Posts by Maria Crăciun

Maria Crăciun

Abstract

In this note we consider an approximation operator of Kantorovich type in which expression appears a basic sequence for a delta operator and a Sheffer sequence for the same delta operator.

We give a convergence theorem for this operator and we find its Lipschitz constant.

Authors

Maria Craciun
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Keywords

approximation operators of Kantorovich type; Sheffer sequences; Lipschitz constants.

References

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https://ictp.acad.ro/craciun/papers/craciun-2002-ANTA.pdf

Cite this paper as:

M. Crăciun, On an approximating operator and its Lipschitz constant, Rev. Anal. Numér. Théor. Approx., vol. 31 (2002), no. 1, 55-60.

About this paper

Journal

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

Publisher Name

Editura Academiei Române

Paper on the journal website

https://ictp.acad.ro/jnaat/journal/article/view/2002-vol31-no1-art7

Print ISSN

1222-9024

Online ISSN

2457-8126

Google Scholar Profile

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[6] Mihesan, V., Approximation of continuous functions by means of linear positive operators, Ph.D. Thesis, Cluj-Napoca, 1997 (in Romanian).

[7] Moldovan, G., Discrete convolutions and linear positive operators, Ann. Univ. Sci. Budapest R. E¨otv¨os, 15, pp. 31–44, 1972.

[8] Popoviciu, T., Remarques sur les polynomes binomiaux, Bull. Soc. Math. Cluj, 6, pp. 146–148, 1931.

[9] Rota, G.-C., Kahaner, D. and Odlyzko, A., On the foundations of combinatorial theory. VIII. Finite operator calculus, J. Math. Anal. Appl., 42, pp. 684–760, 1973.

[10] Sablonniere, P., Positive Bernstein-Sheffer operators, J. Approx. Theory, 83, pp. 330–341, 1995.

[11] Stancu, D. D., Approximation of functions by a new class of linear positive operators, Rev. Roum. Math. Pures Appl., 13, pp. 1173–1194, 1968.

[12] Stancu, D. D., On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu, Rev. Anal. Numer. Theor. Approx., 30, pp. 95–105, 2001.

[13] Stancu, D. D. and Occorsio, M. R., On approximation by binomial operators of Tiberiu Popoviciu type, Rev. Anal. Numer. Theor. Approx., 27, pp. 167–181, 1998.

Paper (preprint) in HTML form

craciun-2002-ANTA

ON AN APPROXIMATION OPERATOR AND ITS LIPSCHITZ CONSTANT

MARIA CRĂCIUN*

Abstract

In this note we consider an approximation operator of Kantorovich type in which expression appears a basic sequence for a delta operator and a Sheffer sequence for the same delta operator. We give a convergence theorem for this operator and we find its Lipschitz constant.

MSC 2000. 41A36.
Keywords. approximation operators of Kantorovich type, Sheffer sequences, Lipschitz constants.

1. INTRODUCTION

In this section we will remind some basic notions and results.
Let P P PPP be the linear space of all polynomials with real coefficients.
A polynomial sequence is a sequence of polynomials ( p n ) p n (p_(n))\left(p_{n}\right)(pn) with deg p n = n deg p n = n degp_(n)=n\operatorname{deg} p_{n}=ndegpn=n for all n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N.
A sequence of binomial type (a binomial sequence) is a polynomial sequence which satisfies the binomial identity
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 real 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,….
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 with T E a = E a T T E a = E a T TE^(a)=E^(a)TT E^{a}=E^{a} TTEa=EaT for all real a a aaa is called a shift invariant operator.
We recall 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.
A delta operator is a shift invariant operator for which Q x = Q x = Qx=Q x=Qx= const. ≠ 0 ≠ 0 !=0\neq 0≠0.
A polynomial sequence ( p n p n p_(n)p_{n}pn ) is called a basic sequence for a delta operator Q Q QQQ if p 0 ( x ) = 1 , p n ( 0 ) = 0 p 0 ( x ) = 1 , p n ( 0 ) = 0 p_(0)(x)=1,p_(n)(0)=0p_{0}(x)=1, p_{n}(0)=0p0(x)=1,pn(0)=0 and Q p n = n p n − 1 , n = 1 , 2 , … Q p n = n p n − 1 , n = 1 , 2 , … Qp_(n)=np_(n-1),n=1,2,dotsQ p_{n}=n p_{n-1}, n=1,2, \ldotsQpn=npn−1,n=1,2,….
Proposition 1. [9]. i) Every delta operator has a unique basic sequence.
ii) A polynomial sequence is a binomial sequence if and only if it is a basic sequence for a delta operator Q Q QQQ.
The Pincherle derivative of an operator T T TTT is defined by T ′ = T X − X T T ′ = T X − X T T^(')=TX-XTT^{\prime}=T X-X TT′=TX−XT, where X X XXX is the multiplication operator, 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).
The Pincherle derivative of a shift invariant operator is also a shift invariant operator and the Pincherle derivative of a delta operator is an invertible operator.
A polynomial sequence ( s n ) n ≥ 0 s n n ≥ 0 (s_(n))_(n >= 0)\left(s_{n}\right)_{n \geq 0}(sn)n≥0 is called a Sheffer sequence relative to a delta operator Q Q QQQ if s 0 ( x ) = s 0 ( x ) = s_(0)(x)=s_{0}(x)=s0(x)= const ≠ 0 ≠ 0 !=0\neq 0≠0 and 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 Appell sequence is a Sheffer sequence relative to the derivative D D DDD.
Proposition 2. [9]. Let Q Q QQQ be a delta operator with the basic sequence ( p n ) p n (p_(n))\left(p_{n}\right)(pn) and ( s n ) s n (s_(n))\left(s_{n}\right)(sn) a polynomial sequence. The following statements are equivalent:
i) s n s n s_(n)s_{n}sn 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 = n n ( n k ) p k ( x ) s n − k ( y ) s n ( x + y ) = ∑ k = n n   ( n k ) p k ( x ) s n − k ( y ) s_(n)(x+y)=sum_(k=n)^(n)((n)/(k))p_(k)(x)s_(n-k)(y)s_{n}(x+y)=\sum_{k=n}^{n}\binom{n}{k} p_{k}(x) s_{n-k}(y)sn(x+y)=∑k=nn(nk)pk(x)sn−k(y)

2. AN APPROXIMATION OPERATOR OF KANTOROVICH TYPE

In our paper [3] we considered some linear approximation operators defined for all 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] by
(1) ( 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 ) (1) 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 {:(1)(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{1} \end{equation*}(1)(LnQ,Sf)(x)=1sn(1)∑k=0n(nk)pk(x)sn−k(1−x)f(kn)
where ( p n p n p_(n)p_{n}pn ) is the basic sequence for a delta operator Q Q QQQ and ( s n s n s_(n)s_{n}sn ) is a Sheffer sequence for the same delta operator, s n ( 1 ) ≠ 0 , ∀ n ∈ N , s n = S − 1 p n s n ( 1 ) ≠ 0 , ∀ n ∈ N , s n = S − 1 p n s_(n)(1)!=0,AA n inN,s_(n)=S^(-1)p_(n)s_{n}(1) \neq 0, \forall n \in \mathbb{N}, s_{n}=S^{-1} p_{n}sn(1)≠0,∀n∈N,sn=S−1pn with S S SSS an invertible shift invariant operator.
We remind that 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 (1) is positive.
In this note we want to introduce an integral operator of Kantorovich type of the form
(2) ( K n Q , S f ) ( x ) = ( n + 1 ) s n ( 1 ) ∑ k = 0 n ( n k ) p k ( x ) s n − k ( 1 − x ) ∫ k n + 1 k + 1 n + 1 f ( t ) d t (2) K n Q , S f ( x ) = ( n + 1 ) s n ( 1 ) ∑ k = 0 n   ( n k ) p k ( x ) s n − k ( 1 − x ) ∫ k n + 1 k + 1 n + 1   f ( t ) d t {:(2)(K_(n)^(Q,S)f)(x)=((n+1))/(s_(n)(1))sum_(k=0)^(n)((n)/(k))p_(k)(x)s_(n-k)(1-x)int_((k)/(n+1))^((k+1)/(n+1))f(t)dt:}\begin{equation*} \left(K_{n}^{Q, S} f\right)(x)=\frac{(n+1)}{s_{n}(1)} \sum_{k=0}^{n}\binom{n}{k} p_{k}(x) s_{n-k}(1-x) \int_{\frac{k}{n+1}}^{\frac{k+1}{n+1}} f(t) \mathrm{d} t \tag{2} \end{equation*}(2)(KnQ,Sf)(x)=(n+1)sn(1)∑k=0n(nk)pk(x)sn−k(1−x)∫kn+1k+1n+1f(t)dt
where f ∈ L 1 ( [ 0 , 1 ] ) , x ∈ [ 0 , 1 ] f ∈ L 1 ( [ 0 , 1 ] ) , x ∈ [ 0 , 1 ] f inL_(1)([0,1]),x in[0,1]f \in L_{1}([0,1]), x \in[0,1]f∈L1([0,1]),x∈[0,1].
We mention that for S = I S = I S=IS=IS=I (that means s n = p n s n = p n s_(n)=p_(n)s_{n}=p_{n}sn=pn ) these operators were considered by O. Agratini in [1] and V. Miheşan in [6].
We recall that the expressions of the operator L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S on the test functions e k ( x ) = x k , k = 0 , 2 ― e k ( x ) = x k , k = 0 , 2 ¯ e_(k)(x)=x^(k),k= bar(0,2)e_{k}(x)=x^{k}, k=\overline{0,2}ek(x)=xk,k=0,2― are (see [3]):
( L n Q , S e 0 ) ( x ) = e 0 ( x ) ( 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 ) 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)],[(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{aligned} & \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) \\ & \left(L_{n}^{Q, S} e_{2}\right)(x)=b_{n} x^{2}+x\left(a_{n}-b_{n}-c_{n}\right) \end{aligned}(LnQ,Se0)(x)=e0(x)(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 ) , 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 ) , 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)),quadb_(n)=(n-1)/(n)(((Q^('))^(-2)s_(n-2))(1))/(s_(n)(1)),quadc_(n)=(n-1)/(n)(((Q^('))^(-2)(S^(-1))^(')Ss_(n-2))(1))/(s_(n)(1))a_{n}=\frac{\left(\left(Q^{\prime}\right)^{-1} s_{n-1}\right)(1)}{s_{n}(1)}, \quad b_{n}=\frac{n-1}{n} \frac{\left(\left(Q^{\prime}\right)^{-2} s_{n-2}\right)(1)}{s_{n}(1)}, \quad 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)}an=((Q′)−1sn−1)(1)sn(1),bn=n−1n((Q′)−2sn−2)(1)sn(1),cn=n−1n((Q′)−2(S−1)′Ssn−2)(1)sn(1) and Q ′ Q ′ Q^(')Q^{\prime}Q′ is the Pincherle derivative of Q Q QQQ.
Lemma 3. If K n Q , S K n Q , S K_(n)^(Q,S)K_{n}^{Q, S}KnQ,S is the linear operator defined by (2) then:
( K n Q , S e 0 ) ( x ) = e 0 ( x ) ( K n Q , S e 1 ) ( x ) = n n + 1 a n e 1 ( x ) + 1 2 ( n + 1 ) ( K n Q , S e 2 ) ( x ) = 1 ( n + 1 ) 2 { x 2 n 2 b n + x [ n 2 ( a n − b n − c n ) + n a n ] + 1 3 } K n Q , S e 0 ( x ) = e 0 ( x ) K n Q , S e 1 ( x ) = n n + 1 a n e 1 ( x ) + 1 2 ( n + 1 ) K n Q , S e 2 ( x ) = 1 ( n + 1 ) 2 x 2 n 2 b n + x n 2 a n − b n − c n + n a n + 1 3 {:[(K_(n)^(Q,S)e_(0))(x)=e_(0)(x)],[(K_(n)^(Q,S)e_(1))(x)=(n)/(n+1)a_(n)e_(1)(x)+(1)/(2(n+1))],[(K_(n)^(Q,S)e_(2))(x)=(1)/((n+1)^(2)){x^(2)n^(2)b_(n)+x[n^(2)(a_(n)-b_(n)-c_(n))+na_(n)]+(1)/(3)}]:}\begin{aligned} & \left(K_{n}^{Q, S} e_{0}\right)(x)=e_{0}(x) \\ & \left(K_{n}^{Q, S} e_{1}\right)(x)=\frac{n}{n+1} a_{n} e_{1}(x)+\frac{1}{2(n+1)} \\ & \left(K_{n}^{Q, S} e_{2}\right)(x)=\frac{1}{(n+1)^{2}}\left\{x^{2} n^{2} b_{n}+x\left[n^{2}\left(a_{n}-b_{n}-c_{n}\right)+n a_{n}\right]+\frac{1}{3}\right\} \end{aligned}(KnQ,Se0)(x)=e0(x)(KnQ,Se1)(x)=nn+1ane1(x)+12(n+1)(KnQ,Se2)(x)=1(n+1)2{x2n2bn+x[n2(an−bn−cn)+nan]+13}
Proof. If we denote s n , k ( x ) = 1 s n ( 1 ) ( n k ) p k ( x ) s n − k ( 1 − x ) s n , k ( x ) = 1 s n ( 1 ) ( n k ) p k ( x ) s n − k ( 1 − x ) s_(n,k)(x)=(1)/(s_(n)(1))((n)/(k))p_(k)(x)s_(n-k)(1-x)s_{n, k}(x)=\frac{1}{s_{n}(1)}\binom{n}{k} p_{k}(x) s_{n-k}(1-x)sn,k(x)=1sn(1)(nk)pk(x)sn−k(1−x) we have
( K n Q , S e 0 ) ( x ) = ( n + 1 ) ∑ k = 0 n s n , k ( x ) ( k + 1 n + 1 − k n + 1 ) = 1 = e 0 ( x ) , ( K n Q , S e 1 ) ( x ) = n n + 1 ∑ k = 0 n s n , k ( x ) k n + 1 2 ( n + 1 ) ∑ k = 0 n s n , k ( x ) = n n + 1 ( L n Q , S e 1 ) ( x ) + 1 2 ( n + 1 ) ( L n Q , S e 0 ) ( x ) = n n + 1 a n e 1 ( x ) + 1 2 ( n + 1 ) , ( K n Q , S e 2 ) ( x ) = n 2 ( n + 1 ) 2 ( L n Q , S e 2 ) ( x ) + n ( n + 1 ) 2 ( L n Q , S e 1 ) ( x ) + 1 3 ( n + 1 ) 2 ( L n Q , S e 0 ) ( x ) = 1 ( n + 1 ) 2 { x 2 n 2 b n + x [ n 2 ( a n − b n − c n ) + n a n ] + 1 3 } . K n Q , S e 0 ( x ) = ( n + 1 ) ∑ k = 0 n   s n , k ( x ) k + 1 n + 1 − k n + 1 = 1 = e 0 ( x ) , K n Q , S e 1 ( x ) = n n + 1 ∑ k = 0 n   s n , k ( x ) k n + 1 2 ( n + 1 ) ∑ k = 0 n   s n , k ( x ) = n n + 1 L n Q , S e 1 ( x ) + 1 2 ( n + 1 ) L n Q , S e 0 ( x ) = n n + 1 a n e 1 ( x ) + 1 2 ( n + 1 ) , K n Q , S e 2 ( x ) = n 2 ( n + 1 ) 2 L n Q , S e 2 ( x ) + n ( n + 1 ) 2 L n Q , S e 1 ( x ) + 1 3 ( n + 1 ) 2 L n Q , S e 0 ( x ) = 1 ( n + 1 ) 2 x 2 n 2 b n + x n 2 a n − b n − c n + n a n + 1 3 . {:[(K_(n)^(Q,S)e_(0))(x)=(n+1)sum_(k=0)^(n)s_(n,k)(x)((k+1)/(n+1)-(k)/(n+1))=1=e_(0)(x)","],[(K_(n)^(Q,S)e_(1))(x)=(n)/(n+1)sum_(k=0)^(n)s_(n,k)(x)(k)/(n)+(1)/(2(n+1))sum_(k=0)^(n)s_(n,k)(x)],[=(n)/(n+1)(L_(n)^(Q,S)e_(1))(x)+(1)/(2(n+1))(L_(n)^(Q,S)e_(0))(x)],[=(n)/(n+1)a_(n)e_(1)(x)+(1)/(2(n+1))","],[(K_(n)^(Q,S)e_(2))(x)=(n^(2))/((n+1)^(2))(L_(n)^(Q,S)e_(2))(x)+(n)/((n+1)^(2))(L_(n)^(Q,S)e_(1))(x)],[+(1)/(3(n+1)^(2))(L_(n)^(Q,S)e_(0))(x)],[=(1)/((n+1)^(2)){x^(2)n^(2)b_(n)+x[n^(2)(a_(n)-b_(n)-c_(n))+na_(n)]+(1)/(3)}.]:}\begin{aligned} \left(K_{n}^{Q, S} e_{0}\right)(x)= & (n+1) \sum_{k=0}^{n} s_{n, k}(x)\left(\frac{k+1}{n+1}-\frac{k}{n+1}\right)=1=e_{0}(x), \\ \left(K_{n}^{Q, S} e_{1}\right)(x)= & \frac{n}{n+1} \sum_{k=0}^{n} s_{n, k}(x) \frac{k}{n}+\frac{1}{2(n+1)} \sum_{k=0}^{n} s_{n, k}(x) \\ = & \frac{n}{n+1}\left(L_{n}^{Q, S} e_{1}\right)(x)+\frac{1}{2(n+1)}\left(L_{n}^{Q, S} e_{0}\right)(x) \\ = & \frac{n}{n+1} a_{n} e_{1}(x)+\frac{1}{2(n+1)}, \\ \left(K_{n}^{Q, S} e_{2}\right)(x)= & \frac{n^{2}}{(n+1)^{2}}\left(L_{n}^{Q, S} e_{2}\right)(x)+\frac{n}{(n+1)^{2}}\left(L_{n}^{Q, S} e_{1}\right)(x) \\ & +\frac{1}{3(n+1)^{2}}\left(L_{n}^{Q, S} e_{0}\right)(x) \\ = & \frac{1}{(n+1)^{2}}\left\{x^{2} n^{2} b_{n}+x\left[n^{2}\left(a_{n}-b_{n}-c_{n}\right)+n a_{n}\right]+\frac{1}{3}\right\} . \end{aligned}(KnQ,Se0)(x)=(n+1)∑k=0nsn,k(x)(k+1n+1−kn+1)=1=e0(x),(KnQ,Se1)(x)=nn+1∑k=0nsn,k(x)kn+12(n+1)∑k=0nsn,k(x)=nn+1(LnQ,Se1)(x)+12(n+1)(LnQ,Se0)(x)=nn+1ane1(x)+12(n+1),(KnQ,Se2)(x)=n2(n+1)2(LnQ,Se2)(x)+n(n+1)2(LnQ,Se1)(x)+13(n+1)2(LnQ,Se0)(x)=1(n+1)2{x2n2bn+x[n2(an−bn−cn)+nan]+13}.
From this Lemma, the central moments of K n Q , S K n Q , S K_(n)^(Q,S)K_{n}^{Q, S}KnQ,S defined by Ω n , k ( x ) = K n Q , S ( ( e 1 − x e 0 ) k , x ) , k ∈ N Ω n , k ( x ) = K n Q , S e 1 − x e 0 k , x , k ∈ N Omega_(n,k)(x)=K_(n)^(Q,S)((e_(1)-xe_(0))^(k),x),k inN\Omega_{n, k}(x)= K_{n}^{Q, S}\left(\left(e_{1}-x e_{0}\right)^{k}, x\right), k \in \mathbb{N}Ωn,k(x)=KnQ,S((e1−xe0)k,x),k∈N are
Ω n , 0 ( x ) = 1 Ω n , 1 ( x ) = 1 n + 1 [ x ( n a n − ( n + 1 ) ) + 1 2 ] and Ω n , 2 ( x ) = 1 ( n + 1 ) 2 { x 2 [ n 2 ( b n − 2 a n + 1 ) + 2 n ( 1 − a n ) ] + x [ n 2 ( a n − b n − c n ) + n ( a n − 1 ) − 1 ] + 1 3 } Ω n , 0 ( x ) = 1 Ω n , 1 ( x ) = 1 n + 1 x n a n − ( n + 1 ) + 1 2  and  Ω n , 2 ( x ) = 1 ( n + 1 ) 2 x 2 n 2 b n − 2 a n + 1 + 2 n 1 − a n + x n 2 a n − b n − c n + n a n − 1 − 1 + 1 3 {:[Omega_(n,0)(x)=1],[Omega_(n,1)(x)=(1)/(n+1)[x(na_(n)-(n+1))+(1)/(2)]" and "],[Omega_(n,2)(x)=(1)/((n+1)^(2)){x^(2)[n^(2)(b_(n)-2a_(n)+1)+2n(1-a_(n))]:}],[{:+x[n^(2)(a_(n)-b_(n)-c_(n))+n(a_(n)-1)-1]+(1)/(3)}]:}\begin{aligned} \Omega_{n, 0}(x)=1 & \\ \Omega_{n, 1}(x)=\frac{1}{n+1}\left[x\left(n a_{n}-(n+1)\right)+\frac{1}{2}\right] \text { and } & \\ \Omega_{n, 2}(x)= & \frac{1}{(n+1)^{2}}\left\{x^{2}\left[n^{2}\left(b_{n}-2 a_{n}+1\right)+2 n\left(1-a_{n}\right)\right]\right. \\ & \left.+x\left[n^{2}\left(a_{n}-b_{n}-c_{n}\right)+n\left(a_{n}-1\right)-1\right]+\frac{1}{3}\right\} \end{aligned}Ωn,0(x)=1Ωn,1(x)=1n+1[x(nan−(n+1))+12] and Ωn,2(x)=1(n+1)2{x2[n2(bn−2an+1)+2n(1−an)]+x[n2(an−bn−cn)+n(an−1)−1]+13}
Theorem 4. Let K n Q , S K n Q , S K_(n)^(Q,S)K_{n}^{Q, S}KnQ,S be the linear operator defined by (2) with p k ′ ( 0 ) ≥ 0 p k ′ ( 0 ) ≥ 0 p_(k)^(')(0) >= 0p_{k}^{\prime}(0) \geq 0pk′(0)≥0 and s k ( 0 ) ≥ 0 , ∀ k ∈ N s k ( 0 ) ≥ 0 , ∀ k ∈ N s_(k)(0) >= 0,AA k inNs_{k}(0) \geq 0, \forall k \in \mathbb{N}sk(0)≥0,∀k∈N.
i) If f ∈ C [ 0 , 1 ] , lim n → ∞ a n = lim n → ∞ b n = 1 f ∈ C [ 0 , 1 ] , lim n → ∞   a n = lim n → ∞   b n = 1 f in C[0,1],lim_(n rarr oo)a_(n)=lim_(n rarr oo)b_(n)=1f \in C[0,1], \lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n}=1f∈C[0,1],limn→∞an=limn→∞bn=1 then K n Q , S K n Q , S K_(n)^(Q,S)K_{n}^{Q, S}KnQ,S converges uniformly to f f fff.
ii) If f ∈ L p [ 0 , 1 ] , lim n → ∞ a n = lim n → ∞ b n = 1 f ∈ L p [ 0 , 1 ] , lim n → ∞   a n = lim n → ∞   b n = 1 f inL_(p)[0,1],lim_(n rarr oo)a_(n)=lim_(n rarr oo)b_(n)=1f \in L_{p}[0,1], \lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n}=1f∈Lp[0,1],limn→∞an=limn→∞bn=1 then ‖ K n Q , S f − f ‖ p = 0 K n Q , S f − f p = 0 ||K_(n)^(Q,S)f-f||_(p)=0\left\|K_{n}^{Q, S} f-f\right\|_{p}=0‖KnQ,Sf−f‖p=0.
Proof. In [3] we proved that 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 , ∀ k ∈ N s k ( 0 ) ≥ 0 , ∀ k ∈ N s_(k)(0) >= 0,AA k inNs_{k}(0) \geq 0, \forall k \in \mathbb{N}sk(0)≥0,∀k∈N then 0 ≤ c n ≤ min { ( 1 − b n ) / 2 , a n − a n 2 } 0 ≤ c n ≤ min 1 − b n / 2 , a n − a n 2 0 <= c_(n) <= min{(1-b_(n))//2,a_(n)-a_(n)^(2)}0 \leq c_{n} \leq \min \left\{\left(1-b_{n}\right) / 2, a_{n}-a_{n}^{2}\right\}0≤cn≤min{(1−bn)/2,an−an2}, so from 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 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. Using Lemma 3 it results that lim n → ∞ ( K n Q , S e i ) ( x ) = e i ( x ) lim n → ∞   K n Q , S e i ( x ) = e i ( x ) lim_(n rarr oo)(K_(n)^(Q,S)e_(i))(x)=e_(i)(x)\lim _{n \rightarrow \infty}\left(K_{n}^{Q, S} e_{i}\right)(x)=e_{i}(x)limn→∞(KnQ,Sei)(x)=ei(x) for i = 0 , 1 , 2 i = 0 , 1 , 2 i=0,1,2i=0,1,2i=0,1,2, and applying the convergence criterion of Bohman-Korovkin we obtain the first affirmation.
The second assertion follows immediately because the Korovkin subspaces in C [ 0 , 1 ] C [ 0 , 1 ] C[0,1]C[0,1]C[0,1] are also Korovkin subspaces in L p [ 0 , 1 ] L p [ 0 , 1 ] L_(p)[0,1]L_{p}[0,1]Lp[0,1].

3. LIPSCHITZ CONSTANTS FOR L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S AND K n Q , S K n Q , S K_(n)^(Q,S)K_{n}^{Q, S}KnQ,S

In this section we want to find the Lipschitz constants for L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S and K n Q , S K n Q , S K_(n)^(Q,S)K_{n}^{Q, S}KnQ,S if f ∈ Lip M α f ∈ Lip M α f inLip_(M)alphaf \in \operatorname{Lip}_{M} \alphaf∈LipMα.
In [2] B.M. Brown, D. Elliot and D.F. Paget proved that the Bernstein operator ( B n = L n D , I B n = L n D , I B_(n)=L_(n)^(D,I)B_{n}=L_{n}^{D, I}Bn=LnD,I ) preserves the Lipschitz constant of the function f f fff for α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1]. V. Miheşan showed in [6] that all positive binomial operators (which can be obtained by L n Q , S L n Q , S L_(n)^(Q,S)L_{n}^{Q, S}LnQ,S when S = I S = I S=IS=IS=I ) preserve the Lipschitz constant of the function f f fff for α ∈ ( 0 , 1 ] α ∈ ( 0 , 1 ] alpha in(0,1]\alpha \in(0,1]α∈(0,1] and if f ∈ Lip M ∗ ( α , [ 0 , 1 ] ) f ∈ Lip M ∗ ( α , [ 0 , 1 ] ) f inLip_(M)^(**)(alpha,[0,1])f \in \operatorname{Lip}_{M}^{*}(\alpha,[0,1])f∈LipM∗(α,[0,1]) then L n Q , I f ∈ Lip 2 M ∗ ( α , [ 0 , 1 ] ) L n Q , I f ∈ Lip 2 M ∗ ( α , [ 0 , 1 ] ) L_(n)^(Q,I)f inLip_(2M)^(**)(alpha,[0,1])L_{n}^{Q, I} f \in \operatorname{Lip}_{2 M}^{*}(\alpha,[0,1])LnQ,If∈Lip2M∗(α,[0,1]), where
Lip M ∗ ( α , [ 0 , 1 ] ) = { f ∈ C [ 0 , 1 ] , ω 2 ( f , h ) ≤ M h α , 0 < h ≤ 1 2 } . Lip M ∗ ( α , [ 0 , 1 ] ) = f ∈ C [ 0 , 1 ] , ω 2 ( f , h ) ≤ M h α , 0 < h ≤ 1 2 . Lip_(M)^(**)(alpha,[0,1])={f in C[0,1],omega_(2)(f,h) <= Mh^(alpha),0 < h <= (1)/(2)}.\operatorname{Lip}_{M}^{*}(\alpha,[0,1])=\left\{f \in C[0,1], \omega_{2}(f, h) \leq M h^{\alpha}, 0<h \leq \frac{1}{2}\right\} .LipM∗(α,[0,1])={f∈C[0,1],ω2(f,h)≤Mhα,0<h≤12}.
Theorem 5. If f ∈ Lip M α , α ∈ ( 0 , 1 ] f ∈ Lip M α , α ∈ ( 0 , 1 ] f inLip_(M)alpha,alpha in(0,1]f \in \operatorname{Lip}_{M} \alpha, \alpha \in(0,1]f∈LipMα,α∈(0,1], then L n Q , S f ∈ Lip M a n α α L n Q , S f ∈ Lip M a n α α L_(n)^(Q,S)f inLip_(Ma_(n)^(alpha))alphaL_{n}^{Q, S} f \in \operatorname{Lip}_{M a_{n}^{\alpha}} \alphaLnQ,Sf∈LipManαα.
Proof. Let x ≤ y x ≤ y x <= yx \leq yx≤y be any two points of [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1]. Using the binomial identity for p n p n p_(n)p_{n}pn we can write
( L n Q , S f ) ( y ) = 1 s n ( 1 ) ∑ j = 0 n ( n j ) p j ( x + ( y − x ) ) s n − j ( 1 − y ) f ( j n ) = 1 s n ( 1 ) ∑ j = 0 n ( n j ) s n − j ( 1 − y ) f ( j n ) ∑ k = 0 j ( j k ) p k ( x ) p j − k ( y − x ) L n Q , S f ( y ) = 1 s n ( 1 ) ∑ j = 0 n   ( n j ) p j ( x + ( y − x ) ) s n − j ( 1 − y ) f j n = 1 s n ( 1 ) ∑ j = 0 n   ( n j ) s n − j ( 1 − y ) f j n ∑ k = 0 j   ( j k ) p k ( x ) p j − k ( y − x ) {:[(L_(n)^(Q,S)f)(y)=(1)/(s_(n)(1))sum_(j=0)^(n)((n)/(j))p_(j)(x+(y-x))s_(n-j)(1-y)f((j)/(n))],[=(1)/(s_(n)(1))sum_(j=0)^(n)((n)/(j))s_(n-j)(1-y)f((j)/(n))sum_(k=0)^(j)((j)/(k))p_(k)(x)p_(j-k)(y-x)]:}\begin{aligned} \left(L_{n}^{Q, S} f\right)(y) & =\frac{1}{s_{n}(1)} \sum_{j=0}^{n}\binom{n}{j} p_{j}(x+(y-x)) s_{n-j}(1-y) f\left(\frac{j}{n}\right) \\ & =\frac{1}{s_{n}(1)} \sum_{j=0}^{n}\binom{n}{j} s_{n-j}(1-y) f\left(\frac{j}{n}\right) \sum_{k=0}^{j}\binom{j}{k} p_{k}(x) p_{j-k}(y-x) \end{aligned}(LnQ,Sf)(y)=1sn(1)∑j=0n(nj)pj(x+(y−x))sn−j(1−y)f(jn)=1sn(1)∑j=0n(nj)sn−j(1−y)f(jn)∑k=0j(jk)pk(x)pj−k(y−x)
If we change the order of summation and note j − k = l j − k = l j-k=lj-k=lj−k=l then we obtain
(3) ( L n Q , S f ) ( y ) = = 1 s n ( 1 ) ∑ k = 0 n ∑ l = 0 n − k n ! k ! l ! ( n − k − l ) ! p k ( x ) p l ( y − x ) s n − k − l ( 1 − y ) f ( k + l n ) ( L n Q , S f ) ( x ) = 1 s n ( 1 ) ∑ k = 0 n ( n k ) p k ( x ) s n − k ( ( y − x ) + ( 1 − y ) ) f ( k n ) = 1 s n ( 1 ) ∑ k = 0 n ( n k ) p k ( x ) f ( k n ) ∑ l = 0 n − k ( n − k l ) p l ( y − x ) s n − k − l ( 1 − y ) f ( k n ) (4) ( L n Q , S f ) ( x ) = = 1 s n ( 1 ) ∑ k = 0 n ∑ l = 0 n − k n ! k ! l ! ( n − k − l ) ! p k ( x ) p l ( y − x ) s n − k − l ( 1 − y ) f ( k n ) (3) L n Q , S f ( y ) = = 1 s n ( 1 ) ∑ k = 0 n   ∑ l = 0 n − k   n ! k ! l ! ( n − k − l ) ! p k ( x ) p l ( y − x ) s n − k − l ( 1 − y ) f k + l n L n Q , S f ( x ) = 1 s n ( 1 ) ∑ k = 0 n   ( n k ) p k ( x ) s n − k ( ( y − x ) + ( 1 − y ) ) f k n = 1 s n ( 1 ) ∑ k = 0 n   ( n k ) p k ( x ) f k n ∑ l = 0 n − k   ( n − k l ) p l ( y − x ) s n − k − l ( 1 − y ) f k n (4) L n Q , S f ( x ) = = 1 s n ( 1 ) ∑ k = 0 n   ∑ l = 0 n − k   n ! k ! l ! ( n − k − l ) ! p k ( x ) p l ( y − x ) s n − k − l ( 1 − y ) f k n {:[(3)(L_(n)^(Q,S)f)(y)=],[=(1)/(s_(n)(1))sum_(k=0)^(n)sum_(l=0)^(n-k)(n!)/(k!l!(n-k-l)!)p_(k)(x)p_(l)(y-x)s_(n-k-l)(1-y)f((k+l)/(n))],[(L_(n)^(Q,S)f)(x)=(1)/(s_(n)(1))sum_(k=0)^(n)((n)/(k))p_(k)(x)s_(n-k)((y-x)+(1-y))f((k)/(n))],[=(1)/(s_(n)(1))sum_(k=0)^(n)((n)/(k))p_(k)(x)f((k)/(n))sum_(l=0)^(n-k)((n-k)/(l))p_(l)(y-x)s_(n-k-l)(1-y)f((k)/(n))],[(4)quad(L_(n)^(Q,S)f)(x)=],[quad=(1)/(s_(n)(1))sum_(k=0)^(n)sum_(l=0)^(n-k)(n!)/(k!l!(n-k-l)!)p_(k)(x)p_(l)(y-x)s_(n-k-l)(1-y)f((k)/(n))]:}\begin{gather*} \left(L_{n}^{Q, S} f\right)(y)= \tag{3}\\ =\frac{1}{s_{n}(1)} \sum_{k=0}^{n} \sum_{l=0}^{n-k} \frac{n!}{k!l!(n-k-l)!} p_{k}(x) p_{l}(y-x) s_{n-k-l}(1-y) f\left(\frac{k+l}{n}\right) \\ \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}((y-x)+(1-y)) f\left(\frac{k}{n}\right) \\ =\frac{1}{s_{n}(1)} \sum_{k=0}^{n}\binom{n}{k} p_{k}(x) f\left(\frac{k}{n}\right) \sum_{l=0}^{n-k}\binom{n-k}{l} p_{l}(y-x) s_{n-k-l}(1-y) f\left(\frac{k}{n}\right) \\ \quad\left(L_{n}^{Q, S} f\right)(x)= \tag{4}\\ \quad=\frac{1}{s_{n}(1)} \sum_{k=0}^{n} \sum_{l=0}^{n-k} \frac{n!}{k!l!(n-k-l)!} p_{k}(x) p_{l}(y-x) s_{n-k-l}(1-y) f\left(\frac{k}{n}\right) \end{gather*}(3)(LnQ,Sf)(y)==1sn(1)∑k=0n∑l=0n−kn!k!l!(n−k−l)!pk(x)pl(y−x)sn−k−l(1−y)f(k+ln)(LnQ,Sf)(x)=1sn(1)∑k=0n(nk)pk(x)sn−k((y−x)+(1−y))f(kn)=1sn(1)∑k=0n(nk)pk(x)f(kn)∑l=0n−k(n−kl)pl(y−x)sn−k−l(1−y)f(kn)(4)(LnQ,Sf)(x)==1sn(1)∑k=0n∑l=0n−kn!k!l!(n−k−l)!pk(x)pl(y−x)sn−k−l(1−y)f(kn)
From (3) and (4) we have
| ( L n Q , S f ) ( y ) − ( L n Q , S f ) ( x ) | = = 1 s n ( 1 ) ∑ k = 0 n ∑ l = 0 n − k n ! k ! l ! ( n − k − l ) ! p k ( x ) p l ( y − x ) s n − k − l ( 1 − y ) | f ( k + l n ) − f ( k n ) | L n Q , S f ( y ) − L n Q , S f ( x ) = = 1 s n ( 1 ) ∑ k = 0 n   ∑ l = 0 n − k   n ! k ! l ! ( n − k − l ) ! p k ( x ) p l ( y − x ) s n − k − l ( 1 − y ) f k + l n − f k n {:[|(L_(n)^(Q,S)f)(y)-(L_(n)^(Q,S)f)(x)|=],[=(1)/(s_(n)(1))sum_(k=0)^(n)sum_(l=0)^(n-k)(n!)/(k!l!(n-k-l)!)p_(k)(x)p_(l)(y-x)s_(n-k-l)(1-y)|f((k+l)/(n))-f((k)/(n))|]:}\begin{aligned} & \left|\left(L_{n}^{Q, S} f\right)(y)-\left(L_{n}^{Q, S} f\right)(x)\right|= \\ & =\frac{1}{s_{n}(1)} \sum_{k=0}^{n} \sum_{l=0}^{n-k} \frac{n!}{k!l!(n-k-l)!} p_{k}(x) p_{l}(y-x) s_{n-k-l}(1-y)\left|f\left(\frac{k+l}{n}\right)-f\left(\frac{k}{n}\right)\right| \end{aligned}|(LnQ,Sf)(y)−(LnQ,Sf)(x)|==1sn(1)∑k=0n∑l=0n−kn!k!l!(n−k−l)!pk(x)pl(y−x)sn−k−l(1−y)|f(k+ln)−f(kn)|
Because f ∈ Lip M α f ∈ Lip M α f inLip_(M)alphaf \in \operatorname{Lip}_{M} \alphaf∈LipMα we have | f ( k + l n ) − f ( k n ) | ≤ M ( l n ) α f k + l n − f k n ≤ M l n α |f((k+l)/(n))-f((k)/(n))| <= M((l)/(n))^(alpha)\left|f\left(\frac{k+l}{n}\right)-f\left(\frac{k}{n}\right)\right| \leq M\left(\frac{l}{n}\right)^{\alpha}|f(k+ln)−f(kn)|≤M(ln)α so we obtain
| ( L n Q , S f ) ( y ) − ( L n Q , S f ) ( x ) | ≤ ≤ M s n ( 1 ) ∑ k = 0 n ∑ l = 0 n − k n ! k ! ! ! ( n − k − l ) ! p k ( x ) p l ( y − x ) s n − k − l ( 1 − y ) ( l n ) α = M s n ( 1 ) ∑ l = 0 n ∑ k = 0 n − l ( n − l k ) p k ( x ) s n − k − l ( 1 − y ) ( n l ) p l ( y − x ) ( l n ) α = M s n ( 1 ) ∑ l = 0 n ( n l ) p l ( y − x ) s n − l ( x + 1 − y ) ( l n ) α = M L n Q , S ( x α ; y − x ) L n Q , S f ( y ) − L n Q , S f ( x ) ≤ ≤ M s n ( 1 ) ∑ k = 0 n   ∑ l = 0 n − k   n ! k ! ! ! ( n − k − l ) ! p k ( x ) p l ( y − x ) s n − k − l ( 1 − y ) l n α = M s n ( 1 ) ∑ l = 0 n   ∑ k = 0 n − l   ( n − l k ) p k ( x ) s n − k − l ( 1 − y ) ( n l ) p l ( y − x ) l n α = M s n ( 1 ) ∑ l = 0 n   ( n l ) p l ( y − x ) s n − l ( x + 1 − y ) l n α = M L n Q , S x α ; y − x {:[|(L_(n)^(Q,S)f)(y)-(L_(n)^(Q,S)f)(x)| <= ],[ <= (M)/(s_(n)(1))sum_(k=0)^(n)sum_(l=0)^(n-k)(n!)/(k!!!(n-k-l)!)p_(k)(x)p_(l)(y-x)s_(n-k-l)(1-y)((l)/(n))^(alpha)],[=(M)/(s_(n)(1))sum_(l=0)^(n)sum_(k=0)^(n-l)((n-l)/(k))p_(k)(x)s_(n-k-l)(1-y)((n)/(l))p_(l)(y-x)((l)/(n))^(alpha)],[=(M)/(s_(n)(1))sum_(l=0)^(n)((n)/(l))p_(l)(y-x)s_(n-l)(x+1-y)((l)/(n))^(alpha)],[=ML_(n)^(Q,S)(x^(alpha);y-x)]:}\begin{aligned} & \left|\left(L_{n}^{Q, S} f\right)(y)-\left(L_{n}^{Q, S} f\right)(x)\right| \leq \\ & \leq \frac{M}{s_{n}(1)} \sum_{k=0}^{n} \sum_{l=0}^{n-k} \frac{n!}{k!!!(n-k-l)!} p_{k}(x) p_{l}(y-x) s_{n-k-l}(1-y)\left(\frac{l}{n}\right)^{\alpha} \\ & =\frac{M}{s_{n}(1)} \sum_{l=0}^{n} \sum_{k=0}^{n-l}\binom{n-l}{k} p_{k}(x) s_{n-k-l}(1-y)\binom{n}{l} p_{l}(y-x)\left(\frac{l}{n}\right)^{\alpha} \\ & =\frac{M}{s_{n}(1)} \sum_{l=0}^{n}\binom{n}{l} p_{l}(y-x) s_{n-l}(x+1-y)\left(\frac{l}{n}\right)^{\alpha} \\ & =M L_{n}^{Q, S}\left(x^{\alpha} ; y-x\right) \end{aligned}|(LnQ,Sf)(y)−(LnQ,Sf)(x)|≤≤Msn(1)∑k=0n∑l=0n−kn!k!!!(n−k−l)!pk(x)pl(y−x)sn−k−l(1−y)(ln)α=Msn(1)∑l=0n∑k=0n−l(n−lk)pk(x)sn−k−l(1−y)(nl)pl(y−x)(ln)α=Msn(1)∑l=0n(nl)pl(y−x)sn−l(x+1−y)(ln)α=MLnQ,S(xα;y−x)
We remind that for a convex function f we have 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) (see [3]). Since the function g ( x ) = − x α , α ∈ ( 0 , 1 ] g ( x ) = − x α , α ∈ ( 0 , 1 ] g(x)=-x^(alpha),alpha in(0,1]g(x)=-x^{\alpha}, \alpha \in(0,1]g(x)=−xα,α∈(0,1], is convex on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] we obtain
( L n Q , S x α ; y − x ) ≤ ( a n ( y − x ) ) α L n Q , S x α ; y − x ≤ a n ( y − x ) α (L_(n)^(Q,S)x^(alpha);y-x) <= (a_(n)(y-x))^(alpha)\left(L_{n}^{Q, S} x^{\alpha} ; y-x\right) \leq\left(a_{n}(y-x)\right)^{\alpha}(LnQ,Sxα;y−x)≤(an(y−x))α
and we get
| ( L n Q , S f ) ( y ) − ( L n Q , S f ) ( x ) | ≤ M a n α ( y − x ) α L n Q , S f ( y ) − L n Q , S f ( x ) ≤ M a n α ( y − x ) α |(L_(n)^(Q,S)f)(y)-(L_(n)^(Q,S)f)(x)| <= Ma_(n)^(alpha)(y-x)^(alpha)\left|\left(L_{n}^{Q, S} f\right)(y)-\left(L_{n}^{Q, S} f\right)(x)\right| \leq M a_{n}^{\alpha}(y-x)^{\alpha}|(LnQ,Sf)(y)−(LnQ,Sf)(x)|≤Manα(y−x)α
Therefore L n Q , S f ∈ Lip M a n α α L n Q , S f ∈ Lip M a n α α L_(n)^(Q,S)f inLip_(M_(a_(n)^(alpha)))alphaL_{n}^{Q, S} f \in \operatorname{Lip}_{M_{a_{n}^{\alpha}}} \alphaLnQ,Sf∈LipManαα.
Theorem 6. If f ∈ Lip M α , α ∈ ( 0 , 1 ] f ∈ Lip M α , α ∈ ( 0 , 1 ] f inLip_(M)alpha,alpha in(0,1]f \in \operatorname{Lip}_{M} \alpha, \alpha \in(0,1]f∈LipMα,α∈(0,1], then K n Q , S f ∈ Lip N n α K n Q , S f ∈ Lip N n α K_(n)^(Q,S)f inLip_(N_(n))alphaK_{n}^{Q, S} f \in \operatorname{Lip}_{N_{n}} \alphaKnQ,Sf∈LipNnα, where N n = M ( n a n n + 1 ) α N n = M n a n n + 1 α N_(n)=M((na_(n))/(n+1))^(alpha)N_{n}=M\left(\frac{n a_{n}}{n+1}\right)^{\alpha}Nn=M(nann+1)α.
Proof. We can write K n Q , S f = L n Q , S h n K n Q , S f = L n Q , S h n K_(n)^(Q,S)f=L_(n)^(Q,S)h_(n)K_{n}^{Q, S} f=L_{n}^{Q, S} h_{n}KnQ,Sf=LnQ,Shn, where
h n ( x ) = ∫ 0 1 f ( t + n x n + 1 ) d t | h n ( x ) − h n ( y ) | = | ∫ 0 1 [ f ( t + n x n + 1 ) − f ( t + n y n + 1 ) ] d t | ≤ M | t + n x n + 1 − t + n y n + 1 | α ≤ M ( n n + 1 ) α | x − y | α h n ( x ) = ∫ 0 1   f t + n x n + 1 d t h n ( x ) − h n ( y ) = ∫ 0 1   f t + n x n + 1 − f t + n y n + 1 d t ≤ M t + n x n + 1 − t + n y n + 1 α ≤ M n n + 1 α | x − y | α {:[h_(n)(x)=int_(0)^(1)f((t+nx)/(n+1))dt],[|h_(n)(x)-h_(n)(y)|=|int_(0)^(1)[f((t+nx)/(n+1))-f((t+ny)/(n+1))]dt|],[ <= M|(t+nx)/(n+1)-(t+ny)/(n+1)|^(alpha) <= M((n)/(n+1))^(alpha)|x-y|^(alpha)]:}\begin{aligned} h_{n}(x) & =\int_{0}^{1} f\left(\frac{t+n x}{n+1}\right) \mathrm{d} t \\ \left|h_{n}(x)-h_{n}(y)\right| & =\left|\int_{0}^{1}\left[f\left(\frac{t+n x}{n+1}\right)-f\left(\frac{t+n y}{n+1}\right)\right] \mathrm{d} t\right| \\ & \leq M\left|\frac{t+n x}{n+1}-\frac{t+n y}{n+1}\right|^{\alpha} \leq M\left(\frac{n}{n+1}\right)^{\alpha}|x-y|^{\alpha} \end{aligned}hn(x)=∫01f(t+nxn+1)dt|hn(x)−hn(y)|=|∫01[f(t+nxn+1)−f(t+nyn+1)]dt|≤M|t+nxn+1−t+nyn+1|α≤M(nn+1)α|x−y|α
So, f ∈ Lip M α f ∈ Lip M α f inLip_(M)alphaf \in \operatorname{Lip}_{M} \alphaf∈LipMα implies h n ∈ Lip M ( n n + 1 ) α α h n ∈ Lip M n n + 1 α α h_(n)inLip_(M((n)/(n+1))^(alpha))alphah_{n} \in \operatorname{Lip}_{M\left(\frac{n}{n+1}\right)^{\alpha}} \alphahn∈LipM(nn+1)αα. From K n Q , S f = L n Q , S h n K n Q , S f = L n Q , S h n K_(n)^(Q,S)f=L_(n)^(Q,S)h_(n)K_{n}^{Q, S} f=L_{n}^{Q, S} h_{n}KnQ,Sf=LnQ,Shn and the previous theorem we obtain the conclusion.

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Received by the editors: September 27, 2001.
    • "T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Romania, e-mail: craciun@ictp.acad.ro.

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