1 s n ( 1 ) ( n k ) p k ( x ) s n − k ( 1 − x ) c k = 1 s n ( 1 ) ( n k ) p k ( x ) s n − k ( 1 − x ) c_(k)=(1)/(s_(n)(1))((n)/(k))p_(k)(x)s_(n-k)(1-x)c_{k}=\frac{1}{s_{n}(1)}\binom{n}{k} p_{k}(x) s_{n-k}(1-x)ck=1sn(1)(nk)pk(x)sn−k(1−x) and x k = k n , k = 0 , 1 , … , n x k = k n , k = 0 , 1 , … , n x_(k)=(k)/(n),k=0,1,dots,nx_{k}=\frac{k}{n}, k=0,1, \ldots, nxk=kn,k=0,1,…,n.
We have c k ≥ 0 , ∑ k = 0 n c k = 1 c k ≥ 0 , ∑ k = 0 n   c k = 1 c_(k) >= 0,sum_(k=0)^(n)c_(k)=1c_{k} \geq 0, \sum_{k=0}^{n} c_{k}=1ck≥0,∑k=0nck=1 and x k > 0 , ∀ k ∈ N x k > 0 , ∀ k ∈ N x_(k) > 0,AA k inNx_{k}>0, \forall k \in \mathbb{N}xk>0,∀k∈N. If f f fff is a convex function then f ( ∑ k = 0 n c k x k ) ≤ ∑ k = 0 n c k f ( x k ) f ∑ k = 0 n   c k x k ≤ ∑ k = 0 n   c k f x k f(sum_(k=0)^(n)c_(k)x_(k)) <= sum_(k=0)^(n)c_(k)f(x_(k))f\left(\sum_{k=0}^{n} c_{k} x_{k}\right) \leq \sum_{k=0}^{n} c_{k} f\left(x_{k}\right)f(∑k=0nckxk)≤∑k=0nckf(xk); but
∑ k = 0 n c k x k = ( L n Q , S e 1 ) ( x ) = a n x and ∑ k = 0 n c k f ( x k ) = ( L n Q , S f ) ( x ) ∑ k = 0 n   c k x k = L n Q , S e 1 ( x ) = a n x  and  ∑ k = 0 n   c k f x k = L n Q , S f ( x ) sum_(k=0)^(n)c_(k)x_(k)=(L_(n)^(Q,S)e_(1))(x)=a_(n)x quad" and "quadsum_(k=0)^(n)c_(k)f(x_(k))=(L_(n)^(Q,S)f)(x)\sum_{k=0}^{n} c_{k} x_{k}=\left(L_{n}^{Q, S} e_{1}\right)(x)=a_{n} x \quad \text { and } \quad \sum_{k=0}^{n} c_{k} f\left(x_{k}\right)=\left(L_{n}^{Q, S} f\right)(x)∑k=0nckxk=(LnQ,Se1)(x)=anx and ∑k=0nckf(xk)=(LnQ,Sf)(x)
therefore we get f ( a n x ) ≤ ( L n Q , S f ) ( x ) f a n x ≤ L n Q , S f ( x ) f(a_(n)x) <= (L_(n)^(Q,S)f)(x)f\left(a_{n} x\right) \leq\left(L_{n}^{Q, S} f\right)(x)f(anx)≤(LnQ,Sf)(x).
If we consider the formula
f ( a n x ) = ( L n Q , S f ) ( x ) + ( R n f ) ( x ) f a n x = L n Q , S f ( x ) + R n f ( x ) f(a_(n)x)=(L_(n)^(Q,S)f)(x)+(R_(n)f)(x)f\left(a_{n} x\right)=\left(L_{n}^{Q, S} f\right)(x)+\left(R_{n} f\right)(x)f(anx)=(LnQ,Sf)(x)+(Rnf)(x)
we have ( R n f ) ≤ 0 R n f ≤ 0 (R_(n)f) <= 0\left(R_{n} f\right) \leq 0(Rnf)≤0 for every convex function f f fff.
Since ( R n e i ) ( x ) = 0 R n e i ( x ) = 0 (R_(n)e_(i))(x)=0\left(R_{n} e_{i}\right)(x)=0(Rnei)(x)=0 for i = 0 , 1 i = 0 , 1 i=0,1i=0,1i=0,1, the degree of exactness of the previous formula is one and then there exist θ 1 n , θ 2 n , θ 3 n ∈ [ 0 , 1 ] θ 1 n , θ 2 n , θ 3 n ∈ [ 0 , 1 ] theta_(1n),theta_(2n),theta_(3n)in[0,1]\theta_{1 n}, \theta_{2 n}, \theta_{3 n} \in[0,1]θ1n,θ2n,θ3n∈[0,1] such that the remainder can be represented in the following form
( R n f ) ( x ) = ( R n e 2 ) ( x ) [ θ 1 n , θ 2 n , θ 3 n ; f ] R n f ( x ) = R n e 2 ( x ) θ 1 n , θ 2 n , θ 3 n ; f (R_(n)f)(x)=(R_(n)e_(2))(x)[theta_(1n),theta_(2n),theta_(3n);f]\left(R_{n} f\right)(x)=\left(R_{n} e_{2}\right)(x)\left[\theta_{1 n}, \theta_{2 n}, \theta_{3 n} ; f\right](Rnf)(x)=(Rne2)(x)[θ1n,θ2n,θ3n;f]
where ( R n e 2 ) ( x ) = x 2 ( a n 2 − b n ) + x ( b n + c n − a n ) R n e 2 ( x ) = x 2 a n 2 − b n + x b n + c n − a n (R_(n)e_(2))(x)=x^(2)(a_(n)^(2)-b_(n))+x(b_(n)+c_(n)-a_(n))\left(R_{n} e_{2}\right)(x)=x^{2}\left(a_{n}^{2}-b_{n}\right)+x\left(b_{n}+c_{n}-a_{n}\right)(Rne2)(x)=x2(an2−bn)+x(bn+cn−an), so we obtain the conclusion.

4. EXAMPLES

  1. If S = I S = I S=IS=IS=I then s n = p n s n = p n s_(n)=p_(n)s_{n}=p_{n}sn=pn and in this case the operator defined by (2) becomes the binomial operator (1) introduced by Tiberiu Popoviciu in 9 .
    1.1. For Q = D Q = D Q=DQ=DQ=D the basic sequence is p n ( x ) = x n p n ( x ) = x n p_(n)(x)=x^(n)p_{n}(x)=x^{n}pn(x)=xn and L n D , I L n D , I L_(n)^(D,I)L_{n}^{D, I}LnD,I is the Bernstein operator B n B n B_(n)B_{n}Bn.
    1.2. If Q Q QQQ is Abel operator A = E − β D A = E − β D A=E^(-beta)DA=E^{-\beta} DA=E−βD we have p n ( x ) = x ( x + n β ) n − 1 p n ( x ) = x ( x + n β ) n − 1 p_(n)(x)=x(x+n beta)^(n-1)p_{n}(x)=x(x+n \beta)^{n-1}pn(x)=x(x+nβ)n−1 and L n A , I L n A , I L_(n)^(A,I)L_{n}^{A, I}LnA,I is the second operator introduced by Cheney and Sharma in [1,
( L n A , I f ) ( x ) = = 1 ( 1 + n β ) n − 1 ∑ k = 0 n ( n k ) x ( x + k β ) k − 1 ( 1 − x ) ( 1 − x + ( n − k ) β ) n − k − 1 f ( k n ) L n A , I f ( x ) = = 1 ( 1 + n β ) n − 1 ∑ k = 0 n   ( n k ) x ( x + k β ) k − 1 ( 1 − x ) ( 1 − x + ( n − k ) β ) n − k − 1 f k n {:[(L_(n)^(A,I)f)(x)=],[=(1)/((1+n beta)^(n-1))sum_(k=0)^(n)((n)/(k))x(x+k beta)^(k-1)(1-x)(1-x+(n-k)beta)^(n-k-1)f((k)/(n))]:}\begin{aligned} & \left(L_{n}^{A, I} f\right)(x)= \\ & =\frac{1}{(1+n \beta)^{n-1}} \sum_{k=0}^{n}\binom{n}{k} x(x+k \beta)^{k-1}(1-x)(1-x+(n-k) \beta)^{n-k-1} f\left(\frac{k}{n}\right) \end{aligned}(LnA,If)(x)==1(1+nβ)n−1∑k=0n(nk)x(x+kβ)k−1(1−x)(1−x+(n−k)β)n−k−1f(kn)
1.3. For Laguerre delta operator L = D D + I L = D D + I L=(D)/(D+I)L=\frac{D}{D+I}L=DD+I the basic sequence is l n ( x ) = ∑ k = 0 n ( n k ) ( n − 1 ) ! ( k − 1 ) ! x k l n ( x ) = ∑ k = 0 n   ( n k ) ( n − 1 ) ! ( k − 1 ) ! x k l_(n)(x)=sum_(k=0)^(n)((n)/(k))((n-1)!)/((k-1)!)x^(k)l_{n}(x)= \sum_{k=0}^{n}\binom{n}{k} \frac{(n-1)!}{(k-1)!} x^{k}ln(x)=∑k=0n(nk)(n−1)!(k−1)!xk and the coresponding binomial operator has been considered by T. Popoviciu.
1.4. The delta operator Q = 1 α ∇ α = 1 α ( I − E − α ) Q = 1 α ∇ α = 1 α I − E − α Q=(1)/(alpha)grad_(alpha)=(1)/(alpha)(I-E^(-alpha))Q=\frac{1}{\alpha} \nabla_{\alpha}=\frac{1}{\alpha}\left(I-E^{-\alpha}\right)Q=1α∇α=1α(I−E−α) has the basic sequence p n ( x ) = x [ n , − α ] = x ( x + α ) … ( x + ( n − 1 ) α ) p n ( x ) = x [ n , − α ] = x ( x + α ) … ( x + ( n − 1 ) α ) p_(n)(x)=x^([n,-alpha])=x(x+alpha)dots(x+(n-1)alpha)p_{n}(x)=x^{[n,-\alpha]}=x(x+\alpha) \ldots(x+(n-1) \alpha)pn(x)=x[n,−α]=x(x+α)…(x+(n−1)α) and in this case we obtain the operator
( S n f ) ( x ) = 1 1 [ n , − α ] ∑ k = 0 n ( n k ) x [ k , − α ] ( 1 − x ) [ n − k , − α ] f ( k n ) S n f ( x ) = 1 1 [ n , − α ] ∑ k = 0 n   ( n k ) x [ k , − α ] ( 1 − x ) [ n − k , − α ] f k n (S_(n)f)(x)=(1)/(1^([n,-alpha]))sum_(k=0)^(n)((n)/(k))x^([k,-alpha])(1-x)^([n-k,-alpha])f((k)/(n))\left(S_{n} f\right)(x)=\frac{1}{1^{[n,-\alpha]}} \sum_{k=0}^{n}\binom{n}{k} x^{[k,-\alpha]}(1-x)^{[n-k,-\alpha]} f\left(\frac{k}{n}\right)(Snf)(x)=11[n,−α]∑k=0n(nk)x[k,−α](1−x)[n−k,−α]f(kn)
which has been introduced and investigated in detail by D. D. Stancu in [14], [16] and other papers.
1.5. The exponential polynomials t n ( x ) = ∑ k = 0 n S ( n , k ) x k = e − x ∑ k = 0 ∞ k n x k k ! t n ( x ) = ∑ k = 0 n   S ( n , k ) x k = e − x ∑ k = 0 ∞   k n x k k ! t_(n)(x)=sum_(k=0)^(n)S(n,k)x^(k)=e^(-x)sum_(k=0)^(oo)(k^(n)x^(k))/(k!)t_{n}(x)=\sum_{k=0}^{n} S(n, k) x^{k}= e^{-x} \sum_{k=0}^{\infty} \frac{k^{n} x^{k}}{k!}tn(x)=∑k=0nS(n,k)xk=e−x∑k=0∞knxkk!, where S ( n , k ) S ( n , k ) S(n,k)S(n, k)S(n,k) denote the Stirling numbers of the second kind, are basic polynomials for the delta operator T = ln ⁡ ( I + D ) T = ln ⁡ ( I + D ) T=ln(I+D)T=\ln (I+D)T=ln⁡(I+D). The approximation operator construct by means of the exponential polynomials
( L n T f ) ( x ) = 1 t n ( 1 ) ∑ k = 0 n ( n k ) t k ( x ) t n − k ( 1 − x ) f ( k n ) L n T f ( x ) = 1 t n ( 1 ) ∑ k = 0 n   ( n k ) t k ( x ) t n − k ( 1 − x ) f k n (L_(n)^(T)f)(x)=(1)/(t_(n)(1))sum_(k=0)^(n)((n)/(k))t_(k)(x)t_(n-k)(1-x)f((k)/(n))\left(L_{n}^{T} f\right)(x)=\frac{1}{t_{n}(1)} \sum_{k=0}^{n}\binom{n}{k} t_{k}(x) t_{n-k}(1-x) f\left(\frac{k}{n}\right)(LnTf)(x)=1tn(1)∑k=0n(nk)tk(x)tn−k(1−x)f(kn)
was studied by C. Manole in [5].
1.6. If we take the delta operator Q = G = 1 α E − β ∇ α = 1 α ( E − β − E − α − β Q = G = 1 α E − β ∇ α = 1 α E − β − E − α − β Q=G=(1)/(alpha)E^(-beta)grad_(alpha)=(1)/(alpha)(E^(-beta)-:}E^(-alpha-beta)Q=G=\frac{1}{\alpha} E^{-\beta} \nabla_{\alpha}=\frac{1}{\alpha}\left(E^{-\beta}-\right. E^{-\alpha-\beta}Q=G=1αE−β∇α=1α(E−β−E−α−β ) its basic sequence is p n ( x ) = x ( x + α + n β ) [ n − 1 , − α ] p n ( x ) = x ( x + α + n β ) [ n − 1 , − α ] p_(n)(x)=x(x+alpha+n beta)^([n-1,-alpha])p_{n}(x)=x(x+\alpha+n \beta)^{[n-1,-\alpha]}pn(x)=x(x+α+nβ)[n−1,−α] and the operator
( L n G f ) ( x ) = 1 ( 1 + n β ) [ n , − α ] ⋅ ∑ k = 0 n ( n k ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x ) ( 1 − x + ( n − k ) β ) [ n − k , − α ] f ( k n ) L n G f ( x ) = 1 ( 1 + n β ) [ n , − α ] ⋅ ∑ k = 0 n   ( n k ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x ) ( 1 − x + ( n − k ) β ) [ n − k , − α ] f k n {:[(L_(n)^(G)f)(x)=(1)/((1+n beta)^([n,-alpha]))],[*sum_(k=0)^(n)((n)/(k))x(x+alpha+k beta)^([k-1,-alpha])(1-x)(1-x+(n-k)beta)^([n-k,-alpha])f((k)/(n))]:}\begin{aligned} & \left(L_{n}^{G} f\right)(x)=\frac{1}{(1+n \beta)^{[n,-\alpha]}} \\ & \cdot \sum_{k=0}^{n}\binom{n}{k} x(x+\alpha+k \beta)^{[k-1,-\alpha]}(1-x)(1-x+(n-k) \beta)^{[n-k,-\alpha]} f\left(\frac{k}{n}\right) \end{aligned}(LnGf)(x)=1(1+nβ)[n,−α]⋅∑k=0n(nk)x(x+α+kβ)[k−1,−α](1−x)(1−x+(n−k)β)[n−k,−α]f(kn)
was investigated by D. D. Stancu, G. Moldovan. In [18] D. D. Stancu and M. R. Occorsio have studied this operator with the nodes k + γ n + δ , 0 ≤ γ ≤ δ k + γ n + δ , 0 ≤ γ ≤ δ (k+gamma)/(n+delta),0 <= gamma <= delta\frac{k+\gamma}{n+\delta}, 0 \leq \gamma \leq \deltak+γn+δ,0≤γ≤δ.
2. If Q = D Q = D Q=DQ=DQ=D and S S SSS is an invertible shift invariant operator then p n ( x ) = x n p n ( x ) = x n p_(n)(x)=x^(n)p_{n}(x)=x^{n}pn(x)=xn and s n = A n = S − 1 x n s n = A n = S − 1 x n s_(n)=A_(n)=S^(-1)x^(n)s_{n}=A_{n}=S^{-1} x^{n}sn=An=S−1xn is an Appell set. The operator of the form
( L n D , S f ) ( x ) = 1 A n ( 1 ) ∑ k = 0 n ( n k ) x k A n − k ( 1 − x ) f ( k n ) L n D , S f ( x ) = 1 A n ( 1 ) ∑ k = 0 n   ( n k ) x k A n − k ( 1 − x ) f k n (L_(n)^(D,S)f)(x)=(1)/(A_(n)(1))sum_(k=0)^(n)((n)/(k))x^(k)A_(n-k)(1-x)f((k)/(n))\left(L_{n}^{D, S} f\right)(x)=\frac{1}{A_{n}(1)} \sum_{k=0}^{n}\binom{n}{k} x^{k} A_{n-k}(1-x) f\left(\frac{k}{n}\right)(LnD,Sf)(x)=1An(1)∑k=0n(nk)xkAn−k(1−x)f(kn)
was introduced and investigated by C. Manole in [5].
2.1. If S = ( I + D ) − 1 S = ( I + D ) − 1 S=(I+D)^(-1)S=(I+D)^{-1}S=(I+D)−1 the coresponding Appell set is A n ( x ) = x n + n x n − 1 A n ( x ) = x n + n x n − 1 A_(n)(x)=x^(n)+nx^(n-1)A_{n}(x)=x^{n}+n x^{n-1}An(x)=xn+nxn−1 and then
( L n D , ( I + D ) − 1 f ) ( x ) = 1 n + 1 ∑ k = 0 n ( n k ) x k ( 1 − x ) n − k ( n − k + 1 − x ) f ( k n ) L n D , ( I + D ) − 1 f ( x ) = 1 n + 1 ∑ k = 0 n   ( n k ) x k ( 1 − x ) n − k ( n − k + 1 − x ) f k n (L_(n)^(D,(I+D)^(-1))f)(x)=(1)/(n+1)sum_(k=0)^(n)((n)/(k))x^(k)(1-x)^(n-k)(n-k+1-x)f((k)/(n))\left(L_{n}^{D,(I+D)^{-1}} f\right)(x)=\frac{1}{n+1} \sum_{k=0}^{n}\binom{n}{k} x^{k}(1-x)^{n-k}(n-k+1-x) f\left(\frac{k}{n}\right)(LnD,(I+D)−1f)(x)=1n+1∑k=0n(nk)xk(1−x)n−k(n−k+1−x)f(kn)
  1. If we take Q = A = E − β D Q = A = E − β D Q=A=E^(-beta)DQ=A=E^{-\beta} DQ=A=E−βD and S = E β Q ′ = I − β D S = E β Q ′ = I − β D S=E^(beta)Q^(')=I-beta DS=E^{\beta} Q^{\prime}=I-\beta DS=EβQ′=I−βD then p n ( x ) = x ( x + n β ) n − 1 p n ( x ) = x ( x + n β ) n − 1 p_(n)(x)=x(x+n beta)^(n-1)p_{n}(x)= x(x+n \beta)^{n-1}pn(x)=x(x+nβ)n−1 is the basic sequence for Q Q QQQ and s n ( x ) = ( x + n β ) n s n ( x ) = ( x + n β ) n s_(n)(x)=(x+n beta)^(n)s_{n}(x)=(x+n \beta)^{n}sn(x)=(x+nβ)n a Sheffer set for Q Q QQQ we obtain the first operator introduced by Cheney and Sharma in [1]:
( L n A , I − β D f ) ( x ) = 1 ( 1 + n β ) n ∑ k = 0 n ( n k ) x ( x + k β ) k − 1 ( 1 − x + ( n − k ) β ) n − k f ( k n ) L n A , I − β D f ( x ) = 1 ( 1 + n β ) n ∑ k = 0 n   ( n k ) x ( x + k β ) k − 1 ( 1 − x + ( n − k ) β ) n − k f k n (L_(n)^(A,I-beta D)f)(x)=(1)/((1+n beta)^(n))sum_(k=0)^(n)((n)/(k))x(x+k beta)^(k-1)(1-x+(n-k)beta)^(n-k)f((k)/(n))\left(L_{n}^{A, I-\beta D} f\right)(x)=\frac{1}{(1+n \beta)^{n}} \sum_{k=0}^{n}\binom{n}{k} x(x+k \beta)^{k-1}(1-x+(n-k) \beta)^{n-k} f\left(\frac{k}{n}\right)(LnA,I−βDf)(x)=1(1+nβ)n∑k=0n(nk)x(x+kβ)k−1(1−x+(n−k)β)n−kf(kn)
  1. For Q = 1 α E − β ∇ α = 1 α ( E − β − E − α − β ) Q = 1 α E − β ∇ α = 1 α E − β − E − α − β Q=(1)/(alpha)E^(-beta)grad_(alpha)=(1)/(alpha)(E^(-beta)-E^(-alpha-beta))Q=\frac{1}{\alpha} E^{-\beta} \nabla_{\alpha}=\frac{1}{\alpha}\left(E^{-\beta}-E^{-\alpha-\beta}\right)Q=1αE−β∇α=1α(E−β−E−α−β) and S = E α + β Q ′ = 1 α ( ( α + β ) I − β E α ) S = E α + β Q ′ = 1 α ( ( α + β ) I − β E α S=E^(alpha+beta)Q^(')=(1)/(alpha)((alpha+beta)I-{: betaE^(alpha))S=E^{\alpha+\beta} Q^{\prime}=\frac{1}{\alpha}((\alpha+\beta) I- \left.\beta E^{\alpha}\right)S=Eα+βQ′=1α((α+β)I−βEα) we have p n ( x ) = x ( x + α + n β ) [ n − 1 , − α ] p n ( x ) = x ( x + α + n β ) [ n − 1 , − α ] p_(n)(x)=x(x+alpha+n beta)^([n-1,-alpha])p_{n}(x)=x(x+\alpha+n \beta)^{[n-1,-\alpha]}pn(x)=x(x+α+nβ)[n−1,−α] and s n ( x ) = ( x + n β ) [ n , − α ] s n ( x ) = ( x + n β ) [ n , − α ] s_(n)(x)=(x+n beta)^([n,-alpha])s_{n}(x)=(x+n \beta)^{[n,-\alpha]}sn(x)=(x+nβ)[n,−α] therefore the operator L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S in this case is
( L n [ α , β ] f ) ( x ) = = 1 ( 1 + n β ) [ n , − α ] ∑ k = 0 n ( n k ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x + ( n − k ) β ) [ n − k , − α ] f ( k n ) L n [ α , β ] f ( x ) = = 1 ( 1 + n β ) [ n , − α ] ∑ k = 0 n   ( n k ) x ( x + α + k β ) [ k − 1 , − α ] ( 1 − x + ( n − k ) β ) [ n − k , − α ] f k n {:[(L_(n)^([alpha,beta])f)(x)=],[=(1)/((1+n beta)^([n,-alpha]))sum_(k=0)^(n)((n)/(k))x(x+alpha+k beta)^([k-1,-alpha])(1-x+(n-k)beta)^([n-k,-alpha])f((k)/(n))]:}\begin{aligned} & \left(L_{n}^{[\alpha, \beta]} f\right)(x)= \\ & =\frac{1}{(1+n \beta)^{[n,-\alpha]}} \sum_{k=0}^{n}\binom{n}{k} x(x+\alpha+k \beta)^{[k-1,-\alpha]}(1-x+(n-k) \beta)^{[n-k,-\alpha]} f\left(\frac{k}{n}\right) \end{aligned}(Ln[α,β]f)(x)==1(1+nβ)[n,−α]∑k=0n(nk)x(x+α+kβ)[k−1,−α](1−x+(n−k)β)[n−k,−α]f(kn)
If we replace x x xxx with s ( x ) s ( x ) s(x)s(x)s(x) we obtain a operator which has been studied by G . Moldovan in [6]. He has found the value of this operator for the monomials e i e i e_(i)e_{i}ei for i = 1 , 2 i = 1 , 2 i=1,2i=1,2i=1,2 using some generalized identities of Vandermonde type.
We want to find the sequences a n , b n , c n a n , b n , c n a_(n),b_(n),c_(n)a_{n}, b_{n}, c_{n}an,bn,cn, which appears in ( L n [ α , β ] e i ) ( x ) L n [ α , β ] e i ( x ) (L_(n)^([alpha,beta])e_(i))(x)\left(L_{n}^{[\alpha, \beta]} e_{i}\right)(x)(Ln[α,β]ei)(x), using relations (9).
The Pincherle derivative of Q Q QQQ is
Q ′ = − β α E − β + ( 1 + β α ) E − α − β = E − α − β ( I − β α Δ α ) Q ′ = − β α E − β + 1 + β α E − α − β = E − α − β I − β α Δ α Q^(')=-(beta )/(alpha)E^(-beta)+(1+(beta )/(alpha))E^(-alpha-beta)=E^(-alpha-beta)(I-(beta )/(alpha)Delta_(alpha))Q^{\prime}=-\frac{\beta}{\alpha} E^{-\beta}+\left(1+\frac{\beta}{\alpha}\right) E^{-\alpha-\beta}=E^{-\alpha-\beta}\left(I-\frac{\beta}{\alpha} \Delta_{\alpha}\right)Q′=−βαE−β+(1+βα)E−α−β=E−α−β(I−βαΔα)
so
( Q ′ ) − 1 = E α + β ∑ k ≥ 0 β k ( Δ α α ) k Q ′ − 1 = E α + β ∑ k ≥ 0   β k Δ α α k (Q^('))^(-1)=E^(alpha+beta)sum_(k >= 0)beta^(k)((Delta_(alpha))/(alpha))^(k)\left(Q^{\prime}\right)^{-1}=E^{\alpha+\beta} \sum_{k \geq 0} \beta^{k}\left(\frac{\Delta_{\alpha}}{\alpha}\right)^{k}(Q′)−1=Eα+β∑k≥0βk(Δαα)k
Since s n − 1 ( x ) = ( x + ( n − 1 ) β ) [ n − 1 , − α ] = E ( n − 2 ) α + ( n − 1 ) β x [ n − 1 , α ] s n − 1 ( x ) = ( x + ( n − 1 ) β ) [ n − 1 , − α ] = E ( n − 2 ) α + ( n − 1 ) β x [ n − 1 , α ] s_(n-1)(x)=(x+(n-1)beta)^([n-1,-alpha])=E^((n-2)alpha+(n-1)beta)x^([n-1,alpha])s_{n-1}(x)=(x+(n-1) \beta)^{[n-1,-\alpha]}=E^{(n-2) \alpha+(n-1) \beta} x^{[n-1, \alpha]}sn−1(x)=(x+(n−1)β)[n−1,−α]=E(n−2)α+(n−1)βx[n−1,α] and x [ n , α ] = x ( x − 1 ) … ( x − ( n − 1 ) α ) x [ n , α ] = x ( x − 1 ) … ( x − ( n − 1 ) α ) x^([n,alpha])=x(x-1)dots(x-(n-1)alpha)x^{[n, \alpha]}= x(x-1) \ldots(x-(n-1) \alpha)x[n,α]=x(x−1)…(x−(n−1)α) is the basic sequence for the delta operator Δ α α Δ α α (Delta_(alpha))/(alpha)\frac{\Delta_{\alpha}}{\alpha}Δαα, we have ( Δ α α ) k s n − 1 ( x ) = E ( n − 2 ) α + ( n − 1 ) β ( Δ α α ) k x [ n − 1 , α ] = ( n − 1 ) [ k ] E ( n − 2 ) α + ( n − 1 ) β Δ α α k s n − 1 ( x ) = E ( n − 2 ) α + ( n − 1 ) β Δ α α k x [ n − 1 , α ] = ( n − 1 ) [ k ] E ( n − 2 ) α + ( n − 1 ) β ((Delta_(alpha))/(alpha))^(k)s_(n-1)(x)=E^((n-2)alpha+(n-1)beta)((Delta_(alpha))/(alpha))^(k)x^([n-1,alpha])=(n-1)^([k])E^((n-2)alpha+(n-1)beta)\left(\frac{\Delta_{\alpha}}{\alpha}\right)^{k} s_{n-1}(x)=E^{(n-2) \alpha+(n-1) \beta}\left(\frac{\Delta_{\alpha}}{\alpha}\right)^{k} x^{[n-1, \alpha]}=(n-1)^{[k]} E^{(n-2) \alpha+(n-1) \beta}(Δαα)ksn−1(x)=E(n−2)α+(n−1)β(Δαα)kx[n−1,α]=(n−1)[k]E(n−2)α+(n−1)β. ⋅ x [ n − 1 − k , α ] ⋅ x [ n − 1 − k , α ] *x^([n-1-k,alpha])\cdot x^{[n-1-k, \alpha]}⋅x[n−1−k,α]. Because a n = [ ( Q ′ ) − 1 s n − 1 ] ( 1 ) s n ( 1 ) a n = Q ′ − 1 s n − 1 ( 1 ) s n ( 1 ) a_(n)=([(Q^('))^(-1)s_(n-1)](1))/(s_(n)(1))a_{n}=\frac{\left[\left(Q^{\prime}\right)^{-1} s_{n-1}\right](1)}{s_{n}(1)}an=[(Q′)−1sn−1](1)sn(1) we get
a n = ∑ k = 0 n − 1 ( n − 1 k ) k ! β k ( 1 + n β ) [ k + 1 , − α ] a n = ∑ k = 0 n − 1   ( n − 1 k ) k ! β k ( 1 + n β ) [ k + 1 , − α ] a_(n)=sum_(k=0)^(n-1)((n-1)/(k))(k!beta^(k))/((1+n beta)^([k+1,-alpha]))a_{n}=\sum_{k=0}^{n-1}\binom{n-1}{k} \frac{k!\beta^{k}}{(1+n \beta)^{[k+1,-\alpha]}}an=∑k=0n−1(n−1k)k!βk(1+nβ)[k+1,−α]
Since b n = n − 1 n [ ( Q ′ ) − 2 s n − 2 ] ( 1 ) s n ( 1 ) b n = n − 1 n Q ′ − 2 s n − 2 ( 1 ) s n ( 1 ) b_(n)=(n-1)/(n)([(Q^('))^(-2)s_(n-2)](1))/(s_(n)(1))b_{n}=\frac{n-1}{n} \frac{\left[\left(Q^{\prime}\right)^{-2} s_{n-2}\right](1)}{s_{n}(1)}bn=n−1n[(Q′)−2sn−2](1)sn(1) and
( Q ′ ) − 2 = E 2 α + 2 β ( I − β α Δ α ) − 2 = E 2 α + 2 β ∑ k ≥ 0 ( k + 1 ) β k ( Δ α α ) k Q ′ − 2 = E 2 α + 2 β I − β α Δ α − 2 = E 2 α + 2 β ∑ k ≥ 0   ( k + 1 ) β k Δ α α k (Q^('))^(-2)=E^(2alpha+2beta)(I-(beta )/(alpha)Delta_(alpha))^(-2)=E^(2alpha+2beta)sum_(k >= 0)(k+1)beta^(k)((Delta_(alpha))/(alpha))^(k)\left(Q^{\prime}\right)^{-2}=E^{2 \alpha+2 \beta}\left(I-\frac{\beta}{\alpha} \Delta_{\alpha}\right)^{-2}=E^{2 \alpha+2 \beta} \sum_{k \geq 0}(k+1) \beta^{k}\left(\frac{\Delta_{\alpha}}{\alpha}\right)^{k}(Q′)−2=E2α+2β(I−βαΔα)−2=E2α+2β∑k≥0(k+1)βk(Δαα)k
we obtain
b n = n − 1 n ∑ k = 0 n − 2 ( n − 2 k ) ( k + 1 ) ! β k ( 1 + n β ) [ k + 2 , − α ] b n = n − 1 n ∑ k = 0 n − 2   ( n − 2 k ) ( k + 1 ) ! β k ( 1 + n β ) [ k + 2 , − α ] b_(n)=(n-1)/(n)sum_(k=0)^(n-2)((n-2)/(k))((k+1)!beta^(k))/((1+n beta)^([k+2,-alpha]))b_{n}=\frac{n-1}{n} \sum_{k=0}^{n-2}\binom{n-2}{k} \frac{(k+1)!\beta^{k}}{(1+n \beta)^{[k+2,-\alpha]}}bn=n−1n∑k=0n−2(n−2k)(k+1)!βk(1+nβ)[k+2,−α]
The Pincherle derivative of S − 1 S − 1 S^(-1)S^{-1}S−1 may be written in the form
( S − 1 ) ′ = − ( α + β ) E − α − β ( Q ′ ) − 1 − E − α − β ( Q ′ ) − 2 Q ′ ′ = − E − α − β ( Q ′ ) − 2 ( ( α + β ) Q ′ + Q ′ ′ ) . S − 1 ′ = − ( α + β ) E − α − β Q ′ − 1 − E − α − β Q ′ − 2 Q ′ ′ = − E − α − β Q ′ − 2 ( α + β ) Q ′ + Q ′ ′ . {:[(S^(-1))^(')=-(alpha+beta)E^(-alpha-beta)(Q^('))^(-1)-E^(-alpha-beta)(Q^('))^(-2)Q^('')],[=-E^(-alpha-beta)(Q^('))^(-2)((alpha+beta)Q^(')+Q^('')).]:}\begin{aligned} \left(S^{-1}\right)^{\prime} & =-(\alpha+\beta) E^{-\alpha-\beta}\left(Q^{\prime}\right)^{-1}-E^{-\alpha-\beta}\left(Q^{\prime}\right)^{-2} Q^{\prime \prime} \\ & =-E^{-\alpha-\beta}\left(Q^{\prime}\right)^{-2}\left((\alpha+\beta) Q^{\prime}+Q^{\prime \prime}\right) . \end{aligned}(S−1)′=−(α+β)E−α−β(Q′)−1−E−α−β(Q′)−2Q′′=−E−α−β(Q′)−2((α+β)Q′+Q′′).
Because Q ′ ′ = 1 α ( β 2 E − β − ( α + β ) 2 E − α − β ) Q ′ ′ = 1 α β 2 E − β − ( α + β ) 2 E − α − β Q^('')=(1)/(alpha)(beta^(2)E^(-beta)-(alpha+beta)^(2)E^(-alpha-beta))Q^{\prime \prime}=\frac{1}{\alpha}\left(\beta^{2} E^{-\beta}-(\alpha+\beta)^{2} E^{-\alpha-\beta}\right)Q′′=1α(β2E−β−(α+β)2E−α−β) and ( α + β ) Q ′ + Q ′ ′ = − β E − β ( α + β ) Q ′ + Q ′ ′ = − β E − β (alpha+beta)Q^(')+Q^('')=-betaE^(-beta)(\alpha+\beta) Q^{\prime}+Q^{\prime \prime}=-\beta E^{-\beta}(α+β)Q′+Q′′=−βE−β this implies
( Q ′ ) − 2 ( S − 1 ) ′ S = β ( Q ′ ) − 3 E − β = β E 3 α + 2 β ( I − β α Δ α ) − 3 = E 3 α + 2 β ∑ k ≥ 0 ( k + 1 ) ( k + 2 ) 2 β k + 1 ( 1 α Δ α ) k . Q ′ − 2 S − 1 ′ S = β Q ′ − 3 E − β = β E 3 α + 2 β I − β α Δ α − 3 = E 3 α + 2 β ∑ k ≥ 0   ( k + 1 ) ( k + 2 ) 2 β k + 1 1 α Δ α k . {:[(Q^('))^(-2)(S^(-1))^(')S=beta(Q^('))^(-3)E^(-beta)],[=betaE^(3alpha+2beta)(I-(beta )/(alpha)Delta_(alpha))^(-3)],[=E^(3alpha+2beta)sum_(k >= 0)((k+1)(k+2))/(2)beta^(k+1)((1)/(alpha)Delta_(alpha))^(k).]:}\begin{aligned} \left(Q^{\prime}\right)^{-2}\left(S^{-1}\right)^{\prime} S & =\beta\left(Q^{\prime}\right)^{-3} E^{-\beta} \\ & =\beta E^{3 \alpha+2 \beta}\left(I-\frac{\beta}{\alpha} \Delta_{\alpha}\right)^{-3} \\ & =E^{3 \alpha+2 \beta} \sum_{k \geq 0} \frac{(k+1)(k+2)}{2} \beta^{k+1}\left(\frac{1}{\alpha} \Delta_{\alpha}\right)^{k} . \end{aligned}(Q′)−2(S−1)′S=β(Q′)−3E−β=βE3α+2β(I−βαΔα)−3=E3α+2β∑k≥0(k+1)(k+2)2βk+1(1αΔα)k.
From (9) and the previous relation we get
c n = n − 1 2 n ( ( 1 + n α + n β ) ∑ k = 0 n − 3 ( n − 2 k ) ( k + 2 ) ! β k + 1 ( 1 + n β ) [ k + 3 , − α ] + n ! β n − 1 ( 1 + n β ) [ n , − α ] ) c n = n − 1 2 n ( 1 + n α + n β ) ∑ k = 0 n − 3   ( n − 2 k ) ( k + 2 ) ! β k + 1 ( 1 + n β ) [ k + 3 , − α ] + n ! β n − 1 ( 1 + n β ) [ n , − α ] c_(n)=(n-1)/(2n)((1+n alpha+n beta)sum_(k=0)^(n-3)((n-2)/(k))((k+2)!beta^(k+1))/((1+n beta)^([k+3,-alpha]))+(n!beta^(n-1))/((1+n beta)^([n,-alpha])))c_{n}=\frac{n-1}{2 n}\left((1+n \alpha+n \beta) \sum_{k=0}^{n-3}\binom{n-2}{k} \frac{(k+2)!\beta^{k+1}}{(1+n \beta)^{[k+3,-\alpha]}}+\frac{n!\beta^{n-1}}{(1+n \beta)^{[n,-\alpha]}}\right)cn=n−12n((1+nα+nβ)∑k=0n−3(n−2k)(k+2)!βk+1(1+nβ)[k+3,−α]+n!βn−1(1+nβ)[n,−α])

5. EVALUATION OF THE ORDERS OF APPROXIMATION

Now we establish some estimates of the order of approximation of a function f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] by means of the operator L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S, defined by (2).
According to a result of O. Shisha and B. Mond [13], we can write
| f ( x ) − ( L n Q , S f ) ( x ) | ≤ [ 1 + 1 δ 2 L n Q , S ( ( t − x ) 2 ; x ) ] ω 1 ( f ; δ ) , δ ∈ R + f ( x ) − L n Q , S f ( x ) ≤ 1 + 1 δ 2 L n Q , S ( t − x ) 2 ; x ω 1 ( f ; δ ) , δ ∈ R + |f(x)-(L_(n)^(Q,S)f)(x)| <= [1+(1)/(delta^(2))L_(n)^(Q,S)((t-x)^(2);x)]omega_(1)(f;delta),quad delta inR^(+)\left|f(x)-\left(L_{n}^{Q, S} f\right)(x)\right| \leq\left[1+\frac{1}{\delta^{2}} L_{n}^{Q, S}\left((t-x)^{2} ; x\right)\right] \omega_{1}(f ; \delta), \quad \delta \in \mathbb{R}^{+}|f(x)−(LnQ,Sf)(x)|≤[1+1δ2LnQ,S((t−x)2;x)]ω1(f;δ),δ∈R+
Using the relations (8) we have
L n Q , S ( ( t − x ) 2 ; x ) = x 2 ( b n − 2 a n + 1 ) + x ( a n − b n − c n ) L n Q , S ( t − x ) 2 ; x = x 2 b n − 2 a n + 1 + x a n − b n − c n L_(n)^(Q,S)((t-x)^(2);x)=x^(2)(b_(n)-2a_(n)+1)+x(a_(n)-b_(n)-c_(n))L_{n}^{Q, S}\left((t-x)^{2} ; x\right)=x^{2}\left(b_{n}-2 a_{n}+1\right)+x\left(a_{n}-b_{n}-c_{n}\right)LnQ,S((t−x)2;x)=x2(bn−2an+1)+x(an−bn−cn)
so we get
| f ( x ) − ( L n Q , S f ) ( x ) | ≤ [ 1 + 1 δ 2 [ x 2 ( b n − 2 a n + 1 ) + x ( a n − b n − c n ) ] ] ω 1 ( f ; δ ) f ( x ) − L n Q , S f ( x ) ≤ 1 + 1 δ 2 x 2 b n − 2 a n + 1 + x a n − b n − c n ω 1 ( f ; δ ) |f(x)-(L_(n)^(Q,S)f)(x)| <= [1+(1)/(delta^(2))[x^(2)(b_(n)-2a_(n)+1)+x(a_(n)-b_(n)-c_(n))]]omega_(1)(f;delta)\left|f(x)-\left(L_{n}^{Q, S} f\right)(x)\right| \leq\left[1+\frac{1}{\delta^{2}}\left[x^{2}\left(b_{n}-2 a_{n}+1\right)+x\left(a_{n}-b_{n}-c_{n}\right)\right]\right] \omega_{1}(f ; \delta)|f(x)−(LnQ,Sf)(x)|≤[1+1δ2[x2(bn−2an+1)+x(an−bn−cn)]]ω1(f;δ).
One observes that if b n − 2 a n + 1 < 0 b n − 2 a n + 1 < 0 b_(n)-2a_(n)+1 < 0b_{n}-2 a_{n}+1<0bn−2an+1<0 then x 2 ( b n − 2 a n + 1 ) + x ( a n − b n − c n ) ≤ ( a n − b n − c n ) 2 4 ( 2 a n − b n − 1 ) , ∀ x ∈ [ 0 , 1 ] x 2 b n − 2 a n + 1 + x a n − b n − c n ≤ a n − b n − c n 2 4 2 a n − b n − 1 , ∀ x ∈ [ 0 , 1 ] x^(2)(b_(n)-2a_(n)+1)+x(a_(n)-b_(n)-c_(n)) <= ((a_(n)-b_(n)-c_(n))^(2))/(4(2a_(n)-b_(n)-1)),AA x in[0,1]x^{2}\left(b_{n}-2 a_{n}+1\right)+x\left(a_{n}-b_{n}-c_{n}\right) \leq \frac{\left(a_{n}-b_{n}-c_{n}\right)^{2}}{4\left(2 a_{n}-b_{n}-1\right)}, \forall x \in[0,1]x2(bn−2an+1)+x(an−bn−cn)≤(an−bn−cn)24(2an−bn−1),∀x∈[0,1].
By choosing δ = 1 n δ = 1 n delta=(1)/(sqrtn)\delta=\frac{1}{\sqrt{n}}δ=1n we can state
Theorem 5. If f ∈ C [ 0 , 1 ] f ∈ C [ 0 , 1 ] f in C[0,1]f \in C[0,1]f∈C[0,1] and ∃ k ∈ N ∃ k ∈ N EE k inN\exists k \in \mathbb{N}∃k∈N such as b n − 2 a n + 1 < 0 , ∀ n ≥ k b n − 2 a n + 1 < 0 , ∀ n ≥ k b_(n)-2a_(n)+1 < 0,AA n >= kb_{n}-2 a_{n}+1<0, \forall n \geq kbn−2an+1<0,∀n≥k, then we can give the following estimation of the order of approximation, by means of the first modulus of continuity
‖ f − L n Q , S f ‖ ≤ ( 1 + n 4 ( a n − b n − c n ) 2 ( 2 a n − b n − 1 ) ) ω 1 ( f ; 1 n ) , n ≥ k f − L n Q , S f ≤ 1 + n 4 a n − b n − c n 2 2 a n − b n − 1 ω 1 f ; 1 n , n ≥ k ||f-L_(n)^(Q,S)f|| <= (1+(n)/(4)((a_(n)-b_(n)-c_(n))^(2))/((2a_(n)-b_(n)-1)))omega_(1)(f;(1)/(sqrtn)),quad n >= k\left\|f-L_{n}^{Q, S} f\right\| \leq\left(1+\frac{n}{4} \frac{\left(a_{n}-b_{n}-c_{n}\right)^{2}}{\left(2 a_{n}-b_{n}-1\right)}\right) \omega_{1}\left(f ; \frac{1}{\sqrt{n}}\right), \quad n \geq k‖f−LnQ,Sf‖≤(1+n4(an−bn−cn)2(2an−bn−1))ω1(f;1n),n≥k
where a n , b n , c n a n , b n , c n a_(n),b_(n),c_(n)a_{n}, b_{n}, c_{n}an,bn,cn are defined by (9).
In the case of binomial operators of positive type defined by (1), since S ′ = I ′ = O S ′ = I ′ = O S^(')=I^(')=OS^{\prime}=I^{\prime}=OS′=I′=O we have
(19) a n = 1 , c n = 0 , b n = n − 1 n [ ( Q ′ ) − 2 p n − 2 ] ( 1 ) p n ( 1 ) (19) a n = 1 , c n = 0 , b n = n − 1 n Q ′ − 2 p n − 2 ( 1 ) p n ( 1 ) {:(19)a_(n)=1","quadc_(n)=0","quadb_(n)=(n-1)/(n)([(Q^('))^(-2)p_(n-2)](1))/(p_(n)(1)):}\begin{equation*} a_{n}=1, \quad c_{n}=0, \quad b_{n}=\frac{n-1}{n} \frac{\left[\left(Q^{\prime}\right)^{-2} p_{n-2}\right](1)}{p_{n}(1)} \tag{19} \end{equation*}(19)an=1,cn=0,bn=n−1n[(Q′)−2pn−2](1)pn(1)
Then b n − 2 a n + 1 = b n − 1 < 0 , ∀ n ∈ N b n − 2 a n + 1 = b n − 1 < 0 , ∀ n ∈ N b_(n)-2a_(n)+1=b_(n)-1 < 0,AA n inNb_{n}-2 a_{n}+1=b_{n}-1<0, \forall n \in \mathbb{N}bn−2an+1=bn−1<0,∀n∈N therefore the previous inequality reduces to
‖ f − L n f ‖ ≤ ( 5 4 + n 4 d n ) ω 1 ( f ; 1 n ) f − L n f ≤ 5 4 + n 4 d n ω 1 f ; 1 n ||f-L_(n)f|| <= ((5)/(4)+(n)/(4)d_(n))omega_(1)(f;(1)/(sqrtn))\left\|f-L_{n} f\right\| \leq\left(\frac{5}{4}+\frac{n}{4} d_{n}\right) \omega_{1}\left(f ; \frac{1}{\sqrt{n}}\right)‖f−Lnf‖≤(54+n4dn)ω1(f;1n)
where
(20) d n = n − 1 n − b n = n − 1 n ( 1 − [ ( Q ′ ) − 2 p n − 2 ] ( 1 ) p n ( 1 ) ) (20) d n = n − 1 n − b n = n − 1 n 1 − Q ′ − 2 p n − 2 ( 1 ) p n ( 1 ) {:(20)d_(n)=(n-1)/(n)-b_(n)=(n-1)/(n)(1-([(Q^('))^(-2)p_(n-2)](1))/(p_(n)(1))):}\begin{equation*} d_{n}=\frac{n-1}{n}-b_{n}=\frac{n-1}{n}\left(1-\frac{\left[\left(Q^{\prime}\right)^{-2} p_{n-2}\right](1)}{p_{n}(1)}\right) \tag{20} \end{equation*}(20)dn=n−1n−bn=n−1n(1−[(Q′)−2pn−2](1)pn(1))
We mention that this inequality was established by D. D. Stancu in [18].
In order to find an evaluation of the order of approximation using both moduli of smoothness ω 1 ω 1 omega_(1)\omega_{1}ω1 and ω 2 ω 2 omega_(2)\omega_{2}ω2 we can use a result of H. H. Gonska and R. K. Kovacheva included in the following
Lemma 3. [2]. If I = [ a , b ] I = [ a , b ] I=[a,b]I=[a, b]I=[a,b] is a compact interval of the real axis and I 1 = [ a 1 , b 1 ] I 1 = a 1 , b 1 I_(1)=[a_(1),b_(1)]I_{1}= \left[a_{1}, b_{1}\right]I1=[a1,b1] is a subinterval of it, and if we assume that L : C ( I ) → C ( I 1 ) L : C ( I ) → C I 1 L:C(I)rarr C(I_(1))L: C(I) \rightarrow C\left(I_{1}\right)L:C(I)→C(I1) is a positive operator, such that L e 0 = e 0 L e 0 = e 0 Le_(0)=e_(0)L e_{0}=e_{0}Le0=e0 and 0 ≤ δ ≤ 1 2 ( b − a ) 0 ≤ δ ≤ 1 2 ( b − a ) 0 <= delta <= (1)/(2)(b-a)0 \leq \delta \leq \frac{1}{2}(b-a)0≤δ≤12(b−a), then we have
| f ( x ) − L ( f ( t ) ; x ) | ≤ 2 δ | L ( t − x ; x ) | ω 1 ( f ; δ ) + + 3 2 [ 1 + 1 δ | L ( t − x ; x ) | + 1 2 δ 2 L ( ( t − x ) 2 ; x ) ] ω 2 ( f ; δ ) | f ( x ) − L ( f ( t ) ; x ) | ≤ 2 δ | L ( t − x ; x ) | ω 1 ( f ; δ ) + + 3 2 1 + 1 δ | L ( t − x ; x ) | + 1 2 δ 2 L ( t − x ) 2 ; x ω 2 ( f ; δ ) {:[|f(x)-L(f(t);x)| <= (2)/(delta)|L(t-x;x)|omega_(1)(f;delta)+],[+(3)/(2)[1+(1)/(delta)|L(t-x;x)|+(1)/(2delta^(2))L((t-x)^(2);x)]omega_(2)(f;delta)]:}\begin{aligned} |f(x)-L(f(t) ; x)| \leq & \frac{2}{\delta}|L(t-x ; x)| \omega_{1}(f ; \delta)+ \\ & +\frac{3}{2}\left[1+\frac{1}{\delta}|L(t-x ; x)|+\frac{1}{2 \delta^{2}} L\left((t-x)^{2} ; x\right)\right] \omega_{2}(f ; \delta) \end{aligned}|f(x)−L(f(t);x)|≤2δ|L(t−x;x)|ω1(f;δ)++32[1+1δ|L(t−x;x)|+12δ2L((t−x)2;x)]ω2(f;δ)
Using the relations (8) we obtain the inequality
| f ( x ) − ( L n Q , S f ) ( x ) | ≤ 2 δ | ( a n − 1 ) x | ω 1 ( f ; δ ) + + 3 2 [ 1 + 1 δ | ( a n − 1 ) x | + 1 2 δ 2 [ x 2 ( b n − 2 a n + 1 ) + x ( a n − b n − c n ) ] ] ω 2 ( f ; δ ) f ( x ) − L n Q , S f ( x ) ≤ 2 δ a n − 1 x ω 1 ( f ; δ ) + + 3 2 1 + 1 δ a n − 1 x + 1 2 δ 2 x 2 b n − 2 a n + 1 + x a n − b n − c n ω 2 ( f ; δ ) {:[|f(x)-(L_(n)^(Q,S)f)(x)| <= (2)/(delta)|(a_(n)-1)x|omega_(1)(f;delta)+],[+(3)/(2)[1+(1)/(delta)|(a_(n)-1)x|+(1)/(2delta^(2))[x^(2)(b_(n)-2a_(n)+1)+x(a_(n)-b_(n)-c_(n))]]omega_(2)(f;delta)]:}\begin{aligned} & \left|f(x)-\left(L_{n}^{Q, S} f\right)(x)\right| \leq \frac{2}{\delta}\left|\left(a_{n}-1\right) x\right| \omega_{1}(f ; \delta)+ \\ & +\frac{3}{2}\left[1+\frac{1}{\delta}\left|\left(a_{n}-1\right) x\right|+\frac{1}{2 \delta^{2}}\left[x^{2}\left(b_{n}-2 a_{n}+1\right)+x\left(a_{n}-b_{n}-c_{n}\right)\right]\right] \omega_{2}(f ; \delta) \end{aligned}|f(x)−(LnQ,Sf)(x)|≤2δ|(an−1)x|ω1(f;δ)++32[1+1δ|(an−1)x|+12δ2[x2(bn−2an+1)+x(an−bn−cn)]]ω2(f;δ)
If b n − 2 a n + 1 < 0 b n − 2 a n + 1 < 0 b_(n)-2a_(n)+1 < 0b_{n}-2 a_{n}+1<0bn−2an+1<0 the previous inequality implies
‖ f − L n Q , S f ‖ ≤ 2 δ ( 1 − a n ) ω 1 ( f ; δ ) + 3 2 [ 1 + 1 δ ( 1 − a n ) + 1 8 δ 2 ( a n − b n − c n ) 2 ( 2 a n − b n − 1 ) ] ω 2 ( f ; δ ) . f − L n Q , S f ≤ 2 δ 1 − a n ω 1 ( f ; δ ) + 3 2 1 + 1 δ 1 − a n + 1 8 δ 2 a n − b n − c n 2 2 a n − b n − 1 ω 2 ( f ; δ ) . ||f-L_(n)^(Q,S)f|| <= (2)/(delta)(1-a_(n))omega_(1)(f;delta)+(3)/(2)[1+(1)/(delta)(1-a_(n))+(1)/(8delta^(2))((a_(n)-b_(n)-c_(n))^(2))/((2a_(n)-b_(n)-1))]omega_(2)(f;delta).\left\|f-L_{n}^{Q, S} f\right\| \leq \frac{2}{\delta}\left(1-a_{n}\right) \omega_{1}(f ; \delta)+\frac{3}{2}\left[1+\frac{1}{\delta}\left(1-a_{n}\right)+\frac{1}{8 \delta^{2}} \frac{\left(a_{n}-b_{n}-c_{n}\right)^{2}}{\left(2 a_{n}-b_{n}-1\right)}\right] \omega_{2}(f ; \delta) .‖f−LnQ,Sf‖≤2δ(1−an)ω1(f;δ)+32[1+1δ(1−an)+18δ2(an−bn−cn)2(2an−bn−1)]ω2(f;δ).
By choosing δ = 1 n δ = 1 n delta=(1)/(sqrtn)\delta=\frac{1}{\sqrt{n}}δ=1n we get
‖ f − L n Q , S f ‖ ≤ 2 n ( 1 − a n ) ω 1 ( f ; 1 n ) + + 3 2 [ 1 + n ( 1 − a n ) + n 8 ( a n − b n − c n ) 2 ( 2 a n − b n − 1 ) ] ω 2 ( f ; 1 n ) . f − L n Q , S f ≤ 2 n 1 − a n ω 1 f ; 1 n + + 3 2 1 + n 1 − a n + n 8 a n − b n − c n 2 2 a n − b n − 1 ω 2 f ; 1 n . {:[||f-L_(n)^(Q,S)f|| <= 2sqrtn(1-a_(n))omega_(1)(f;(1)/(sqrtn))+],[+(3)/(2)[1+sqrtn(1-a_(n))+(n)/(8)((a_(n)-b_(n)-c_(n))^(2))/((2a_(n)-b_(n)-1))]omega_(2)(f;(1)/(sqrtn)).]:}\begin{aligned} \left\|f-L_{n}^{Q, S} f\right\| \leq & 2 \sqrt{n}\left(1-a_{n}\right) \omega_{1}\left(f ; \frac{1}{\sqrt{n}}\right)+ \\ & +\frac{3}{2}\left[1+\sqrt{n}\left(1-a_{n}\right)+\frac{n}{8} \frac{\left(a_{n}-b_{n}-c_{n}\right)^{2}}{\left(2 a_{n}-b_{n}-1\right)}\right] \omega_{2}\left(f ; \frac{1}{\sqrt{n}}\right) . \end{aligned}‖f−LnQ,Sf‖≤2n(1−an)ω1(f;1n)++32[1+n(1−an)+n8(an−bn−cn)2(2an−bn−1)]ω2(f;1n).
If we consider the binomial operator introduced by Tiberiu Popoviciu, using (19) and the previous relation, we arrive at an inequality which has found by D. D. Stancu (see [18])
‖ f − L n f ‖ ≤ 3 16 ( 9 + n d n ) ω 2 ( f ; 1 n ) , f − L n f ≤ 3 16 9 + n d n ω 2 f ; 1 n , ||f-L_(n)f|| <= (3)/(16)(9+nd_(n))omega_(2)(f;(1)/(sqrtn)),\left\|f-L_{n} f\right\| \leq \frac{3}{16}\left(9+n d_{n}\right) \omega_{2}\left(f ; \frac{1}{\sqrt{n}}\right),‖f−Lnf‖≤316(9+ndn)ω2(f;1n),
where d n d n d_(n)d_{n}dn is defined by (20).

REFERENCES

[1] Cheney, E. W. and Sharma, A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma, 5, pp. 77-82, 1964.
[2] Gonska, H. H. and Kovacheva, R. K., The second order modulus revisited: remarks, applications, problems, Conferenze del Seminario di Matematica Univ. Bari, 257, pp. 132, 1994.
[3] Lupaş, L. and Lupaş, A., Polynomials of binomial type and approximation operators, Studia Univ. Babes-Bolyai, Mathematica, 32, pp. 61-69, 1987.
[4] Lupaş, A., Approximation operators of binomial type, Proc. IDoMAT 98, International Series of Numerical Mathematics, ISNM 132, Birkhäuser Verlag, Basel, pp. 175-198, 1999.
[5] Manole, C., Developments in series of generalized Appell polynomials, with applications to the approximation of functions, Ph.D. Thesis, Cluj-Napoca, Romania, 1984 (in Romanian).
[6] Moldovan, G., Generalizations of the S. N. Bernstein operators, Ph.D. Thesis, ClujNapoca, Romania, 1971 (in Romanian).
[7] Moldovan, G., Discrete convolutions and positive operators I, Annales Univ. Sci. Budapest R. Eötvös, 15, pp. 31-34, 1972.
[8] Mullin, R. and Rota, G. C., On the foundations of combinatorial theory III, theory of binomial enumeration, in: B. Harris, ed., Graph Theory and Its Applications, Academic Press, New York, pp. 167-213, 1970.
[9] Popoviciu, T., Remarques sur les poynômes binomiaux, Bul. Soc. Ştiinte Cluj, 6, pp 146-148, 1931.
[10] Roman, S., Operational formulas, Linear and Multilinear Algebra, 12, pp. 1-20, 1982.
[11] Rota, G. C., Kahaner, D. and Odlyzko, A., Finite operator calculus, J. Math. Anal. Appl., 42, pp. 685-760, 1973.
[12] Sablonnière, P., Positive Bernstein-Sheffer operators, J. Approx. Theory, 83, pp. 330-341, 1995.
[13] Shisha, O. and Mond, B., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 60, pp. 1196-1200, 1968.
[14] Stancu, D. D., Approximation of functions by a new class of linear positive operators, Rev. Roum. Math. Pures Appl., 13, pp. 1173-1194, 1968.
[15] Stancu, D. D., On a generalization of the Bernstein polynomials, Studia Univ. BabeşBolyai, Cluj, 14, pp. 31-45, 1969.
[16] Stancu, D. D., Approximation properties of a class of linear positive operators, Studia Univ. Babeş-Bolyai, Cluj, 15, pp. 31-38, 1970.
[17] Stancu, D. D., Approximation of functions by means of some new classes of positive linear operators, Numerische Methoden der Approximationstheorie, Proc. Conf. Oberwolfach 1971 ISNM 16, Birkhäuser-Verlag, Basel, pp. 187-203, 1972.
[18] Stancu, D. D. and Occorsio, M.R., On approximation by binomial operators of Tiberiu Popoviciu type, Rev. Anal. Numér. Théor. Approx., 27 no.1, pp. 167-181, 1998. 줄
[19] Stancu, D. D. and Cismaşiu, C., On an approximating linear positive operator of Cheney-Sharma, Rev. Anal. Numér. Théor. Approx., 26, pp. 221-227, 1997.«
Received September 12, 2000.
  1. "T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Romania, e-mail: craciun@ictp-acad.math.ubbcluj.ro.

[1] Cheney, E. W. and Sharma, A., On a generalization of Bernstein polynomials, Riv. Mat. Univ. Parma, 5, pp. 77–82, 1964.
[2] Gonska, H. H. and Kovacheva, R. K., The second order modulus revisited: remarks, applications, problems, Conferenze del Seminario di Matematica Univ. Bari, 257, pp.1–32, 1994.
[3] Lupas, L. and Lupas, A., Polynomials of binomial type and approximation operators, Studia Univ. Babes–Bolyai, Mathematica, 32, pp. 61–69, 1987.
[4] Lupas, A., Approximation operators of binomial type, Proc. IDoMAT 98, International Series of Numerical Mathematics, ISNM 132, Birkhauser Verlag, Basel, pp. 175 198, 1999.
[5] Manole, C., Developments in series of generalized Appell polynomials, with applications to the approximation of functions, Ph.D. Thesis, Cluj–Napoca, Romania, 1984 (in Romanian).
[6] Moldovan, G., Generalizations of the S. N. Bernstein operators, Ph.D. Thesis, Cluj–Napoca, Romania, 1971 (in Romanian).
[7] Moldovan, G., Discrete convolutions and positive operators I, Annales Univ. Sci. Budapest R. E¨otv¨os, 15, pp. 31–34, 1972.
[8] Mullin, R. and Rota, G. C., On the foundations of combinatorial theory III, theory of binomial enumeration, in: B. Harris, ed., Graph Theory and Its Applications, Academic Press, New York, pp. 167–213, 1970.
[9] Popoviciu, T., Remarques sur les poynomes binomiaux, Bul. Soc. Stiinte Cluj, 6, pp 146–148, 1931.
[10] Roman, S., Operational formulas, Linear and Multilinear Algebra, 12, pp. 1–20, 1982.
[11] Rota, G. C., Kahaner, D. and Odlyzko, A., Finite operator calculus, J. Math. Anal.
Appl., 42, pp. 685–760, 1973.
[12] Sablonniere, P. ` , Positive Bernstein–Sheffer operators, J. Approx. Theory, 83, pp. 330–341, 1995.
[13] Shisha, O. and Mond, B., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 60, pp. 1196–1200, 1968.
[14] Stancu, D. D., Approximation of functions by a new class of linear positive operators, Rev. Roum. Math. Pures Appl., 13, pp. 1173–1194, 1968.
[15] Stancu, D. D., On a generalization of the Bernstein polynomials, Studia Univ. Babes– Bolyai, Cluj, 14, pp. 31–45, 1969.
[16] Stancu, D. D., Approximation properties of a class of linear positive operators, Studia Univ. Babes–Bolyai, Cluj, 15, pp. 31–38, 1970.
[17] Stancu, D. D., Approximation of functions by means of some new classes of positive linear operators, Numerische Methoden der Approximationstheorie, Proc. Conf. Oberwolfach 1971 ISNM 16, Birkhauser–Verlag, Basel, pp. 187–203, 1972.
[18] Stancu, D. D. and Occorsio, M.R., On approximation by binomial operators of Tiberiu Popoviciu type, Rev. Anal. Numer. Theor. Approx., 27 no.1, pp. 167–181, 1998.
[19] Stancu, D. D. and Cismasiu, C., On an approximating linear positive operator of Cheney–Sharma, Rev. Anal. Numer. Theor. Approx., 26, pp. 221–227, 1997.

2001

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