Sequences of linear operators related to Cesàro-convergent sequences

Mira Anisiu
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Abstract

Given a Cesaro-convergent sequence of real numbers \((a_{n}% )_{n\in \mathbb{N}}\), a sequence \((\varphi_{n})_{n\in \mathbb{N}}\) of operators is defined on the Banach space \(\mathcal{R}(I,F)\) of regular functions defined on \(I=[0,1]\) and having values in a Banach space \(F\),%
\[
\varphi_{n}\left( f\right) =\frac{1}{n}\sum_{k=1}^{n}a_{k}f\left(\tfrac
{k}{n}\right) .
\]

It is proved that if, in addition, the sequence \(\left( \tfrac{\left \vert a_{1}\right \vert +\ldots \left \vert a_{n}\right \vert }{n}\right)\) is bounded, the \(\varphi_{n}\left( f\right)\) converges to \(a\cdot f_{0}^{1}f\), where \(a=lim_{n\infty1}\tfrac{\left \vert a_{1}\right \vert +\ldots+\left \vert a_{n}\right \vert }{n}\). The converse of this statement is also true. Another result is that the supplementary condition can be dropped if the operators are considered on the space \(C^{1}(I,F)\).

Authors

Mira-Cristiana Anisiu
Tiberiu Popoviciu Institute of Numerical Analysis Cluj-Napoca, Romania“

Valeriu Anisiu
Babes-Bolyai University, Faculty of Mathematics and Computer Science, Cluj-Napoca, Romania,

Keywords

Linear operators; Cesaro-convergent sequences.

Paper coordinates

M.-C. Anisiu, V. Anisiu, Sequences of linear operators related to Cesàro-convergent sequences, Rev. Anal. Numér. Théor. Approx. 31 (2) (2002), 139-145??, p.135-141

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https://doi.org/10.33993/jnaat312-717

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[1] Anisiu, V.,A principle of double condensation of singularities using σ-porosity, “Babes- Bolyai” Univ., Fac. of Math., Research Seminaries, Seminar on Math. Analysis, Preprint Nr. 7, pp. 85–88, 1985.
[2] Cobzas, S. and Muntean, I.,Condensation of singularities and divergence results in approximation theory, J. Approx. Theory, 31, pp. 148–153, 1981.
[3] Dieudonne, J.,Fondements de l’analyse moderne, Paris, Gauthier-Villars, 1963.
[4] Dunford, N. and Schwartz, J. T.,Linear Operators. Part 1: General Theory, John Wiley & Sons, New York, 1988.
[5] Trif, T.,On a problem from the Mathematical Contest, County Stage, Gazeta Matematica CVI (11), pp. 394–396, 2001 (in Romanian
jnaat,+Journal+manager,+2002-2-Anisiu-Anisiu-15-10-30

REVUE D'ANALYSE NUMÉRIQUE ET DE THÉORIE DE L'APPROXIMATION

Rev. Anal. Numér. Théor. Approx., vol. 31 (2002) no. 2, pp. 135-141
ictp.acad.ro/jnaat

SEQUENCES OF LINEAR OPERATORS
RELATED TO CESÀRO - CONVERGENT SEQUENCES

MIRA-CRISTIANA ANISIU* and VALERIU ANISIU † † ^(†){ }^{\dagger}†

Abstract

Given a Cesàro-convergent sequence of real numbers ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N, a sequence ( φ n ) n ∈ N φ n n ∈ N (varphi_(n))_(n inN)\left(\varphi_{n}\right)_{n \in \mathbb{N}}(φn)n∈N of operators is defined on the Banach space R ( I , F ) R ( I , F ) R(I,F)\mathcal{R}(I, F)R(I,F) of regular functions defined on I = [ 0 , 1 ] I = [ 0 , 1 ] I=[0,1]I=[0,1]I=[0,1] and having values in a Banach space F F FFF, φ n ( f ) = 1 n ∑ k = 1 n a k f ( k n ) . φ n ( f ) = 1 n ∑ k = 1 n   a k f k n . varphi_(n)(f)=(1)/(n)sum_(k=1)^(n)a_(k)f((k)/(n)).\varphi_{n}(f)=\frac{1}{n} \sum_{k=1}^{n} a_{k} f\left(\frac{k}{n}\right) .φn(f)=1n∑k=1nakf(kn).

It is proved that if, in addition, the sequence ( | a 1 | + … + | a n | n ) n ∈ N a 1 + … + a n n n ∈ N ((|a_(1)|+dots+|a_(n)|)/(n))_(n inN)\left(\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n}\right)_{n \in \mathbb{N}}(|a1|+…+|an|n)n∈N is bounded, then φ n ( f ) φ n ( f ) varphi_(n)(f)\varphi_{n}(f)φn(f) converges to a ⋅ ∫ 0 1 f a ⋅ ∫ 0 1   f a*int_(0)^(1)fa \cdot \int_{0}^{1} fa⋅∫01f, where a = lim n → ∞ a 1 + … + a n n a = lim n → ∞   a 1 + … + a n n a=lim_(n rarr oo)(a_(1)+dots+a_(n))/(n)a=\lim _{n \rightarrow \infty} \frac{a_{1}+\ldots+a_{n}}{n}a=limn→∞a1+…+ann. The converse of this statement is also true. Another result is that the supplementary condition can be dropped if the operators are considered on the space C 1 ( I , F ) C 1 ( I , F ) C^(1)(I,F)\mathcal{C}^{1}(I, F)C1(I,F).

MSC 2000. 47B38, 26E60.
Keywords. Linear operators, Cesàro-convergent sequences.

1. INTRODUCTION

Let ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N be a sequence of real numbers. It will be called Cesàro-convergent if the sequence of its Cesàro (arithmetic) means is convergent, i.e.
lim n → ∞ a 1 + … + a n n ∈ R . lim n → ∞   a 1 + … + a n n ∈ R . lim_(n rarr oo)(a_(1)+dots+a_(n))/(n)inR.\lim _{n \rightarrow \infty} \frac{a_{1}+\ldots+a_{n}}{n} \in \mathbb{R} .limn→∞a1+…+ann∈R.
For x ∈ R , ⌊ x ⌋ x ∈ R , ⌊ x ⌋ x inR,|__ x __|x \in \mathbb{R},\lfloor x\rfloorx∈R,⌊x⌋ will denote the greatest integer number n ≤ x n ≤ x n <= xn \leq xn≤x (the integer part of x ) x ) x)x)x).
Given the interval I = [ 0 , 1 ] I = [ 0 , 1 ] I=[0,1]I=[0,1]I=[0,1] and a Banach space F ≠ { 0 } F ≠ { 0 } F!={0}F \neq\{0\}F≠{0}, we denote by B ( I , F ) B ( I , F ) B(I,F)\mathcal{B}(I, F)B(I,F) the Banach space of bounded functions f : I → F f : I → F f:I rarr Ff: I \rightarrow Ff:I→F endowed with the sup norm. The space B ( I , F ) B ( I , F ) B(I,F)\mathcal{B}(I, F)B(I,F) contains as a subspace the set of "step-functions" E ( I , F ) = { f : I → F : ∃ t 0 , … , t n ∈ I , t 0 = 0 < t 1 < … < t n = 1 , ∃ u k ∈ F E ( I , F ) = f : I → F : ∃ t 0 , … , t n ∈ I , t 0 = 0 < t 1 < … < t n = 1 , ∃ u k ∈ F E(I,F)={f:I rarr F:EEt_(0),dots,t_(n)in I,t_(0)=0 < t_(1) < dots < t_(n)=1,EEu_(k)in F:}\mathcal{E}(I, F)=\left\{f: I \rightarrow F: \exists t_{0}, \ldots, t_{n} \in I, t_{0}=0<t_{1}<\ldots<t_{n}=1, \exists u_{k} \in F\right.E(I,F)={f:I→F:∃t0,…,tn∈I,t0=0<t1<…<tn=1,∃uk∈F so that f | ( t k − 1 , t k ) = u k , k = 1 , … , n } f t k − 1 , t k = u k , k = 1 , … , n {:f|_((t_(k-1),t_(k)))=u_(k),k=1,dots,n}\left.\left.f\right|_{\left(t_{k-1}, t_{k}\right)}=u_{k}, k=1, \ldots, n\right\}f|(tk−1,tk)=uk,k=1,…,n}. In fact each f ∈ E ( I , F ) f ∈ E ( I , F ) f inE(I,F)f \in \mathcal{E}(I, F)f∈E(I,F) is a finite sum of functions having the form χ [ α , β ] ⋅ u χ [ α , β ] ⋅ u chi_([alpha,beta])*u\chi_{[\alpha, \beta]} \cdot uχ[α,β]⋅u, where 0 ≤ α ≤ β ≤ 1 , u ∈ F 0 ≤ α ≤ β ≤ 1 , u ∈ F 0 <= alpha <= beta <= 1,u in F0 \leq \alpha \leq \beta \leq 1, u \in F0≤α≤β≤1,u∈F and χ [ α , β ] χ [ α , β ] chi_([alpha,beta])\chi_{[\alpha, \beta]}χ[α,β] is the characteristic function of the interval [ α , β ] [ α , β ] [alpha,beta][\alpha, \beta][α,β]. We denote by R ( I , F ) R ( I , F ) R(I,F)\mathcal{R}(I, F)R(I,F) the Banach space of regular functions (which admit side limits at each t ∈ I t ∈ I t in It \in It∈I ), endowed with the uniform norm ‖ f ‖ = sup t ∈ [ 0 , 1 ] ‖ f ( t ) ‖ ‖ f ‖ = sup t ∈ [ 0 , 1 ]   ‖ f ( t ) ‖ ||f||=s u p_(t in[0,1])||f(t)||\|f\|=\sup _{t \in[0,1]}\|f(t)\|‖f‖=supt∈[0,1]‖f(t)‖. We mention that R ( I , F ) R ( I , F ) R(I,F)\mathcal{R}(I, F)R(I,F) is the closure in B ( I , F ) B ( I , F ) B(I,F)\mathcal{B}(I, F)B(I,F) of the subspace E ( I , F ) E ( I , F ) E(I,F)\mathcal{E}(I, F)E(I,F), and it contains the
Banach space of continuous functions C ( I , F ) C ( I , F ) C(I,F)\mathcal{C}(I, F)C(I,F). More details on these spaces of functions are to be found in [3, p. 137].
We define a sequence of operators associated to ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N, namely φ n φ n varphi_(n)\varphi_{n}φn : R ( I , F ) → F , n ∈ N R ( I , F ) → F , n ∈ N R(I,F)rarr F,n inN\mathcal{R}(I, F) \rightarrow F, n \in \mathbb{N}R(I,F)→F,n∈N
(1) φ n ( f ) = 1 n ∑ k = 1 n a k f ( k n ) . (1) φ n ( f ) = 1 n ∑ k = 1 n   a k f k n . {:(1)varphi_(n)(f)=(1)/(n)sum_(k=1)^(n)a_(k)f((k)/(n)).:}\begin{equation*} \varphi_{n}(f)=\frac{1}{n} \sum_{k=1}^{n} a_{k} f\left(\frac{k}{n}\right) . \tag{1} \end{equation*}(1)φn(f)=1n∑k=1nakf(kn).
Proposition 1. The operator φ n φ n varphi_(n)\varphi_{n}φn is linear and continuous, and its norm is given by
(2) ‖ φ n ‖ = 1 n ∑ k = 1 n | a k | . (2) φ n = 1 n ∑ k = 1 n   a k . {:(2)||varphi_(n)||=(1)/(n)sum_(k=1)^(n)|a_(k)|.:}\begin{equation*} \left\|\varphi_{n}\right\|=\frac{1}{n} \sum_{k=1}^{n}\left|a_{k}\right| . \tag{2} \end{equation*}(2)‖φn‖=1n∑k=1n|ak|.
Proof. The linearity is straightforward. Because ‖ f ( k n ) ‖ ≤ ‖ f ‖ f k n ≤ ‖ f ‖ ||f((k)/(n))|| <= ||f||\left\|f\left(\frac{k}{n}\right)\right\| \leq\|f\|‖f(kn)‖≤‖f‖, it follows
(3) ‖ φ n ( f ) ‖ ≤ ( 1 n ∑ k = 1 n | a k | ) ⋅ ‖ f ‖ , (3) φ n ( f ) ≤ 1 n ∑ k = 1 n   a k ⋅ ‖ f ‖ , {:(3)||varphi_(n)(f)|| <= ((1)/(n)sum_(k=1)^(n)|a_(k)|)*||f||",":}\begin{equation*} \left\|\varphi_{n}(f)\right\| \leq\left(\frac{1}{n} \sum_{k=1}^{n}\left|a_{k}\right|\right) \cdot\|f\|, \tag{3} \end{equation*}(3)‖φn(f)‖≤(1n∑k=1n|ak|)⋅‖f‖,
hence φ n φ n varphi_(n)\varphi_{n}φn is also continuous. To obtain the norm of φ n φ n varphi_(n)\varphi_{n}φn, we use the inequality (3) and the function
f 0 ( t ) = { ( sign a k ) u , for t = k n , k = 1 , … , n 0 , otherwise f 0 ( t ) = sign a k u ,       for  t = k n , k = 1 , … , n 0 ,       otherwise  f_(0)(t)={[(signa_(k))u","," for "t=(k)/(n)","k=1","dots","n],[0","," otherwise "]:}f_{0}(t)= \begin{cases}\left(\operatorname{sign} a_{k}\right) u, & \text { for } t=\frac{k}{n}, k=1, \ldots, n \\ 0, & \text { otherwise }\end{cases}f0(t)={(signak)u, for t=kn,k=1,…,n0, otherwise 
where u ∈ F u ∈ F u in Fu \in Fu∈F and ‖ u ‖ = 1 ‖ u ‖ = 1 ||u||=1\|u\|=1‖u‖=1. We have f 0 ∈ E ( I , F ) ⊆ R ( I , F ) , ‖ f 0 ‖ = 1 f 0 ∈ E ( I , F ) ⊆ R ( I , F ) , f 0 = 1 f_(0)inE(I,F)subeR(I,F),||f_(0)||=1f_{0} \in \mathcal{E}(I, F) \subseteq \mathcal{R}(I, F),\left\|f_{0}\right\|=1f0∈E(I,F)⊆R(I,F),‖f0‖=1 and φ n ( f 0 ) = ( 1 n ∑ k = 1 n | a k | ) ⋅ u φ n f 0 = 1 n ∑ k = 1 n   a k ⋅ u varphi_(n)(f_(0))=((1)/(n)sum_(k=1)^(n)|a_(k)|)*u\varphi_{n}\left(f_{0}\right)=\left(\frac{1}{n} \sum_{k=1}^{n}\left|a_{k}\right|\right) \cdot uφn(f0)=(1n∑k=1n|ak|)⋅u, hence the equality 22 follows.

2. MAIN RESULTS

We are interested in finding conditions on the sequence ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N in order to obtain the convergence of the sequence of linear operators (1). The theorem below guarantees the convergence of ( φ n ( f ) ) n ∈ N φ n ( f ) n ∈ N (varphi_(n)(f))_(n inN)\left(\varphi_{n}(f)\right)_{n \in \mathbb{N}}(φn(f))n∈N for each regular function f ∈ F ( I , F ) f ∈ F ( I , F ) f inF(I,F)f \in \mathcal{F}(I, F)f∈F(I,F). Beside the condition of Cesàro-convergence for ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N, the boundedness of a certain sequence related to this is imposed.
Theorem 2. Let there be given a regular function f ∈ R ( I , F ) f ∈ R ( I , F ) f inR(I,F)f \in \mathcal{R}(I, F)f∈R(I,F) and a sequence ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N of real numbers satisfying the conditions:
  1. ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N is Cesàro-convergent to a ( lim n → ∞ a 1 + … + a n n = a ) lim n → ∞   a 1 + … + a n n = a (lim_(n rarr oo)(a_(1)+dots+a_(n))/(n)=a)\left(\lim _{n \rightarrow \infty} \frac{a_{1}+\ldots+a_{n}}{n}=a\right)(limn→∞a1+…+ann=a);
  2. the sequence ( | a 1 | + … + | a n | n ) n ∈ N a 1 + … + a n n n ∈ N ((|a_(1)|+dots+|a_(n)|)/(n))_(n inN)\left(\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n}\right)_{n \in \mathbb{N}}(|a1|+…+|an|n)n∈N is bounded.
Then the sequence ( φ n ( f ) ) n ∈ N φ n ( f ) n ∈ N (varphi_(n)(f))_(n inN)\left(\varphi_{n}(f)\right)_{n \in \mathbb{N}}(φn(f))n∈N is convergent and
(4) lim n → ∞ φ n ( f ) = a ⋅ ∫ 0 1 f (4) lim n → ∞   φ n ( f ) = a ⋅ ∫ 0 1   f {:(4)lim_(n rarr oo)varphi_(n)(f)=a*int_(0)^(1)f:}\begin{equation*} \lim _{n \rightarrow \infty} \varphi_{n}(f)=a \cdot \int_{0}^{1} f \tag{4} \end{equation*}(4)limn→∞φn(f)=a⋅∫01f
Proof. At first we shall prove (4) for functions f f fff of the form
(5) f = χ [ α , β ] ⋅ u , where 0 ≤ α ≤ β ≤ 1 , u ∈ F . (5) f = χ [ α , β ] ⋅ u ,  where  0 ≤ α ≤ β ≤ 1 , u ∈ F .  {:(5)f=chi_([alpha,beta])*u","" where "0 <= alpha <= beta <= 1","u in F". ":}\begin{equation*} f=\chi_{[\alpha, \beta]} \cdot u, \text { where } 0 \leq \alpha \leq \beta \leq 1, u \in F \text {. } \tag{5} \end{equation*}(5)f=χ[α,β]⋅u, where 0≤α≤β≤1,u∈F. 
We have
φ n ( f ) = ( 1 n ∑ k ∈ N α n ≤ k ≤ β n a k ) ⋅ u = ( 1 n ∑ k ∈ N k ≤ β n a k ) ⋅ u − ( 1 n ∑ k ∈ N k < α n a k ) ⋅ u . φ n ( f ) = 1 n ∑ k ∈ N α n ≤ k ≤ β n   a k ⋅ u = 1 n ∑ k ∈ N k ≤ β n   a k ⋅ u − 1 n ∑ k ∈ N k < α n   a k ⋅ u . varphi_(n)(f)=((1)/(n)sum_({:[k inN],[alpha n <= k <= beta n]:})a_(k))*u=((1)/(n)sum_({:[k inN],[k <= beta n]:})a_(k))*u-((1)/(n)sum_({:[k inN],[k < alpha n]:})a_(k))*u.\varphi_{n}(f)=\left(\frac{1}{n} \sum_{\substack{k \in \mathbb{N} \\ \alpha n \leq k \leq \beta n}} a_{k}\right) \cdot u=\left(\frac{1}{n} \sum_{\substack{k \in \mathbb{N} \\ k \leq \beta n}} a_{k}\right) \cdot u-\left(\frac{1}{n} \sum_{\substack{k \in \mathbb{N} \\ k<\alpha n}} a_{k}\right) \cdot u .φn(f)=(1n∑k∈Nαn≤k≤βnak)⋅u=(1n∑k∈Nk≤βnak)⋅u−(1n∑k∈Nk<αnak)⋅u.
If α = 0 α = 0 alpha=0\alpha=0α=0 the conclusion follows obviously.
For α > 0 α > 0 alpha > 0\alpha>0α>0 we denote a n ∗ = a 1 + … + a n n a n ∗ = a 1 + … + a n n a_(n)^(**)=(a_(1)+dots+a_(n))/(n)a_{n}^{*}=\frac{a_{1}+\ldots+a_{n}}{n}an∗=a1+…+ann and we write the two sums in the above formula as
∑ k ∈ N k ≤ β n a k = ⌊ β n ⌋ ⋅ a ⌊ β n ⌋ ∗ , ∑ k ∈ N k < α n a k = ⌊ α n ⌋ ⋅ a ⌊ α n ⌋ ∗ − a ⌊ α n ⌋ ⋅ θ n ∑ k ∈ N k ≤ β n   a k = ⌊ β n ⌋ ⋅ a ⌊ β n ⌋ ∗ , ∑ k ∈ N k < α n   a k = ⌊ α n ⌋ ⋅ a ⌊ α n ⌋ ∗ − a ⌊ α n ⌋ ⋅ θ n sum_({:[k inN],[k <= beta n]:})a_(k)=|__ beta n __|*a_(|__ beta n __|)^(**),quadsum_({:[k inN],[k < alpha n]:})a_(k)=|__ alpha n __|*a_(|__ alpha n __|)^(**)-a_(|__ alpha n __|)*theta_(n)\sum_{\substack{k \in \mathbb{N} \\ k \leq \beta n}} a_{k}=\lfloor\beta n\rfloor \cdot a_{\lfloor\beta n\rfloor}^{*}, \quad \sum_{\substack{k \in \mathbb{N} \\ k<\alpha n}} a_{k}=\lfloor\alpha n\rfloor \cdot a_{\lfloor\alpha n\rfloor}^{*}-a_{\lfloor\alpha n\rfloor} \cdot \theta_{n}∑k∈Nk≤βnak=⌊βn⌋⋅a⌊βn⌋∗,∑k∈Nk<αnak=⌊αn⌋⋅a⌊αn⌋∗−a⌊αn⌋⋅θn
where
θ n = { 1 , for α n ∈ N 0 , otherwise θ n = 1 ,  for  α n ∈ N 0 ,  otherwise  theta_(n)={[1","" for "alpha n inN],[0","" otherwise "]:}\theta_{n}=\left\{\begin{array}{l} 1, \text { for } \alpha n \in \mathbb{N} \\ 0, \text { otherwise } \end{array}\right.θn={1, for αn∈N0, otherwise 
We finally obtain
φ n ( f ) = ( ⌊ β n ⌋ n ⋅ a ⌊ β n ⌋ ∗ − ⌊ α n ⌋ n ⋅ a ⌊ α n ⌋ ∗ + a ⌊ α n ⌋ n ⋅ θ n ) ⋅ u . φ n ( f ) = ⌊ β n ⌋ n ⋅ a ⌊ β n ⌋ ∗ − ⌊ α n ⌋ n ⋅ a ⌊ α n ⌋ ∗ + a ⌊ α n ⌋ n ⋅ θ n ⋅ u . varphi_(n)(f)=((|__ beta n __|)/(n)*a_(|__ beta n __|)^(**)-(|__ alpha n __|)/(n)*a_(|__ alpha n __|)^(**)+(a_(|__ alpha n __|))/(n)*theta_(n))*u.\varphi_{n}(f)=\left(\frac{\lfloor\beta n\rfloor}{n} \cdot a_{\lfloor\beta n\rfloor}^{*}-\frac{\lfloor\alpha n\rfloor}{n} \cdot a_{\lfloor\alpha n\rfloor}^{*}+\frac{a_{\lfloor\alpha n\rfloor}}{n} \cdot \theta_{n}\right) \cdot u .φn(f)=(⌊βn⌋n⋅a⌊βn⌋∗−⌊αn⌋n⋅a⌊αn⌋∗+a⌊αn⌋n⋅θn)⋅u.
We have lim n → ∞ a ⌊ α n ⌋ ∗ = a lim n → ∞   a ⌊ α n ⌋ ∗ = a lim_(n rarr oo)a_(|__ alpha n __|)^(**)=a\lim _{n \rightarrow \infty} a_{\lfloor\alpha n\rfloor}^{*}=alimn→∞a⌊αn⌋∗=a; but a n n = a n ∗ − ( 1 − 1 n ) a n − 1 ∗ a n n = a n ∗ − 1 − 1 n a n − 1 ∗ (a_(n))/(n)=a_(n)^(**)-(1-(1)/(n))a_(n-1)^(**)\frac{a_{n}}{n}=a_{n}^{*}-\left(1-\frac{1}{n}\right) a_{n-1}^{*}ann=an∗−(1−1n)an−1∗, hence lim n → ∞ a n n = 0 lim n → ∞   a n n = 0 lim_(n rarr oo)(a_(n))/(n)=0\lim _{n \rightarrow \infty} \frac{a_{n}}{n}=0limn→∞ann=0. It follows that in this case
lim n → ∞ φ n ( f ) = ( β a − α a ) ⋅ u = a ⋅ ∫ 0 1 f lim n → ∞   φ n ( f ) = ( β a − α a ) ⋅ u = a ⋅ ∫ 0 1   f lim_(n rarr oo)varphi_(n)(f)=(beta a-alpha a)*u=a*int_(0)^(1)f\lim _{n \rightarrow \infty} \varphi_{n}(f)=(\beta a-\alpha a) \cdot u=a \cdot \int_{0}^{1} flimn→∞φn(f)=(βa−αa)⋅u=a⋅∫01f
We consider now the general case f ∈ R ( I , F ) f ∈ R ( I , F ) f inR(I,F)f \in \mathcal{R}(I, F)f∈R(I,F). The sequence ( | a 1 | + … + | a n | n ) n ∈ N a 1 + … + a n n n ∈ N ((|a_(1)|+dots+|a_(n)|)/(n))_(n inN)\left(\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n}\right)_{n \in \mathbb{N}}(|a1|+…+|an|n)n∈N being bounded, let us choose M M MMM such that | a 1 | + … + | a n | n ≤ M a 1 + … + a n n ≤ M (|a_(1)|+dots+|a_(n)|)/(n) <= M\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n} \leq M|a1|+…+|an|n≤M for each n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N; let also ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 be an arbitrary constant. From the definition of the space R ( I , F ) R ( I , F ) R(I,F)\mathcal{R}(I, F)R(I,F) it follows the existence of the functions f i , i = 1 , … , p f i , i = 1 , … , p f_(i),i=1,dots,pf_{i}, i=1, \ldots, pfi,i=1,…,p of the type described in (5), with ‖ f − ∑ i = 1 p f i ‖ < ε f − ∑ i = 1 p   f i < ε ||f-sum_(i=1)^(p)f_(i)|| < epsi\left\|f-\sum_{i=1}^{p} f_{i}\right\|<\varepsilon‖f−∑i=1pfi‖<ε. We have
φ n ( f ) − a ∫ 0 1 f = φ n ( f − ∑ i = 1 p f i ) + ∑ i = 1 p ( φ n ( f i ) − a ∫ 0 1 f i ) − a ∫ 0 1 ( f − ∑ i = 1 p f i ) . φ n ( f ) − a ∫ 0 1   f = φ n f − ∑ i = 1 p   f i + ∑ i = 1 p   φ n f i − a ∫ 0 1   f i − a ∫ 0 1   f − ∑ i = 1 p   f i . varphi_(n)(f)-aint_(0)^(1)f=varphi_(n)(f-sum_(i=1)^(p)f_(i))+sum_(i=1)^(p)(varphi_(n)(f_(i))-aint_(0)^(1)f_(i))-aint_(0)^(1)(f-sum_(i=1)^(p)f_(i)).\varphi_{n}(f)-a \int_{0}^{1} f=\varphi_{n}\left(f-\sum_{i=1}^{p} f_{i}\right)+\sum_{i=1}^{p}\left(\varphi_{n}\left(f_{i}\right)-a \int_{0}^{1} f_{i}\right)-a \int_{0}^{1}\left(f-\sum_{i=1}^{p} f_{i}\right) .φn(f)−a∫01f=φn(f−∑i=1pfi)+∑i=1p(φn(fi)−a∫01fi)−a∫01(f−∑i=1pfi).
The norm of φ n φ n varphi_(n)\varphi_{n}φn, as given by (2), is ‖ φ n ‖ = | a 1 | + … + | a n | n φ n = a 1 + … + a n n ||varphi_(n)||=(|a_(1)|+dots+|a_(n)|)/(n)\left\|\varphi_{n}\right\|=\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n}‖φn‖=|a1|+…+|an|n, hence
(6) ‖ φ n ( f − ∑ i = 1 p f i ) ‖ ≤ ‖ φ n ‖ ⋅ ‖ f − ∑ i = 1 p f i ‖ ≤ M ⋅ ε (6) φ n f − ∑ i = 1 p   f i ≤ φ n ⋅ f − ∑ i = 1 p   f i ≤ M ⋅ ε {:(6)||varphi_(n)(f-sum_(i=1)^(p)f_(i))|| <= ||varphi_(n)||*||f-sum_(i=1)^(p)f_(i)|| <= M*epsi:}\begin{equation*} \left\|\varphi_{n}\left(f-\sum_{i=1}^{p} f_{i}\right)\right\| \leq\left\|\varphi_{n}\right\| \cdot\left\|f-\sum_{i=1}^{p} f_{i}\right\| \leq M \cdot \varepsilon \tag{6} \end{equation*}(6)‖φn(f−∑i=1pfi)‖≤‖φn‖⋅‖f−∑i=1pfi‖≤M⋅ε
Taking into account the first part of the proof, for each i = 1 , … , p i = 1 , … , p i=1,dots,pi=1, \ldots, pi=1,…,p there exists n i ∈ N n i ∈ N n_(i)inNn_{i} \in \mathbb{N}ni∈N so that ‖ φ n ( f i ) − a ⋅ ∫ 0 1 f i ‖ < ε p φ n f i − a ⋅ ∫ 0 1   f i < ε p ||varphi_(n)(f_(i))-a*int_(0)^(1)f_(i)|| < (epsi )/(p)\left\|\varphi_{n}\left(f_{i}\right)-a \cdot \int_{0}^{1} f_{i}\right\|<\frac{\varepsilon}{p}‖φn(fi)−a⋅∫01fi‖<εp for n ≥ n i n ≥ n i n >= n_(i)n \geq n_{i}n≥ni. It follows that for
n ≥ max i = 1 , … , p n i n ≥ max i = 1 , … , p   n i n >= max_(i=1,dots,p)n_(i)n \geq \max _{i=1, \ldots, p} n_{i}n≥maxi=1,…,pni we have
(7) ‖ ∑ i = 1 p ( φ n ( f i ) − a ⋅ ∫ 0 1 f i ) ‖ ≤ ε . (7) ∑ i = 1 p   φ n f i − a ⋅ ∫ 0 1   f i ≤ ε . {:(7)||sum_(i=1)^(p)(varphi_(n)(f_(i))-a*int_(0)^(1)f_(i))|| <= epsi.:}\begin{equation*} \left\|\sum_{i=1}^{p}\left(\varphi_{n}\left(f_{i}\right)-a \cdot \int_{0}^{1} f_{i}\right)\right\| \leq \varepsilon . \tag{7} \end{equation*}(7)‖∑i=1p(φn(fi)−a⋅∫01fi)‖≤ε.
But
(8) ‖ a ∫ 0 1 ( f − ∑ i = 1 p f i ) ‖ ≤ | a | ⋅ ε , (8) a ∫ 0 1   f − ∑ i = 1 p   f i ≤ | a | ⋅ ε , {:(8)||aint_(0)^(1)(f-sum_(i=1)^(p)f_(i))|| <= |a|*epsi",":}\begin{equation*} \left\|a \int_{0}^{1}\left(f-\sum_{i=1}^{p} f_{i}\right)\right\| \leq|a| \cdot \varepsilon, \tag{8} \end{equation*}(8)‖a∫01(f−∑i=1pfi)‖≤|a|⋅ε,
and the inequalities (6), (7) and (8) imply that
‖ φ n ( f ) − a ⋅ ∫ 0 1 f ‖ ≤ M ⋅ ε + ε + | a | ⋅ ε , for n ≥ N . φ n ( f ) − a ⋅ ∫ 0 1   f ≤ M ⋅ ε + ε + | a | ⋅ ε ,  for  n ≥ N . ||varphi_(n)(f)-a*int_(0)^(1)f|| <= M*epsi+epsi+|a|*epsi," for "n >= N.\left\|\varphi_{n}(f)-a \cdot \int_{0}^{1} f\right\| \leq M \cdot \varepsilon+\varepsilon+|a| \cdot \varepsilon, \text { for } n \geq \mathbb{N} .‖φn(f)−a⋅∫01f‖≤M⋅ε+ε+|a|⋅ε, for n≥N.
It follows that the conclusion holds also for the general case f ∈ F ( I , F ) f ∈ F ( I , F ) f inF(I,F)f \in \mathcal{F}(I, F)f∈F(I,F).
Remark 1. The Cesàro-convergence of ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N in Theorem 2 does not necessarily imply the boundedness of ( | a 1 | + … + | a n | n ) n ∈ N a 1 + … + a n n n ∈ N ((|a_(1)|+dots+|a_(n)|)/(n))_(n inN)\left(\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n}\right)_{n \in \mathbb{N}}(|a1|+…+|an|n)n∈N. For example, let the sequence be given by
a n = { n , n odd − n − 1 , n even a n = n ,      n  odd  − n − 1 ,      n  even  a_(n)={[sqrtn",",n" odd "],[-sqrt(n-1)",",n" even "]:}a_{n}= \begin{cases}\sqrt{n}, & n \text { odd } \\ -\sqrt{n-1}, & n \text { even }\end{cases}an={n,n odd −n−1,n even 
Then
a n ∗ = { 1 / n , n odd 0 , n even , a n ∗ = 1 / n ,      n  odd  0 ,      n  even  , a_(n)^(**)={[1//sqrtn",",n" odd "],[0",",n" even "","]:}a_{n}^{*}= \begin{cases}1 / \sqrt{n}, & n \text { odd } \\ 0, & n \text { even },\end{cases}an∗={1/n,n odd 0,n even ,
hence lim n → ∞ a n ∗ = 0 lim n → ∞   a n ∗ = 0 lim_(n rarr oo)a_(n)^(**)=0\lim _{n \rightarrow \infty} a_{n}^{*}=0limn→∞an∗=0, but lim n → ∞ | a 1 | + … + | a n | n = lim n → ∞ | a n | = ∞ lim n → ∞   a 1 + … + a n n = lim n → ∞   a n = ∞ lim_(n rarr oo)(|a_(1)|+dots+|a_(n)|)/(n)=lim_(n rarr oo)|a_(n)|=oo\lim _{n \rightarrow \infty} \frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n}=\lim _{n \rightarrow \infty}\left|a_{n}\right|=\inftylimn→∞|a1|+…+|an|n=limn→∞|an|=∞.
The condition of Cesàro-convergence imposed to the sequence ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N in Theorem 22 is a natural one and cannot be relaxed, neither the boundedness of the sequence ( | a 1 | + … + | a n | n ) n ∈ N a 1 + … + a n n n ∈ N ((|a_(1)|+dots+|a_(n)|)/(n))_(n inN)\left(\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n}\right)_{n \in \mathbb{N}}(|a1|+…+|an|n)n∈N. In fact, Theorem 2 does admit the following converse:
Theorem 3. Let ( φ n ) n ∈ N φ n n ∈ N (varphi_(n))_(n inN)\left(\varphi_{n}\right)_{n \in \mathbb{N}}(φn)n∈N be the sequence (1) of linear operators associated to the sequence of real numbers ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N. If lim n → ∞ φ n ( f ) lim n → ∞   φ n ( f ) lim_(n rarr oo)varphi_(n)(f)\lim _{n \rightarrow \infty} \varphi_{n}(f)limn→∞φn(f) exists for every f ∈ C ( I , F ) ⊆ R ( I , F ) f ∈ C ( I , F ) ⊆ R ( I , F ) f inC(I,F)subeR(I,F)f \in \mathcal{C}(I, F) \subseteq \mathcal{R}(I, F)f∈C(I,F)⊆R(I,F), then:
  1. ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N is Cesàro-convergent to a ( lim n → ∞ a 1 + … + a n n = a ) lim n → ∞   a 1 + … + a n n = a (lim_(n rarr oo)(a_(1)+dots+a_(n))/(n)=a)\left(\lim _{n \rightarrow \infty} \frac{a_{1}+\ldots+a_{n}}{n}=a\right)(limn→∞a1+…+ann=a);
  2. the sequence ( | a 1 | + … + | a n | n ) n ∈ N a 1 + … + a n n n ∈ N ((|a_(1)|+dots+|a_(n)|)/(n))_(n inN)\left(\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n}\right)_{n \in \mathbb{N}}(|a1|+…+|an|n)n∈N is bounded.
Proof. The first conclusion follows by taking f ( t ) = u f ( t ) = u f(t)=uf(t)=uf(t)=u for each t ∈ I t ∈ I t in It \in It∈I, with u ∈ F ∖ { 0 } u ∈ F ∖ { 0 } u in F\\{0}u \in F \backslash\{0\}u∈F∖{0}. In this case φ n ( f ) = a 1 + … + a n n u φ n ( f ) = a 1 + … + a n n u varphi_(n)(f)=(a_(1)+dots+a_(n))/(n)u\varphi_{n}(f)=\frac{a_{1}+\ldots+a_{n}}{n} uφn(f)=a1+…+annu.
The norm of the operators φ n φ n varphi_(n)\varphi_{n}φn in the space C ( I , F ) C ( I , F ) C(I,F)\mathcal{C}(I, F)C(I,F) is the same as in (1). Indeed, in the proof of Proposition 1, the function f 0 f 0 f_(0)f_{0}f0 can be modified to a continuous and piecewise affine one which takes also the values ( sign a k ) u sign a k u (signa_(k))u\left(\operatorname{sign} a_{k}\right) u(signak)u on the points k n , k = 1 , … , n k n , k = 1 , … , n (k)/(n),k=1,dots,n\frac{k}{n}, k=1, \ldots, nkn,k=1,…,n. From the principle of uniform boundedness [4, p. 66] the second conclusion follows.
Remark 2. Using a principle of condensation of singularities [2, one can prove that the convergence in (4) does not hold for "typical" continuous functions. Even stronger principles of condensation of singularities [1] may be applied.
In what follows we shall prove that for the class of continuous functions having also a continuous derivative, the condition of boundedness of the sequence ( | a 1 | + … + | a n | n ) n ∈ N a 1 + … + a n n n ∈ N ((|a_(1)|+dots+|a_(n)|)/(n))_(n inN)\left(\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n}\right)_{n \in \mathbb{N}}(|a1|+…+|an|n)n∈N is no longer necessary. In this setting, the principle of uniform boundedness does not work, because C 1 ( I , F ) C 1 ( I , F ) C^(1)(I,F)\mathcal{C}^{1}(I, F)C1(I,F) endowed with the uniform norm is not a Banach space. The norm of φ n φ n varphi_(n)\varphi_{n}φn is still the same. In this case we have
Theorem 4. Let there be given a function f ∈ C 1 ( I , F ) f ∈ C 1 ( I , F ) f inC^(1)(I,F)f \in \mathcal{C}^{1}(I, F)f∈C1(I,F) and a sequence ( a n ) n ∈ N a n n ∈ N (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}(an)n∈N of real numbers which is Cesàro-convergent to a ( lim n → ∞ a 1 + … + a n n = a lim n → ∞   a 1 + … + a n n = a lim_(n rarr oo)(a_(1)+dots+a_(n))/(n)=a\lim _{n \rightarrow \infty} \frac{a_{1}+\ldots+a_{n}}{n}=alimn→∞a1+…+ann=a ). Then
(9) lim n → ∞ φ n ( f ) = a ⋅ ∫ 0 1 f (9) lim n → ∞   φ n ( f ) = a ⋅ ∫ 0 1   f {:(9)lim_(n rarr oo)varphi_(n)(f)=a*int_(0)^(1)f:}\begin{equation*} \lim _{n \rightarrow \infty} \varphi_{n}(f)=a \cdot \int_{0}^{1} f \tag{9} \end{equation*}(9)limn→∞φn(f)=a⋅∫01f
Proof. We write φ n ( f ) φ n ( f ) varphi_(n)(f)\varphi_{n}(f)φn(f) successively as
φ n ( f ) = 1 n ∑ k = 1 n ( k a k ∗ − ( k − 1 ) a k − 1 ∗ ) f ( k n ) = 1 n ∑ k = 1 n k a k ∗ f ( k n ) − 1 n ∑ k = 1 n − 1 k a k ∗ f ( k + 1 n ) = ∑ k = 1 n − 1 a k ∗ k n ( f ( k n ) − f ( k + 1 n ) ) + a n ∗ f ( 1 ) . φ n ( f ) = 1 n ∑ k = 1 n   k a k ∗ − ( k − 1 ) a k − 1 ∗ f k n = 1 n ∑ k = 1 n   k a k ∗ f k n − 1 n ∑ k = 1 n − 1   k a k ∗ f k + 1 n = ∑ k = 1 n − 1   a k ∗ k n f k n − f k + 1 n + a n ∗ f ( 1 ) . {:[varphi_(n)(f)=(1)/(n)sum_(k=1)^(n)(ka_(k)^(**)-(k-1)a_(k-1)^(**))f((k)/(n))],[=(1)/(n)sum_(k=1)^(n)ka_(k)^(**)f((k)/(n))-(1)/(n)sum_(k=1)^(n-1)ka_(k)^(**)f((k+1)/(n))],[=sum_(k=1)^(n-1)a_(k)^(**)(k)/(n)(f((k)/(n))-f((k+1)/(n)))+a_(n)^(**)f(1).]:}\begin{aligned} \varphi_{n}(f) & =\frac{1}{n} \sum_{k=1}^{n}\left(k a_{k}^{*}-(k-1) a_{k-1}^{*}\right) f\left(\frac{k}{n}\right) \\ & =\frac{1}{n} \sum_{k=1}^{n} k a_{k}^{*} f\left(\frac{k}{n}\right)-\frac{1}{n} \sum_{k=1}^{n-1} k a_{k}^{*} f\left(\frac{k+1}{n}\right) \\ & =\sum_{k=1}^{n-1} a_{k}^{*} \frac{k}{n}\left(f\left(\frac{k}{n}\right)-f\left(\frac{k+1}{n}\right)\right)+a_{n}^{*} f(1) . \end{aligned}φn(f)=1n∑k=1n(kak∗−(k−1)ak−1∗)f(kn)=1n∑k=1nkak∗f(kn)−1n∑k=1n−1kak∗f(k+1n)=∑k=1n−1ak∗kn(f(kn)−f(k+1n))+an∗f(1).
We bring now into the scene the continuous function g g ggg given by g ( t ) = t f ′ ( t ) g ( t ) = t f ′ ( t ) g(t)=tf^(')(t)g(t)=t f^{\prime}(t)g(t)=tf′(t) and express φ n ( f ) φ n ( f ) varphi_(n)(f)\varphi_{n}(f)φn(f) in the form
(10) φ n ( f ) = − ∑ k = 1 n − 1 a k ∗ k n ( f ( k + 1 n ) − f ( k n ) − 1 n f ′ ( k n ) ) − 1 n ∑ k = 1 n − 1 a k ∗ k n f ′ ( k n ) + a n ∗ f ( 1 ) = − ∑ k = 1 n − 1 a k ∗ k n ( f ( k + 1 n ) − f ( k n ) − 1 n f ′ ( k n ) ) − 1 n ∑ k = 1 n a k ∗ g ( k n ) + 1 n a n ∗ f ′ ( 1 ) + a n ∗ f ( 1 ) . (10) φ n ( f ) = − ∑ k = 1 n − 1   a k ∗ k n f k + 1 n − f k n − 1 n f ′ k n − 1 n ∑ k = 1 n − 1   a k ∗ k n f ′ k n + a n ∗ f ( 1 ) = − ∑ k = 1 n − 1   a k ∗ k n f k + 1 n − f k n − 1 n f ′ k n − 1 n ∑ k = 1 n   a k ∗ g k n + 1 n a n ∗ f ′ ( 1 ) + a n ∗ f ( 1 ) . {:[(10)varphi_(n)(f)=-sum_(k=1)^(n-1)a_(k)^(**)(k)/(n)(f((k+1)/(n))-f((k)/(n))-(1)/(n)f^(')((k)/(n)))-(1)/(n)sum_(k=1)^(n-1)a_(k)^(**)(k)/(n)f^(')((k)/(n))+a_(n)^(**)f(1)],[=-sum_(k=1)^(n-1)a_(k)^(**)(k)/(n)(f((k+1)/(n))-f((k)/(n))-(1)/(n)f^(')((k)/(n)))-(1)/(n)sum_(k=1)^(n)a_(k)^(**)g((k)/(n))+(1)/(n)a_(n)^(**)f^(')(1)+a_(n)^(**)f(1).]:}\begin{gather*} \varphi_{n}(f)=-\sum_{k=1}^{n-1} a_{k}^{*} \frac{k}{n}\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)-\frac{1}{n} f^{\prime}\left(\frac{k}{n}\right)\right)-\frac{1}{n} \sum_{k=1}^{n-1} a_{k}^{*} \frac{k}{n} f^{\prime}\left(\frac{k}{n}\right)+a_{n}^{*} f(1) \tag{10}\\ =-\sum_{k=1}^{n-1} a_{k}^{*} \frac{k}{n}\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)-\frac{1}{n} f^{\prime}\left(\frac{k}{n}\right)\right)-\frac{1}{n} \sum_{k=1}^{n} a_{k}^{*} g\left(\frac{k}{n}\right)+\frac{1}{n} a_{n}^{*} f^{\prime}(1)+a_{n}^{*} f(1) . \end{gather*}(10)φn(f)=−∑k=1n−1ak∗kn(f(k+1n)−f(kn)−1nf′(kn))−1n∑k=1n−1ak∗knf′(kn)+an∗f(1)=−∑k=1n−1ak∗kn(f(k+1n)−f(kn)−1nf′(kn))−1n∑k=1nak∗g(kn)+1nan∗f′(1)+an∗f(1).
Applying Theorem 2 for the function g g ggg and for the sequence ( a n ∗ ) n ∈ N a n ∗ n ∈ N (a_(n)^(**))_(n inN)\left(a_{n}^{*}\right)_{n \in \mathbb{N}}(an∗)n∈N convergent to a a aaa, for which obviously lim n → ∞ a 1 ∗ + … + a n ∗ n = a lim n → ∞   a 1 ∗ + … + a n ∗ n = a lim_(n rarr oo)(a_(1)^(**)+dots+a_(n)^(**))/(n)=a\lim _{n \rightarrow \infty} \frac{a_{1}^{*}+\ldots+a_{n}^{*}}{n}=alimn→∞a1∗+…+an∗n=a and ( | a 1 ∗ | + … + | a n ∗ | n ) n ∈ N a 1 ∗ + … + a n ∗ n n ∈ N ((|a_(1)^(**)|+dots+|a_(n)^(**)|)/(n))_(n inN)\left(\frac{\left|a_{1}^{*}\right|+\ldots+\left|a_{n}^{*}\right|}{n}\right)_{n \in \mathbb{N}}(|a1∗|+…+|an∗|n)n∈N is bounded (because of the convergence of ( a n ∗ ) n ∈ N a n ∗ n ∈ N (a_(n)^(**))_(n inN)\left(a_{n}^{*}\right)_{n \in \mathbb{N}}(an∗)n∈N ) we get
lim n → ∞ 1 n ∑ k = 1 n a k ∗ g ( k n ) = a ⋅ ∫ 0 1 g = a ⋅ f ( 1 ) − a ⋅ ∫ 0 1 f lim n → ∞   1 n ∑ k = 1 n   a k ∗ g k n = a ⋅ ∫ 0 1   g = a ⋅ f ( 1 ) − a ⋅ ∫ 0 1   f lim_(n rarr oo)(1)/(n)sum_(k=1)^(n)a_(k)^(**)g((k)/(n))=a*int_(0)^(1)g=a*f(1)-a*int_(0)^(1)f\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} a_{k}^{*} g\left(\frac{k}{n}\right)=a \cdot \int_{0}^{1} g=a \cdot f(1)-a \cdot \int_{0}^{1} flimn→∞1n∑k=1nak∗g(kn)=a⋅∫01g=a⋅f(1)−a⋅∫01f
(the last equality is a consequence of an integration by parts). The function f ′ f ′ f^(')f^{\prime}f′ being uniformly continuous on I I III, given ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0 and n n nnn sufficiently large, we
obtain as a consequence of a mean theorem [3, p. 154]
‖ f ( k + 1 n ) − f ( k n ) − 1 n f ′ ( k n ) ‖ ≤ 1 n sup t ∈ ( k n , k + 1 n ) ‖ f ′ ( t ) − f ′ ( k n ) ‖ < ε n , f k + 1 n − f k n − 1 n f ′ k n ≤ 1 n sup t ∈ k n , k + 1 n   f ′ ( t ) − f ′ k n < ε n , ||f((k+1)/(n))-f((k)/(n))-(1)/(n)f^(')((k)/(n))|| <= (1)/(n)s u p_(t in((k)/(n),(k+1)/(n)))||f^(')(t)-f^(')((k)/(n))|| < (epsi )/(n),\left\|f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)-\frac{1}{n} f^{\prime}\left(\frac{k}{n}\right)\right\| \leq \frac{1}{n} \sup _{t \in\left(\frac{k}{n}, \frac{k+1}{n}\right)}\left\|f^{\prime}(t)-f^{\prime}\left(\frac{k}{n}\right)\right\|<\frac{\varepsilon}{n},‖f(k+1n)−f(kn)−1nf′(kn)‖≤1nsupt∈(kn,k+1n)‖f′(t)−f′(kn)‖<εn,
hence
‖ ∑ k = 1 n − 1 a k ∗ k n ( f ( k + 1 n ) − f ( k n ) − 1 n f ′ ( k n ) ) ‖ ≤ ∑ k = 1 n − 1 M ε n 2 k = n − 1 2 n M ε ≤ M ε , ∑ k = 1 n − 1   a k ∗ k n f k + 1 n − f k n − 1 n f ′ k n ≤ ∑ k = 1 n − 1   M ε n 2 k = n − 1 2 n M ε ≤ M ε , ||sum_(k=1)^(n-1)a_(k)^(**)(k)/(n)(f((k+1)/(n))-f((k)/(n))-(1)/(n)f^(')((k)/(n)))|| <= sum_(k=1)^(n-1)(M epsi)/(n^(2))k=(n-1)/(2n)M epsi <= M epsi,\left\|\sum_{k=1}^{n-1} a_{k}^{*} \frac{k}{n}\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)-\frac{1}{n} f^{\prime}\left(\frac{k}{n}\right)\right)\right\| \leq \sum_{k=1}^{n-1} \frac{M \varepsilon}{n^{2}} k=\frac{n-1}{2 n} M \varepsilon \leq M \varepsilon,‖∑k=1n−1ak∗kn(f(k+1n)−f(kn)−1nf′(kn))‖≤∑k=1n−1Mεn2k=n−12nMε≤Mε,
where M M MMM is a upper bound for the convergent sequence ( | a n ∗ | ) n ∈ N a n ∗ n ∈ N (|a_(n)^(**)|)_(n inN)\left(\left|a_{n}^{*}\right|\right)_{n \in \mathbb{N}}(|an∗|)n∈N. It follows that
lim n → ∞ ∑ k = 1 n − 1 a k ∗ k n ( f ( k + 1 n ) − f ( k n ) − 1 n f ′ ( k n ) ) = 0 lim n → ∞   ∑ k = 1 n − 1   a k ∗ k n f k + 1 n − f k n − 1 n f ′ k n = 0 lim_(n rarr oo)sum_(k=1)^(n-1)a_(k)^(**)(k)/(n)(f((k+1)/(n))-f((k)/(n))-(1)/(n)f^(')((k)/(n)))=0\lim _{n \rightarrow \infty} \sum_{k=1}^{n-1} a_{k}^{*} \frac{k}{n}\left(f\left(\frac{k+1}{n}\right)-f\left(\frac{k}{n}\right)-\frac{1}{n} f^{\prime}\left(\frac{k}{n}\right)\right)=0limn→∞∑k=1n−1ak∗kn(f(k+1n)−f(kn)−1nf′(kn))=0
We take the limit in (10) and get the conclusion.
As an application of Theorem 2 we obtain a somehow surprising result, proved directly for differentiable functions with bounded derivative in [5]: For each a ∈ [ 0 , 1 ] a ∈ [ 0 , 1 ] a in[0,1]a \in[0,1]a∈[0,1], there exist ε n ∈ { 0 , 1 } ε n ∈ { 0 , 1 } epsi_(n)in{0,1}\varepsilon_{n} \in\{0,1\}εn∈{0,1} such that
lim n → ∞ 1 n ∑ k = 1 n ε k f ( k n ) = a ⋅ ∫ 0 1 f , ∀ f ∈ R ( I , F ) lim n → ∞   1 n ∑ k = 1 n   ε k f k n = a ⋅ ∫ 0 1   f , ∀ f ∈ R ( I , F ) lim_(n rarr oo)(1)/(n)sum_(k=1)^(n)epsi_(k)f((k)/(n))=a*int_(0)^(1)f,quad AA f inR(I,F)\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n} \varepsilon_{k} f\left(\frac{k}{n}\right)=a \cdot \int_{0}^{1} f, \quad \forall f \in \mathcal{R}(I, F)limn→∞1n∑k=1nεkf(kn)=a⋅∫01f,∀f∈R(I,F)
To prove this equality, we choose ε n = a n = ⌊ ( n + 1 ) a ⌋ − ⌊ n a ⌋ , n ∈ N ε n = a n = ⌊ ( n + 1 ) a ⌋ − ⌊ n a ⌋ , n ∈ N epsi_(n)=a_(n)=|__(n+1)a __|-|__ na __|,n inN\varepsilon_{n}=a_{n}=\lfloor(n+1) a\rfloor-\lfloor n a\rfloor, n \in \mathbb{N}εn=an=⌊(n+1)a⌋−⌊na⌋,n∈N which satisfy ε n ∈ { 0 , 1 } ε n ∈ { 0 , 1 } epsi_(n)in{0,1}\varepsilon_{n} \in\{0,1\}εn∈{0,1} and lim n → ∞ a 1 + … + a n n = a lim n → ∞   a 1 + … + a n n = a lim_(n rarr oo)(a_(1)+dots+a_(n))/(n)=a\lim _{n \rightarrow \infty} \frac{a_{1}+\ldots+a_{n}}{n}=alimn→∞a1+…+ann=a.
Open question. It would be interesting to find out if the conclusion of Theorem 2 also holds for a class of functions more general than the regular ones as, for example, the Riemann integrable real-valued functions. For the class of Lebesgue integrable functions the result does not hold, as the function of Dirichlet type f : I → F = R f : I → F = R f:I rarr F=Rf: I \rightarrow F=\mathbb{R}f:I→F=R,
f ( t ) = { arbitrary, t ∈ [ 0 , 1 ] ∩ Q 0 , otherwise f ( t ) =  arbitrary,       t ∈ [ 0 , 1 ] ∩ Q 0 ,       otherwise  f(t)={[" arbitrary, ",t in[0","1]nnQ],[0","," otherwise "]:}f(t)= \begin{cases}\text { arbitrary, } & t \in[0,1] \cap \mathbb{Q} \\ 0, & \text { otherwise }\end{cases}f(t)={ arbitrary, t∈[0,1]∩Q0, otherwise 
shows.

REFERENCES

[1] Anisiu, V., A principle of double condensation of singularities using σ σ sigma\sigmaσ-porosity, "BabeşBolyai" Univ., Fac. of Math., Research Seminaries, Seminar on Math. Analysis, Preprint Nr. 7, pp. 85-88, 1985.
[2] Cobzaş, S. and Muntean, I., Condensation of singularities and divergence results in approximation theory, J. Approx. Theory, 31, pp. 148-153, 1981.
[3] Dieudonné, J., Fondements de l'analyse moderne, Paris, Gauthier-Villars, 1963.
[4] Dunford, N. and Schwartz, J. T., Linear Operators. Part 1: General Theory, John Wiley & Sons, New York, 1988.
[5] Trif, T., On a problem from the Mathematical Contest, County Stage, Gazeta Matematică CVI (11), pp. 394-396, 2001 (in Romanian).
  1. *"T. Popoviciu" Institute of Numerical Analysis, P.O. Box 68-1, 3400 Cluj-Napoca, Romania (mira@ictp.acad.ro).
    †"Babeş-Bolyai" University, Faculty of Mathematics and Computer Science, st. Kogălniceanu 1, 3400 Cluj-Napoca, Romania, e-mail: anisiu@math.ubbcluj.ro.
2002

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