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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[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
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
ictp.acad.ro/jnaat
SEQUENCES OF LINEAR OPERATORS
RELATED TO CESÀRO - CONVERGENT SEQUENCES
Abstract
Given a Cesàro-convergent sequence of real numbers (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}, a sequence (varphi_(n))_(n inN)\left(\varphi_{n}\right)_{n \in \mathbb{N}} of operators is defined on the Banach space R(I,F)\mathcal{R}(I, F) of regular functions defined on I=[0,1]I=[0,1] and having values in a Banach space FF, 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) .
It is proved that if, in addition, the sequence ((|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}} is bounded, then varphi_(n)(f)\varphi_{n}(f) converges to a*int_(0)^(1)fa \cdot \int_{0}^{1} f, where a=lim_(n rarr oo)(a_(1)+dots+a_(n))/(n)a=\lim _{n \rightarrow \infty} \frac{a_{1}+\ldots+a_{n}}{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)\mathcal{C}^{1}(I, F).
MSC 2000. 47B38, 26E60.
Keywords. Linear operators, Cesàro-convergent sequences.
Keywords. Linear operators, Cesàro-convergent sequences.
1. INTRODUCTION
Let (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{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 rarr oo)(a_(1)+dots+a_(n))/(n)inR.\lim _{n \rightarrow \infty} \frac{a_{1}+\ldots+a_{n}}{n} \in \mathbb{R} .
For x inR,|__ x __|x \in \mathbb{R},\lfloor x\rfloor will denote the greatest integer number n <= xn \leq x (the integer part of x)x).
Given the interval I=[0,1]I=[0,1] and a Banach space F!={0}F \neq\{0\}, we denote by B(I,F)\mathcal{B}(I, F) the Banach space of bounded functions f:I rarr Ff: I \rightarrow F endowed with the sup norm. The space B(I,F)\mathcal{B}(I, F) contains as a subspace the set of "step-functions" 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. so that {: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\}. In fact each f inE(I,F)f \in \mathcal{E}(I, F) is a finite sum of functions having the form chi_([alpha,beta])*u\chi_{[\alpha, \beta]} \cdot u, where 0 <= alpha <= beta <= 1,u in F0 \leq \alpha \leq \beta \leq 1, u \in F and chi_([alpha,beta])\chi_{[\alpha, \beta]} is the characteristic function of the interval [alpha,beta][\alpha, \beta]. We denote by R(I,F)\mathcal{R}(I, F) the Banach space of regular functions (which admit side limits at each t in It \in I ), endowed with the uniform norm ||f||=s u p_(t in[0,1])||f(t)||\|f\|=\sup _{t \in[0,1]}\|f(t)\|. We mention that R(I,F)\mathcal{R}(I, F) is the closure in B(I,F)\mathcal{B}(I, F) of the subspace E(I,F)\mathcal{E}(I, F), and it contains the
Banach space of continuous functions C(I,F)\mathcal{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 inN)\left(a_{n}\right)_{n \in \mathbb{N}}, namely varphi_(n)\varphi_{n} : R(I,F)rarr F,n inN\mathcal{R}(I, F) \rightarrow F, n \in \mathbb{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*}
Proposition 1. The operator varphi_(n)\varphi_{n} is linear and continuous, and its norm is given by
{:(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*}
Proof. The linearity is straightforward. Because ||f((k)/(n))|| <= ||f||\left\|f\left(\frac{k}{n}\right)\right\| \leq\|f\|, it follows
{:(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*}
hence varphi_(n)\varphi_{n} is also continuous. To obtain the norm of varphi_(n)\varphi_{n}, we use the inequality (3) and the function
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}
where u in Fu \in F and ||u||=1\|u\|=1. We have 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\|=1 and 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 u, hence the equality 22 follows.
2. MAIN RESULTS
We are interested in finding conditions on the sequence (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}} in order to obtain the convergence of the sequence of linear operators (1). The theorem below guarantees the convergence of (varphi_(n)(f))_(n inN)\left(\varphi_{n}(f)\right)_{n \in \mathbb{N}} for each regular function f inF(I,F)f \in \mathcal{F}(I, F). Beside the condition of Cesàro-convergence for (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}, the boundedness of a certain sequence related to this is imposed.
Theorem 2. Let there be given a regular function f inR(I,F)f \in \mathcal{R}(I, F) and a sequence (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}} of real numbers satisfying the conditions:
- (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}} is Cesàro-convergent to 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);
- the sequence ((|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}} is bounded.
Then the sequence (varphi_(n)(f))_(n inN)\left(\varphi_{n}(f)\right)_{n \in \mathbb{N}} is convergent and
{:(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*}
Proof. At first we shall prove (4) for functions ff of the form
{:(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*}
We have
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 .
If alpha=0\alpha=0 the conclusion follows obviously.
For alpha > 0\alpha>0 we denote a_(n)^(**)=(a_(1)+dots+a_(n))/(n)a_{n}^{*}=\frac{a_{1}+\ldots+a_{n}}{n} and we write the two sums in the above formula as
For alpha > 0\alpha>0 we denote a_(n)^(**)=(a_(1)+dots+a_(n))/(n)a_{n}^{*}=\frac{a_{1}+\ldots+a_{n}}{n} and we write the two sums in the above formula as
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}
where
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.
We finally obtain
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 .
We have lim_(n rarr oo)a_(|__ alpha n __|)^(**)=a\lim _{n \rightarrow \infty} a_{\lfloor\alpha n\rfloor}^{*}=a; but (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}^{*}, hence lim_(n rarr oo)(a_(n))/(n)=0\lim _{n \rightarrow \infty} \frac{a_{n}}{n}=0. It follows that in this case
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} f
We consider now the general case f inR(I,F)f \in \mathcal{R}(I, F). The sequence ((|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}} being bounded, let us choose MM such that (|a_(1)|+dots+|a_(n)|)/(n) <= M\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n} \leq M for each n inNn \in \mathbb{N}; let also epsi > 0\varepsilon>0 be an arbitrary constant. From the definition of the space R(I,F)\mathcal{R}(I, F) it follows the existence of the functions f_(i),i=1,dots,pf_{i}, i=1, \ldots, p of the type described in (5), with ||f-sum_(i=1)^(p)f_(i)|| < epsi\left\|f-\sum_{i=1}^{p} f_{i}\right\|<\varepsilon. We have
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) .
The norm of varphi_(n)\varphi_{n}, as given by (2), is ||varphi_(n)||=(|a_(1)|+dots+|a_(n)|)/(n)\left\|\varphi_{n}\right\|=\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n}, hence
{:(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*}
Taking into account the first part of the proof, for each i=1,dots,pi=1, \ldots, p there exists n_(i)inNn_{i} \in \mathbb{N} so that ||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} for n >= n_(i)n \geq n_{i}. It follows that for
n >= max_(i=1,dots,p)n_(i)n \geq \max _{i=1, \ldots, p} n_{i} we have
n >= max_(i=1,dots,p)n_(i)n \geq \max _{i=1, \ldots, p} n_{i} we have
{:(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*}
But
{:(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*}
and the inequalities (6), (7) and (8) imply that
||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} .
It follows that the conclusion holds also for the general case f inF(I,F)f \in \mathcal{F}(I, F).
Remark 1. The Cesàro-convergence of (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}} in Theorem 2 does not necessarily imply the boundedness of ((|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}}. For example, let the sequence be given by
Remark 1. The Cesàro-convergence of (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}} in Theorem 2 does not necessarily imply the boundedness of ((|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}}. For example, let the sequence be given by
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}
Then
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}
hence lim_(n rarr oo)a_(n)^(**)=0\lim _{n \rightarrow \infty} a_{n}^{*}=0, but 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|=\infty.
The condition of Cesàro-convergence imposed to the sequence (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}} in Theorem 22 is a natural one and cannot be relaxed, neither the boundedness of the sequence ((|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}}. In fact, Theorem 2 does admit the following converse:
The condition of Cesàro-convergence imposed to the sequence (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}} in Theorem 22 is a natural one and cannot be relaxed, neither the boundedness of the sequence ((|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}}. In fact, Theorem 2 does admit the following converse:
Theorem 3. Let (varphi_(n))_(n inN)\left(\varphi_{n}\right)_{n \in \mathbb{N}} be the sequence (1) of linear operators associated to the sequence of real numbers (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}}. If lim_(n rarr oo)varphi_(n)(f)\lim _{n \rightarrow \infty} \varphi_{n}(f) exists for every f inC(I,F)subeR(I,F)f \in \mathcal{C}(I, F) \subseteq \mathcal{R}(I, F), then:
- (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}} is Cesàro-convergent to 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);
- the sequence ((|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}} is bounded.
Proof. The first conclusion follows by taking f(t)=uf(t)=u for each t in It \in I, with u in F\\{0}u \in F \backslash\{0\}. In this case varphi_(n)(f)=(a_(1)+dots+a_(n))/(n)u\varphi_{n}(f)=\frac{a_{1}+\ldots+a_{n}}{n} u.
The norm of the operators varphi_(n)\varphi_{n} in the space C(I,F)\mathcal{C}(I, F) is the same as in (1). Indeed, in the proof of Proposition 1, the function f_(0)f_{0} can be modified to a continuous and piecewise affine one which takes also the values (signa_(k))u\left(\operatorname{sign} a_{k}\right) u on the points (k)/(n),k=1,dots,n\frac{k}{n}, k=1, \ldots, 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)|+dots+|a_(n)|)/(n))_(n inN)\left(\frac{\left|a_{1}\right|+\ldots+\left|a_{n}\right|}{n}\right)_{n \in \mathbb{N}} is no longer necessary. In this setting, the principle of uniform boundedness does not work, because C^(1)(I,F)\mathcal{C}^{1}(I, F) endowed with the uniform norm is not a Banach space. The norm of varphi_(n)\varphi_{n} is still the same. In this case we have
Theorem 4. Let there be given a function f inC^(1)(I,F)f \in \mathcal{C}^{1}(I, F) and a sequence (a_(n))_(n inN)\left(a_{n}\right)_{n \in \mathbb{N}} of real numbers which is Cesàro-convergent to a ( lim_(n rarr oo)(a_(1)+dots+a_(n))/(n)=a\lim _{n \rightarrow \infty} \frac{a_{1}+\ldots+a_{n}}{n}=a ). Then
{:(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*}
Proof. We write varphi_(n)(f)\varphi_{n}(f) successively as
{:[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}
We bring now into the scene the continuous function gg given by g(t)=tf^(')(t)g(t)=t f^{\prime}(t) and express varphi_(n)(f)\varphi_{n}(f) in the form
{:[(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*}
Applying Theorem 2 for the function gg and for the sequence (a_(n)^(**))_(n inN)\left(a_{n}^{*}\right)_{n \in \mathbb{N}} convergent to aa, for which obviously lim_(n rarr oo)(a_(1)^(**)+dots+a_(n)^(**))/(n)=a\lim _{n \rightarrow \infty} \frac{a_{1}^{*}+\ldots+a_{n}^{*}}{n}=a and ((|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}} is bounded (because of the convergence of (a_(n)^(**))_(n inN)\left(a_{n}^{*}\right)_{n \in \mathbb{N}} ) we get
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} f
(the last equality is a consequence of an integration by parts). The function f^(')f^{\prime} being uniformly continuous on II, given epsi > 0\varepsilon>0 and nn sufficiently large, we
obtain as a consequence of a mean theorem [3, p. 154]
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)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},
hence
||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,
where MM is a upper bound for the convergent sequence (|a_(n)^(**)|)_(n inN)\left(\left|a_{n}^{*}\right|\right)_{n \in \mathbb{N}}. It follows that
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)=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 in[0,1]a \in[0,1], there exist epsi_(n)in{0,1}\varepsilon_{n} \in\{0,1\} such that
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 in[0,1]a \in[0,1], there exist epsi_(n)in{0,1}\varepsilon_{n} \in\{0,1\} such that
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)
To prove this equality, we choose 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} which satisfy epsi_(n)in{0,1}\varepsilon_{n} \in\{0,1\} and lim_(n rarr oo)(a_(1)+dots+a_(n))/(n)=a\lim _{n \rightarrow \infty} \frac{a_{1}+\ldots+a_{n}}{n}=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 rarr F=Rf: I \rightarrow F=\mathbb{R},
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}
shows.
REFERENCES
[1] Anisiu, V., A principle of double condensation of singularities using sigma\sigma-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).
[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).
- *"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.