Abstract
As a consequence of a general result, we prove that in the case of singular integrals the set of convergence consists only of the two functions \(\textbf{1}\) and \(\cos \). We prove also a multivariate version of this result and apply it to find the necessary and sufficient conditions for the convergence of the sequences of positive linear operators associated to the rectangular and triangular summation.
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1 Introduction and Notation
The Korovkin theorem for the space of all continuous \(2\pi \)-periodic functions on the real line asserts that if \(L_{n}:C_{2\pi }\left( \mathbb {R} \right) \rightarrow C_{2\pi }\left( \mathbb {R}\right) \) is a sequence of positive linear operators, then \(\lim \nolimits _{n\rightarrow \infty }L_{n}\left( f\right) =f\) uniformly on \(\mathbb {R}\) for all \(f\in C_{2\pi }\left( \mathbb {R}\right) \) if and only if \(\lim \nolimits _{n\rightarrow \infty }L_{n}\left( f\right) =f\) uniformly on \(\mathbb {R}\) for all \(f\in \left\{ \textbf{1},\cos ,\sin \right\} \), see [8]. For this result and related questions the reader can consult the book of Korovkin [8] and the books of Butzer and Nessel [3] and Altomare and Campiti [1]. In the present paper we will prove that in the case of singular integrals the test set of convergence consists only of two functions \(\textbf{1}\) and \(\cos \), Theorem 2 and moreover, a multivariate version of this result, see Theorem 3. We use Theorem 3 to extend an old result of Korovkin in the univariate case and a result of A. A. Fomin in the bivariate case, to the so called multivariate rectangular and triangular summation, see Theorems 4 and 5. Our approach in the multivariate case is different than that of A. A. Fomin. Let us fix some notation and notions. Let k be a natural number. A function \(f:\mathbb {R}^{k}\rightarrow \mathbb { R}\) is called \(2\pi \)-periodic with respect to every variable if for all \( \left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\) we have \(f\left( t_{1}+2\pi ,\ldots ,t_{k}\right) =f\left( t_{1},\ldots ,t_{k}\right) \), ..., \( f\left( t_{1},\ldots ,t_{k-1},t_{k}+2\pi \right) =f\left( t_{1},\ldots ,t_{k-1},t_{k}\right) \). We write \(C_{2\pi }\left( \mathbb {R} ^{k}\right) \) (\(C_{2\pi }\left( \mathbb {R}\right) \) for \(k=1\)) to denote the real Banach space of the all continuous functions \(f:\mathbb {R} ^{k}\rightarrow \mathbb {R}\) (\(f:\mathbb {R}\rightarrow \mathbb {R}\) for \(k=1\)) which are \(2\pi \)-periodic with respect to every variable endowed with the norm \(\left\| f\right\| =\sup \nolimits _{t\in \left[ -\pi ,\pi \right] ^{k}}\left| f\left( t\right) \right| =\sup \nolimits _{t\in \mathbb {R} ^{k}}\left| f\left( t\right) \right| \) (\(\left\| f\right\| =\sup \nolimits _{t\in \left[ -\pi ,\pi \right] }\left| f\left( t\right) \right| =\sup \nolimits _{t\in \mathbb {R}}\left| f\left( t\right) \right| \) for \(k=1\)). If \(f\in C_{2\pi }\left( \mathbb {R}^{k}\right) \) and \(\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\) we define \( f_{t_{1},\ldots ,t_{k}}:\mathbb {R}^{k}\rightarrow \mathbb {R}\) by
Note that \(f_{t_{1},\ldots ,t_{k}}\in C_{2\pi }\left( \mathbb {R}^{k}\right) \). A function \(f\in C_{2\pi }\left( \mathbb {R}^{k}\right) \) is called positive and we write, as usual, \(f\ge 0\) if \(f\left( t_{1},\ldots ,t_{k}\right) \ge 0\) , \(\forall \left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\) and also if \( f,g\in C_{2\pi }\left( \mathbb {R}^{k}\right) \) the notation \(f\le g\) means \( g-f\ge 0\). An operator \(L:C_{2\pi }\left( \mathbb {R}^{k}\right) \rightarrow C_{2\pi }\left( \mathbb {R}^{k}\right) \) is called positive if \(f\ge 0\) implies \(L\left( f\right) \ge 0\). By \(\textbf{1}\) we denote the constant function \(\textbf{1}:\mathbb {R}^{k}\rightarrow \mathbb {R}\), \(\textbf{1} \left( t_{1},\ldots ,t_{k}\right) =1\). For every \(1\le i\le k\), \(pr_{i}: \mathbb {R}^{k}\rightarrow \mathbb {R}\) are the canonical projections \( pr_{i}\left( x_{1},\ldots ,x_{k}\right) =x_{i}\). All notation and notion used and not defined are standard, see [1].
2 A New Convergence Result
We need the following result, see [12, Corollary 1] and also [9, 11]. The case \(k=1\) can be found in [8, page 15–16 and 24].
Lemma 1
Let \(f\in C_{2\pi }\left( \mathbb {R}^{k}\right) \). Then for all \(\varepsilon >0\) there exists \(\eta _{\varepsilon }>0\) such that for all \(\left( s_{1},\ldots ,s_{k}\right) \in \mathbb {R}^{k}\), all \( \left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\) we have
The following technical lemma is the key ingredient for the proof of the main result of this paper.
Lemma 2
Let \(f\in C_{2\pi }\left( \mathbb {R}^{k}\right) \). Then for all \(\varepsilon >0\) there exists \(\eta _{\varepsilon }>0\) such that for all linear positive functionals \(x^{*}:C_{2\pi }\left( \mathbb {R}^{k}\right) \rightarrow \mathbb {R}\) and all \(\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R} ^{k}\) the following relation holds
Proof
Let \(\varepsilon >0\). Then from Lemma 1 there exists \(\eta _{\varepsilon }>0\) such that for all \(\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\), all \(\left( \theta _{1},\ldots ,\theta _{k}\right) \in \mathbb {R}^{k}\) we have
that is for all \(\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\),
If \(x^{*}:C_{2\pi }\left( \mathbb {R}^{k}\right) \rightarrow \mathbb {R}\) is a linear positive functional we deduce that
Then
\(\square \)
The next result is the main result of this paper.
Theorem 1
Let \(x_{n}^{*}:C_{2\pi }\left( \mathbb {R} ^{k}\right) \rightarrow \mathbb {R}\) be a sequence of linear positive functionals. Then the following assertions are equivalent:
-
(i)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{k}\right) \) we have \( \lim \nolimits _{n\rightarrow \infty }x_{n}^{*}\left( f_{t_{1},\ldots ,t_{k}}\right) =f\left( t_{1},\ldots ,t_{k}\right) \) uniformly with respect to \(\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\).
-
(ii)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{k}\right) \) and all \(\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\) we have \(\lim \nolimits _{n \rightarrow \infty }x_{n}^{*}\left( f_{t_{1},\ldots ,t_{k}}\right) =f\left( t_{1},\ldots ,t_{k}\right) \).
-
(iii)
\(\lim \nolimits _{n\rightarrow \infty }x_{n}^{*}\left( \textbf{1} \right) =1\) and \(\lim \nolimits _{n\rightarrow \infty }x_{n}^{*}\left( \cos \circ pr_{j}\right) =1\) for all \(j=1\), ..., k.
Proof
(i)\(\Rightarrow \)(ii) is trivial. (ii)\(\Rightarrow \)(iii). From (ii) we have, in particular, \(\lim \nolimits _{n\rightarrow \infty }x_{n}^{*}\left( f_{0,\ldots ,0}\right) =f\left( 0,\ldots ,0\right) \) for all \(f\in C_{2\pi }\left( \mathbb {R}^{k}\right) \). Then since for \(f\in \{ \textbf{1},\cos \circ pr_{1},\ldots ,\cos \circ pr_{k}\} \) we have \(f_{0,\ldots ,0}=f\) and \(f\left( 0,\ldots ,0\right) =1\) we get (iii).
(iii)\(\Rightarrow \)(i) Let \(f\in C_{2\pi }\left( \mathbb {R}^{k}\right) \) and \(\varepsilon >0\). From Lemma 2 there exists \(\delta _{\varepsilon }>0\) such that for all \(n\in \mathbb {N}\) and all \(\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\) the following relation holds
Since \(\lim \nolimits _{n\rightarrow \infty }x_{n}^{*}\left( \textbf{1} \right) =1<2\) there exists \(n_{0}\in \mathbb {N}\) such that for all \(n\ge n_{0}\) we have \(x_{n}^{*}\left( \textbf{1}\right) <2\) and there exists \( mss_{\varepsilon }\in \mathbb {N}\) such that for all \(n\ge m_{\varepsilon }\) we have \(\left| x_{n}^{*}\left( \textbf{1}\right) -1\right| < \frac{\varepsilon }{3\left( \left\| f\right\| +1\right) }\). From (iii) \(\lim \nolimits _{n\rightarrow \infty }\sum \nolimits _{j=1}^{k}\left[ x_{n}^{*}\left( \textbf{1}\right) -x_{n}^{*}\left( \cos \circ pr_{j}\right) \right] =0\), hence there exists \(p_{\varepsilon }\in \mathbb {N}\) such that for all \(n\ge p_{\varepsilon }\) we have \(\sum \nolimits _{j=1}^{k}\left[ x_{n}^{*}\left( \textbf{1}\right) -x_{n}^{*}\right. \left. \left( \cos \circ pr_{j}\right) \right] <\frac{\varepsilon }{3\delta _{\varepsilon }}\). We deduce that for all \(n\ge \max \left( n_{0},m_{\varepsilon },p_{\varepsilon }\right) \) and all \(\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\) we have \(\left| x_{n}^{*}\left( f_{t_{1},\ldots ,t_{k}}\right) -f\left( t_{1},\ldots ,t_{k}\right) \right| <\varepsilon \) that is (i). \(\square \)
Let us state as an explicit result the univariate case of Theorem 1.
Corollary 1
Let \(x_{n}^{*}:C_{2\pi }\left( \mathbb {R}\right) \rightarrow \mathbb {R}\) be a sequence of linear positive functionals. Then the following assertions are equivalent:
-
(i)
For all \(f\in C_{2\pi }\left( \mathbb {R}\right) \) we have \( \lim \nolimits _{n\rightarrow \infty }x_{n}^{*}\left( f_{t}\right) =f\left( t\right) \) uniformly with respect to \(t\in \mathbb {R}\).
-
(ii)
For all \(f\in C_{2\pi }\left( \mathbb {R}\right) \) and all \(t\in \mathbb { R}\) we have \(\lim \nolimits _{n\rightarrow \infty }x_{n}^{*}\left( f_{t}\right) =f\left( t\right) \).
-
(iii)
\(\lim \nolimits _{n\rightarrow \infty }x_{n}^{*}\left( \textbf{1} \right) =1\) and \(\lim \nolimits _{n\rightarrow \infty }x_{n}^{*}\left( \cos \right) =1\).
We prove in the sequel a new convergence result for univariate singular integrals. The novelty of this result is that the test set of convergence consists only of two functions \(\textbf{1}\) and \(\cos \). Let us mention that in all the results which appear in the literature the test set of convergence consist of the three functions \(\textbf{1}\), \(\cos \) and \(\sin \) , see the book of Korovkin [8], or Butzer and Nessel [3, page 58, Theorem 1.3.7 and Corollary 1.3.8]. For an operator version of the classical Korovkin theorem we recommend the reader to consult our very recent paper [13].
Theorem 2
Let \(K_{n}:\mathbb {R}\rightarrow \left[ 0,\infty \right) \) be a sequence of continuous functions. Then the following assertions are equivalent:
-
(i)
For all \(f\in C_{2\pi }\left( \mathbb {R}\right) \) we have
$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{2\pi }\int _{-\pi }^{\pi }K_{n}\left( \theta \right) f\left( t-\theta \right) \textrm{d}\theta =f\left( t\right) \text { uniformly with respect to }t\in \mathbb {R}. \end{aligned}$$ -
(ii)
For all \(f\in C_{2\pi }\left( \mathbb {R}\right) \) and all \(t\in \mathbb { R}\) we have
$$\begin{aligned}{} & {} \lim \limits _{n\rightarrow \infty }\frac{1}{2\pi }\int _{-\pi }^{\pi }K_{n}\left( \theta \right) f\left( t-\theta \right) \textrm{d}\theta \nonumber \\ {}{} & {} =f\left( t\right) . \end{aligned}$$ -
(iii)
\(\lim \nolimits _{n\rightarrow \infty }\frac{1}{2\pi }\int _{-\pi }^{\pi }K_{n}\left( t\right) \textrm{d}t=1\) and \(\lim \nolimits _{n\rightarrow \infty }\frac{1}{ 2\pi }\int _{-\pi }^{\pi }K_{n}\left( t\right) \cos tdt=1\).
Proof
Let us define \(x_{n}^{*}:C_{2\pi }\left( \mathbb {R}\right) \rightarrow \mathbb {R}\) by \(x_{n}^{*}\left( f\right) =\frac{1}{2\pi }\int _{-\pi }^{\pi }K_{n}\left( \theta \right) f\left( \theta \right) \textrm{d}\theta \) and note that \(x_{n}^{*}\left( f_{t}\right) =\frac{1}{2\pi }\int _{-\pi }^{\pi }K_{n}\left( \theta \right) f_{t}\left( \theta \right) \textrm{d}\theta =\frac{1}{ 2\pi }\int _{-\pi }^{\pi }K_{n}\left( \theta \right) f\left( t-\theta \right) \textrm{d}\theta \). Then the equivalences from the statement follow from Corollary 1. \(\square \)
Our next objective is to prove a multivariate version of Theorem 2. Let us introduce some common notations.
Definition 1
Let \(k\ge 2\) be a natural number and \(K: \mathbb {R}^{k}\rightarrow \mathbb {R}\) a continuous function. We define \( K^{\left\langle 1\right\rangle },\ldots ,K^{\left\langle k\right\rangle }: \mathbb {R}\rightarrow \mathbb {R}\) by
Let us note that by Fubini’s theorem for every \(j=1\), ..., k we have
Theorem 3
Let \(k\ge 2\) be a natural number and \(K_{n}:\mathbb {R} ^{k}\rightarrow \left[ 0,\infty \right) \) a sequence of continuous functions. Then the following assertions are equivalent:
-
(i)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{k}\right) \) we have
$$\begin{aligned}{} & {} \lim \limits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{k}}\int _{ \left[ -\pi ,\pi \right] ^{k}}K_{n}\left( \theta _{1},\ldots ,\theta _{k}\right) f\left( t_{1}-\theta _{1},\ldots ,t_{k}-\theta _{k}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{k} \\{} & {} \quad =f\left( t_{1},\ldots ,t_{k}\right) \text { uniformly with respect to }\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}. \end{aligned}$$ -
(ii)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{k}\right) \) and all \(\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\) we have
$$\begin{aligned}{} & {} \lim \limits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{k}}\int _{ \left[ -\pi ,\pi \right] ^{k}}K_{n}\left( \theta _{1},\ldots ,\theta _{k}\right) f\left( t_{1}-\theta _{1},\ldots ,t_{k}-\theta _{k}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{k}\\{} & {} \qquad =f\left( t_{1},\ldots ,t_{k}\right) .\text { } \end{aligned}$$ -
(iii)
$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{k}}\int _{ \left[ -\pi ,\pi \right] ^{k}}K_{n}\left( \theta _{1},\ldots ,\theta _{k}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{k}=1 \end{aligned}$$
and
$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{2\pi }\int _{-\pi }^{\pi }K_{n}^{\left\langle j\right\rangle }\left( \theta _{j}\right) \cos \theta _{j}\textrm{d}\theta _{j}=1\text { for every }j=1,\ldots ,k. \end{aligned}$$
Proof
Let us define \(x_{n}^{*}:C_{2\pi }\left( \mathbb {R}^{k}\right) \rightarrow \mathbb {R}\) by \(x_{n}^{*}\left( f\right) =\frac{1}{\left( 2\pi \right) ^{k}}\int _{\left[ -\pi ,\pi \right] ^{k}}K_{n}\left( \theta _{1},\ldots ,\theta _{k}\right) f\left( \theta _{1},\ldots ,\theta _{k}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{k}\) and note that for all \(\left( t_{1},\ldots ,t_{k}\right) \in \mathbb {R}^{k}\) we have
Then the equivalences from the statement follow from Theorem 1. \(\square \)
3 Some Calculations
In the rest of the paper we will apply the general result in Theorem 3 in the study of convergence of some multivariate operators, see Theorems 4 and 5. This kind of operators are multivariate versions of the operators studied by Korovkin in the paper [7, Theorem 1] and in the bivariate case by Fomin in [4, Theorem 1]; related questions are treated in [2,3,4,5,6].
Proposition 1
Let \(h:\left\{ 0,\ldots ,n\right\} ^{2}\rightarrow \mathbb {C}\) and \(f:\left\{ -n,\ldots ,-1,0,1,\ldots ,n\right\} \rightarrow \mathbb {C}\). Then:
-
(i)
\(\sum \nolimits _{k,l=0,k>l}^{n}h\left( k,l\right) f\left( k-l\right) =\nu _{1}f\left( 1\right) +\nu _{2}f\left( 2\right) +\nu _{3}f\left( 3\right) +\cdot \cdot \cdot +\nu _{n}f\left( n\right) \) where
$$\begin{aligned} \nu _{1}= & {} h\left( 1,0\right) +\ h\left( 2,1\right) +\ h\left( 3,2\right) +\cdot \cdot \cdot +h\left( n-1,n-2\right) +h\left( n,n-1\right) , \\ \nu _{2}= & {} h\left( 2,0\right) +\ h\left( 3,1\right) +\ h\left( 4,2\right) +\cdot \cdot \cdot +h\left( n,n-2\right) ,\ldots , \\ \nu _{n-1}= & {} h\left( n-1,0\right) +h\left( n,1\right) ,\nu _{n}=h\left( n,0\right) . \end{aligned}$$ -
(ii)
\(\sum \nolimits _{k,l=0,k<l}^{n}h\left( k,l\right) f\left( k-l\right) =\lambda _{1}f\left( -1\right) +\lambda _{2}f\left( -2\right) +\lambda _{3}f\left( -3\right) +\cdot \cdot \cdot +\lambda _{n}f\left( -n\right) \), where
$$\begin{aligned} \lambda _{1}= & {} h\left( 0,1\right) +\ h\left( 1,2\right) +\ h\left( 2,3\right) +\cdot \cdot \cdot +h\left( n-2,n-1\right) +h\left( n-1,n\right) ,\\ \lambda _{2}= & {} h\left( 0,2\right) +\ h\left( 1,3\right) +\ h\left( 2,4\right) +\cdot \cdot \cdot +h\left( n-2,n\right) ,\ldots , \\ \lambda _{n-1}= & {} h\left( 0,n-1\right) +h\left( 1,n\right) ,\lambda _{n}=h\left( 0,n\right) . \end{aligned}$$
Proof
(i) We have
(ii) Let us define \(g\left( k,l\right) =h\left( l,k\right) \) and \(\varphi \left( k\right) =f\left( -k\right) \). Then \(\sum \nolimits _{k,l=0,k<l}^{n}h \left( k,l\right) f\left( k-l\right) =\sum \nolimits _{k,l=0,l>k}^{n}g\left( l,k\right) \varphi \left( l-k\right) \) and by (i) \(\sum \nolimits _{k,l=0,l>k}^{n}g\left( l,k\right) \varphi \left( l-k\right) =\lambda _{1}\varphi \left( 1\right) +\lambda _{2}\varphi \left( 2\right) +\cdot \cdot \cdot +\lambda _{n}\varphi \left( n\right) =\lambda _{1}f\left( -1\right) +\lambda _{2}f\left( -2\right) +\lambda _{3}f\left( -3\right) +\cdot \cdot \cdot +\lambda _{n}f\left( -n\right) \) where
\(\square \)
Corollary 2
(i) Let \(h:\left\{ 0,\ldots ,n\right\} ^{2}\rightarrow \mathbb {C}\) be such that \(h\left( k,l\right) =h\left( l,k\right) \) for all \(\left( k,l\right) \in \left\{ 0,\ldots ,n\right\} ^{2}\) and \(f:\left\{ -n,\ldots ,-1,0,1,\ldots ,n\right\} \rightarrow \mathbb {C}\) an even function. Then
where \(\lambda _{1}=h\left( 0,1\right) +h\left( 1,2\right) +\cdot \cdot \cdot +h\left( n-1,n\right) \), \(\lambda _{2}=h\left( 0,2\right) +h\left( 1,3\right) +\cdot \cdot \cdot +h\left( n-2,n\right) \), ..., \(\lambda _{n-1}=h\left( 0,n-1\right) +h\left( 1,n\right) \), \(\lambda _{n}=h\left( 0,n\right) \).
(ii) Let \(h:\left\{ 0,\ldots ,n\right\} ^{2}\rightarrow \mathbb {C}\) be such that \( h\left( k,l\right) =-h\left( l,k\right) \) for all \(\left( k,l\right) \in \left\{ 0,\ldots ,n\right\} ^{2}\) and \(f:\left\{ -n,\ldots ,-1,0,1,\ldots ,n\right\} \rightarrow \mathbb {C}\) an odd function. Then
where \(\nu _{1}=h\left( 1,0\right) +\ h\left( 2,1\right) +\ h\left( 3,2\right) +\cdot \cdot \cdot +h\left( n-1,n-2\right) +h\left( n,n-1\right) \), \(\nu _{2}=h\left( 2,0\right) +\ h\left( 3,1\right) +\ h\left( 4,2\right) +\cdot \cdot \cdot +h\left( n,n-2\right) \), ..., \(\nu _{n-1}=h\left( n-1,0\right) +h\left( n,1\right) \), \(\nu _{n}=h\left( n,0\right) \).
Proof
(i) Since f is even, by Proposition 1(ii)\( \sum \nolimits _{k,l=0,k<l}^{n}h\left( k,l\right) f\left( k-l\right) =\lambda _{1}f\left( 1\right) +\lambda _{2}f\left( 2\right) +\cdot \cdot \cdot +\lambda _{n}f\left( n\right) \). By Proposition 1(i) \( \sum \nolimits _{k,l=0,k>l}^{n}h\left( k,l\right) f\left( k-l\right) =\nu _{1}f\left( 1\right) +\nu _{2}f\left( 2\right) +\cdot \cdot \cdot +\nu _{n}f\left( n\right) \). Since \(h\left( k,l\right) =h\left( l,k\right) \), \( \forall \left( k,l\right) \in \left\{ 0,\ldots ,n\right\} ^{2}\) we deduce that \( \nu _{1}=\lambda _{1}\), ..., \(\nu _{n}=\lambda _{n}\) and hence \( \sum \nolimits _{k,l=0,k>l}^{n}h\left( k,l\right) f\left( k-l\right) =\lambda _{1}f\left( 1\right) +\lambda _{2}f\left( 2\right) +\cdot \cdot \cdot +\lambda _{n}f\left( n\right) \). To finish the proof let us note that \( \sum \nolimits _{k,l=0,k\ne l}^{n}h\left( k,l\right) f\left( k-l\right) =\sum \nolimits _{k,l=0,k>l}^{n}h\left( k,l\right) f\left( k-l\right) +\sum \nolimits _{k,l=0,k<l}^{n}h\left( k,l\right) f\left( k-l\right) \).
(ii) Since f is odd from Proposition 1(ii)
\(\sum \nolimits _{k,l=0,k<l}^{n}h\left( k,l\right) f\left( k-l\right) =-\left( \lambda _{1}f\left( 1\right) +\lambda _{2}f\left( 2\right) +\cdot \cdot \cdot +\lambda _{n}f\left( n\right) \right) \). Since \(h\left( k,l\right) =-h\left( l,k\right) \) we get \(\lambda _{1}=-\nu _{1}\), ..., \(\lambda _{n}=-\nu _{n}\) and hence \(\sum \nolimits _{k,l=0,k<l}^{n}h\left( k,l\right) f\left( k-l\right) =\nu _{1}f\left( 1\right) +\nu _{2}f\left( 2\right) +\cdot \cdot \cdot +\nu _{n}f\left( n\right) \). Also by Proposition 1(i) \(\sum \nolimits _{k,l=0,k>l}^{n}h\left( k,l\right) f\left( k-l\right) =\nu _{1}f\left( 1\right) +\nu _{2}f\left( 2\right) +\cdot \cdot \cdot +\nu _{n}f\left( n\right) \) and to finish the proof let us note that \( \sum \nolimits _{k,l=0,k\ne l}^{n}h\left( k,l\right) f\left( k-l\right) =\sum \nolimits _{k,l=0,k>l}^{n}h\left( k,l\right) f\left( k-l\right) +\sum \nolimits _{k,l=0,k<l}^{n}h\left( k,l\right) f\left( k-l\right) \). \(\square \)
If \(z\in \mathbb {C}\) then \(\mathfrak {R}\left( z\right) \) denotes the real part of z and \(\mathfrak {I}m\left( z\right) \) denotes the imaginary part of z.
Proposition 2
Let \(\varphi _{n}:\left\{ 0,\ldots ,n\right\} \rightarrow \mathbb {C}\). Then for all \(\theta \in \mathbb {R}\) the following equality holds
where \(\alpha _{n0}=\sum \nolimits _{k=0}^{n}\left| \varphi _{n}\left( k\right) \right| ^{2}\), \(\alpha _{n1}=\sum \nolimits _{k=0}^{n-1}\mathfrak {R} \left( \varphi _{n}\left( k\right) \overline{\varphi _{n}\left( k+1\right) } \right) \), ..., \(\alpha _{nn-1}=\mathfrak {R}\left( \varphi _{n}\left( 0\right) \overline{\varphi _{n}\left( n-1\right) }\right) +\mathfrak {R} \left( \varphi _{n}\left( 1\right) \overline{\varphi _{n}\left( n\right) } \right) \), \(\alpha _{nn}=\mathfrak {R}\left( \varphi _{n}\left( 0\right) \overline{\varphi _{n}\left( n\right) }\right) \), \(\beta _{n1}=\sum \nolimits _{k=0}^{n-1}\mathfrak {I}m\left( \varphi _{n}\left( k+1\right) \overline{\varphi _{n}\left( k\right) }\right) \), ..., \(\beta _{nn-1}=\mathfrak {I}m\left( \varphi _{n}\left( n-1\right) \overline{\varphi _{n}\left( 0\right) }\right) +\mathfrak {I}m\left( \varphi _{n}\left( n\right) \overline{\varphi _{n}\left( 1\right) }\right) \), \(\beta _{nn}= \mathfrak {I}m\left( \varphi _{n}\left( n\right) \overline{\varphi _{n}\left( 0\right) }\right) \).
Proof
We have
and since the left member is real we deduce that
From the equality \(\mathfrak {R}\left( z\overline{w}\right) =\left( \mathfrak { R}z\right) \left( \mathfrak {R}w\right) +\left( \mathfrak {I}mz\right) \left( \mathfrak {I}mw\right) \), \(z,w\in \mathbb {C}\) we get
and thus
Since \(h\left( k,l\right) =\mathfrak {R}\left( \varphi _{n}\left( k\right) \overline{\varphi _{n}\left( l\right) }\right) \) has the property \(h\left( k,l\right) =h\left( l,k\right) \) and \(\cos \) is an even function, from Corollary 2(i)
Since \(h\left( k,l\right) =\mathfrak {I}m\left( \varphi _{n}\left( k\right) \overline{\varphi _{n}\left( l\right) }\right) \) has the property \(h\left( k,l\right) =-h\left( l,k\right) \) (use \(\mathfrak {I}m\left( z\overline{w} \right) =\left( \mathfrak {I}mz\right) \left( \mathfrak {R}w\right) -\left( \mathfrak {R}z\right) \left( \mathfrak {I}mw\right) \), \(z,w\in \mathbb {C}\)) and \(\sin \) is an odd function, from Corollary 2(ii) we get
\(\square \)
The real case of Proposition 2, that is \(\varphi _{n}\) takes the real values, is a well-known result, see [10, Problem 39 page 77 with solution at pages 258–259].
We need the following well-known result. For the sake of completeness we include its proof.
Proposition 3
Let m be a natural number, \(\beta _{n},\gamma _{n}:\left\{ 0,1,\ldots ,n\right\} ^{m}\rightarrow \mathbb {C}\) and \( B_{n},C_{n}:\mathbb {R}^{m}\rightarrow \mathbb {C}\) defined by
Then
In particular,
Proof
From \(B_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) \overline{C_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) }= \sum \nolimits _{k_{1},\ldots ,k_{m}=0,l_{1},\ldots ,l_{m}=0}^{n}\beta _{n}\left( k_{1},\ldots ,k_{m}\right) \overline{\gamma _{n}\left( l_{1},\ldots ,l_{m}\right) } e^{i\left( k_{1}-l_{1}\right) \theta _{1}}\cdot \cdot \cdot e^{i\left( k_{m}-l_{m}\right) \theta _{m}}\) and Fubini’s theorem we have
Above \(\delta _{kl}=1\) if \(k=l\); 0 if \(k\ne l\). \(\square \)
4 A Technical Lemma
The following technical lemma will be essential in what follows. We recall that the scalar product in the space \(L_{2}\left[ -\pi ,\pi \right] \) is defined by \(\left\langle f,g\right\rangle _{L_{2}\left[ -\pi ,\pi \right] }= \frac{1}{2\pi }\int _{-\pi }^{\pi }f\left( t\right) \overline{g\left( t\right) }\textrm{d}t\). Moreover, we use that the system of functions \(e_{k}\left( t\right) =e^{ikt}\) is orthonormal in \(L_{2}\left[ -\pi ,\pi \right] \), \( \left\langle e_{k},e_{l}\right\rangle _{L_{2}\left[ -\pi ,\pi \right] }=\delta _{kl}\).
Lemma 3
Let \(m\ge 2\) be a natural number, \(\varphi _{n}:\left\{ 0,1,\ldots ,n\right\} ^{m}\rightarrow \mathbb {R}\). For every \(j=1\), ..., m and all \(k_{j}=0,\ldots ,n\) let \(A_{k_{j}}:\mathbb {R}^{m-1}\rightarrow \mathbb {C}\) be defined by
and \(\alpha _{n\text { }1}^{\left( j\right) }:\mathbb {R}^{m-1}\rightarrow \mathbb {R}\) defined by
Then
Proof
We prove the case \(j=m\), the others being similar. We have
For \(m=2\) from the Proposition 3 we have
and by (1) we get the equality stated.
If \(m\ge 3\) then, by Fubini’s theorem
Let us fix \(\left( \theta _{1},\ldots ,\theta _{m-2}\right) \in \left[ -\pi ,\pi \right] ^{m-2}\). For all \(\theta _{m-1}\in \left[ -\pi ,\pi \right] \) we have
where
Thus \(A_{k_{m}}\left( \theta _{1},\ldots ,\theta _{m-2},\cdot \right) =\sum \nolimits _{k_{m-1}=0}^{n}A_{k_{m-1}k_{m}}\left( \theta _{1},\ldots ,\theta _{m-2}\right) e_{k_{m-1}}\). The orthonormality of the functions \(\left( e_{k_{m-1}}\right) _{0\le k_{m-1}\le n}\) gives us that
Then
Since
the Proposition 3 gives us that
From the relations (3) and (4) we deduce that
To finish the proof just combine the relations (1), (2) and (5). \(\square \)
5 The Case of Rectangular Summation
In this section, we apply the above general results to find the necessary and sufficient conditions that some natural multivariate operators associated to some summation methods to be convergent. For various other methods of summation of trigonometric series the reader can consult the book of Weiss [14].
Theorem 4
Let \(m\ge 2\) be a natural number, \( \varphi _{n}:\left\{ 0,1,\ldots ,n\right\} ^{m}\rightarrow \mathbb {R}\), \(K_{n}: \mathbb {R}^{m}\rightarrow \left[ 0,\infty \right) \) defined by
Then the following assertions are equivalent:
-
(i)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{m}\right) \) we have
$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{m}}\int _{ \left[ -\pi ,\pi \right] ^{m}}K_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) f\left( t_{1}{-}\theta _{1},\ldots ,t_{m}-\theta _{m}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{m}{=}f\left( t\right) \end{aligned}$$uniformly with respect to \(t=\left( t_{1},\ldots ,t_{m}\right) \in \mathbb {R} ^{m} \).
-
(ii)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{m}\right) \) and all \(t=\left( t_{1},\ldots ,t_{m}\right) \in \mathbb {R}^{m}\) we have
$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{m}}\int _{ \left[ -\pi ,\pi \right] ^{m}}K_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) f\left( t_{1}{-}\theta _{1},\ldots ,t_{m}{-}\theta _{m}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{m}{=}f\left( t\right) . \end{aligned}$$ -
(iii)
\(\lim \nolimits _{n\rightarrow \infty }\sum \nolimits _{k_{1},\ldots ,k_{m}=0}^{n} \left[ \varphi _{n}\left( k_{1},\ldots ,k_{m}\right) \right] ^{2}=1\) and
$$\begin{aligned}{} & {} \lim \limits _{n\rightarrow \infty }\sum \limits _{k_{1}=0}^{n-1}\left( \sum \limits _{k_{2},\ldots ,k_{m}=0}^{n}\varphi _{n}\left( k_{1},k_{2},\ldots ,k_{m}\right) \varphi _{n}\left( k_{1}+1,k_{2},\ldots ,k_{m}\right) \right) =1,\\{} & {} \lim \limits _{n\rightarrow \infty }\sum \limits _{k_{2}=0}^{n-1}\left( \sum \limits _{k_{1},k_{3},\ldots ,k_{m}=0}^{n}\varphi _{n}\left( k_{1},k_{2},k_{3},\ldots ,k_{m}\right) \varphi _{n}\left( k_{1},k_{2}+1,k_{3},\ldots ,k_{m}\right) \right) \\{} & {} \quad =1,\ldots ,\\{} & {} \lim \limits _{n\rightarrow \infty }\sum \limits _{k_{m}=0}^{n-1}\left( \sum \limits _{k_{1},\ldots ,k_{m-1}=0}^{n}\varphi _{n}\left( k_{1},\ldots ,k_{m-1},k_{m}\right) \varphi _{n}\left( k_{1},\ldots ,k_{m-1},k_{m}+1\right) \right) = 1. \end{aligned}$$
Proof
By Theorem 3 the conditions (i), (ii) are equivalent to \( \lim \nolimits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{m}}\int _{ \left[ -\pi ,\pi \right] ^{m}}K_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{m}=1\) and \(\lim \nolimits _{n\rightarrow \infty }\frac{1}{2\pi }\int _{-\pi }^{\pi }K_{n}^{\left\langle j\right\rangle }\left( \theta _{j}\right) \cos \theta _{j}\textrm{d}\theta _{j}=1\) for every \( j=1,\ldots ,m\). Since by Proposition 3
we obtain \(\lim \nolimits _{n\rightarrow \infty }\sum \nolimits _{k_{1},\ldots ,k_{m}=0}^{n}\left[ \varphi _{n}\left( k_{1},\ldots ,k_{m}\right) \right] ^{2}=1\). For all \(k_{m}=0\), ..., n let us define \(A_{k_{m}}:\mathbb {R}^{m-1}\rightarrow \mathbb {C}\) by \( A_{k_{m}}\left( \theta _{1},\ldots ,\theta _{m-1}\right) =\sum \nolimits _{k_{1},\ldots ,k_{m-1}=0}^{n}\varphi _{n}( k_{1},\ldots ,k_{m-1},k_{m}) e^{ik_{1}\theta _{1}}\cdot \cdot \cdot e^{ik_{m-1}\theta _{m-1}}\). By Proposition 2 we have
where \(\alpha _{n\text { }1}^{\left( m\right) }\left( \theta _{1},\ldots ,\theta _{m-1}\right) {=}\sum \nolimits _{k_{m}=0}^{n}\mathfrak {R}\left( A_{k_{m}}\left( \theta _{1},\ldots ,\theta _{m-1}\right) \right. \left. \overline{A_{k_{m}+1}\left( \theta _{1},\ldots ,\theta _{m-1}\right) }\right) \). We deduce that
where
Let us note that for all \(n\in \mathbb {N}\) we have \(\frac{1}{2\pi } \int _{-\pi }^{\pi }K_{n}^{\left\langle m\right\rangle }\left( \theta _{m}\right) \cos \theta _{m}\textrm{d}\theta _{m}=\nu _{n\text { }1}^{\left( m\right) } \). The condition of the convergence for \(K_{n}^{\left\langle m\right\rangle } \), that is
becomes
But, since from the Lemma 3
we deduce that the condition of the convergence for \(K_{n}^{\left\langle m\right\rangle }\) is equivalent to
Similar, from the conditions of the convergence for \(K_{n}^{\left\langle 1\right\rangle }\), ..., \(K_{n}^{\left\langle m-1\right\rangle }\) we get the remaining conditions stated in (iii). \(\square \)
6 The Case of Triangular Summation
We use now Theorem 4 to prove a similar result for the triangular summation.
Theorem 5
Let \(m\ge 2\) be a natural number, \(\varphi _{n}:\left\{ 0,1,\ldots ,n\right\} ^{m}\rightarrow \mathbb {R}\), \(K_{n}:\mathbb {R} ^{m}\rightarrow \left[ 0,\infty \right) \) defined by
Then the following assertions are equivalent:
-
(i)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{m}\right) \) we have
$$\begin{aligned}{} & {} \lim \limits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{m}}\int _{ \left[ -\pi ,\pi \right] ^{m}}K_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) f\left( t_{1}-\theta _{1},\ldots ,t_{m}-\theta _{m}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{m}\\{} & {} \qquad =f\left( t_{1},\ldots ,t_{m}\right) \end{aligned}$$uniformly with respect to \(\left( t_{1},\ldots ,t_{m}\right) \in \mathbb {R}^{m}\).
-
(ii)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{m}\right) \) and all \(t=\left( t_{1},\ldots ,t_{m}\right) \in \mathbb {R}^{m}\) we have
$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{m}}\int _{ \left[ -\pi ,\pi \right] ^{m}}K_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) f\left( t_{1}{-}\theta _{1},\ldots ,t_{m}-\theta _{m}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{m}{=}f\left( t\right) . \end{aligned}$$ -
(iii)
\(\lim \nolimits _{n\rightarrow \infty }\sum \nolimits _{k_{1}+\cdot \cdot \cdot +k_{m}\le n}\left[ \varphi _{n}\left( k_{1},\ldots ,k_{m}\right) \right] ^{2}=1\) and
$$\begin{aligned}{} & {} \lim \limits _{n\rightarrow \infty }\sum \limits _{k_{1}=0}^{n-1}\left( \sum \limits _{k_{2}+\cdot \cdot \cdot +k_{m}\le n-k_{1}-1}\varphi _{n}\left( k_{1},k_{2},\ldots ,k_{m}\right) \varphi _{n}\left( k_{1}+1,k_{2},\ldots ,k_{m}\right) \right) =1, \\{} & {} \lim \limits _{n\rightarrow \infty }\sum \limits _{k_{2}=0}^{n-1}\left( \sum \limits _{k_{1}+k_{3}+\cdot \cdot \cdot +k_{m}\le n-k_{2}-1}\varphi _{n}\left( k_{1},k_{2},k_{3},\ldots ,k_{m}\right) \varphi _{n}\left( k_{1},k_{2}+1,k_{3},\ldots ,k_{m}\right) \right) \\{} & {} \qquad =1,\ldots ,\\{} & {} \lim \limits _{n\rightarrow \infty }\sum \limits _{k_{m}=0}^{n-1}\left( \sum \limits _{k_{1}+\cdot \cdot \cdot +k_{m-1}\le n-k_{m}-1}\varphi _{n}\left( k_{1},\ldots ,k_{m-1},k_{m}\right) \varphi _{n}\left( k_{1},\ldots ,k_{m-1},k_{m}+1\right) \right) \\{} & {} \qquad =1. \end{aligned}$$
Proof
For every \(n\in \mathbb {N}\) let \(\Delta _{n}=\{ \left( k_{1},\ldots ,k_{m}\right) \in \left\{ 0,1,\ldots ,n\right\} ^{m}\mid k_{1}+\cdot \cdot \cdot +k_{m}\le n\} \) and \(\beta _{n}:\left\{ 0,1,\ldots ,n\right\} ^{m}\rightarrow \mathbb {R}\) defined by
Then \(K_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) =\left| \sum \nolimits _{k_{1},\ldots ,k_{m}=0}^{n}\beta _{n}\left( k_{1},\ldots ,k_{m}\right) e^{ik_{1}\theta _{1}}\cdot \cdot \cdot e^{ik_{m}\theta _{m}}\right| ^{2}\) . From Theorem 4 (i), (ii) are equivalent to \(\lim \nolimits _{n\rightarrow \infty }\sum \nolimits _{k_{1},\ldots ,k_{m}=0}^{n} \left[ \beta _{n}\left( k_{1},\ldots ,k_{m}\right) \right] ^{2}=1\) and
Since
we get the first condition in (iii). Now let us note the equality
If \(k_{2}+\cdot \cdot \cdot +k_{m}\ge n-k_{1}\) then \(\left( k_{1}+1\right) +k_{2}+\cdot \cdot \cdot +k_{m}\ge n+1\) i.e., \(\left( k_{1}+1,k_{2},\ldots ,k_{m}\right) \notin \Delta _{n}\), so \(\beta _{n}\left( k_{1}+1,k_{2},\ldots ,k_{m}\right) =0\) and
We deduce that
In the same way we get the remaining conditions in (ii). \(\square \)
7 Two Examples
Proposition 4
Let \(m\ge 2\) be a natural number, \( \varphi :\left[ 0,1\right] ^{m}\rightarrow \mathbb {R}\) a continuous function and \(K_{n}:\mathbb {R}^{m}\rightarrow \left[ 0,\infty \right) \) defined by
Then the following assertions are equivalent:
-
(i)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{m}\right) \) we have
$$\begin{aligned}{} & {} \lim \limits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{m}}\int _{ \left[ -\pi ,\pi \right] ^{m}}K_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) f\left( t_{1}-\theta _{1},\ldots ,t_{m}-\theta _{m}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{m}\\{} & {} \qquad =f\left( t\right) \end{aligned}$$uniformly with respect to \(t=\left( t_{1},\ldots ,t_{m}\right) \in \mathbb {R} ^{m} \).
-
(ii)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{m}\right) \) and all \(t=\left( t_{1},\ldots ,t_{m}\right) \in \mathbb {R}^{m}\) we have
$$\begin{aligned}{} & {} \lim \limits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{m}}\int _{ \left[ -\pi ,\pi \right] ^{m}}K_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) f\left( t_{1}-\theta _{1},\ldots ,t_{m}-\theta _{m}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{m}\\{} & {} \qquad =f\left( t\right) . \end{aligned}$$ -
(iii)
\(\int _{\left[ 0,1\right] ^{m}}\varphi ^{2}\left( t_{1},\ldots ,t_{m}\right) \textrm{d}t_{1}\cdot \cdot \cdot \textrm{d}t_{m}=1\).
Proof
Let us define \(\varphi _{n}:\left\{ 0,1,\ldots ,n\right\} ^{m}\rightarrow \mathbb {R}\) by \(\varphi _{n}\left( k_{1},\ldots ,k_{m}\right) =\frac{1}{\sqrt{ n^{m}}}\varphi \left( \frac{k_{1}}{n},\ldots ,\frac{k_{m}}{n}\right) \). We denote \(S_{n,0}=\sum \nolimits _{k_{1},\ldots ,k_{m}=0}^{n}\left[ \varphi _{n}\left( k_{1},\ldots ,k_{m}\right) \right] ^{2}\),
The condition \(\lim \nolimits _{n\rightarrow \infty }\sum \nolimits _{k_{1},\ldots ,k_{m}=0}^{n}\left[ \varphi _{n}\left( k_{1},\ldots ,k_{m}\right) \right] ^{2}=1\) is equivalent to \(\lim \nolimits _{n \rightarrow \infty }S_{n,0}=\lim \nolimits _{n\rightarrow \infty }\frac{1}{n^{m}} \sum \nolimits _{k_{1},\ldots ,k_{m}=0}^{n}\varphi ^{2}\left( \frac{k_{1}}{n},\ldots , \frac{k_{m}}{n}\right) =1\), that is \(\int _{\left[ 0,1\right] ^{m}}\varphi ^{2}\left( t_{1},\ldots ,t_{m}\right) \textrm{d}t_{1}\cdot \cdot \cdot \textrm{d}t_{m}=1\). Let \( \varepsilon >0\). Since \(\varphi \) is uniformly continuous, there exists \( \delta _{\varepsilon }>0\) such that if \(\max \nolimits _{1\le i\le n}\left| t_{i}-u_{i}\right| <\delta _{\varepsilon }\) we have \( \left| \varphi \left( t_{1},\ldots ,t_{m}\right) -\varphi \left( v_{1},\ldots ,v_{m}\right) \right| <\varepsilon \). There exists \( n_{\varepsilon }\in \mathbb {N}\) such that \(\frac{1}{n}<\delta _{\varepsilon } \) for all \(n\ge n_{\varepsilon }\). Let \(n\ge n_{\varepsilon }\). Then for all \(0\le k_{1}\le n-1\), \(1\le k_{2},\ldots ,k_{m}\le n\) we have
and hence
where \(\left\| \varphi \right\| =\sup \nolimits _{\left( t_{1},\ldots ,t_{m}\right) \in \left[ 0,1\right] ^{m}}\left| \varphi \left( t_{1},\ldots ,t_{m}\right) \right| \). We deduce that \(\lim \nolimits _{n \rightarrow \infty }[ S_{n,1}-S_{n,0}] =0\) and so \( \lim \nolimits _{n\rightarrow \infty }S_{n,1}=\lim \nolimits _{n\rightarrow \infty }S_{n,0}=\int _{\left[ 0,1\right] ^{m}}\varphi ^{2}\left( t_{1},\ldots ,t_{m}\right) \textrm{d}t_{1}\cdot \cdot \cdot \textrm{d}t_{m}=1\). By a similar reasoning we verify all the conditions (iii) from Theorem 4(iii). \(\square \)
Proposition 5
Let \(m\ge 2\) be a natural number, \( T_{m}=\{ \left( t_{1},\ldots ,t_{m}\right) \in \left[ 0,1\right] ^{m}\mid t_{1}+\cdot \cdot \cdot +t_{m}\le 1\} \), \(\varphi :T_{m}\rightarrow \mathbb {R}\) a continuous function and \(K_{n}:\mathbb {R}^{m}\rightarrow \left[ 0,\infty \right) \) defined by
Then the following assertions are equivalent:
-
(i)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{m}\right) \) we have
$$\begin{aligned}{} & {} \lim \limits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{m}}\int _{ \left[ -\pi ,\pi \right] ^{m}}K_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) f\left( t_{1}-\theta _{1},\ldots ,t_{m}-\theta _{m}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{m}\\{} & {} \qquad =f\left( t_{1},\ldots ,t_{m}\right) \end{aligned}$$uniformly with respect to \(\left( t_{1},\ldots ,t_{m}\right) \in \mathbb {R}^{m}\).
-
(ii)
For all \(f\in C_{2\pi }\left( \mathbb {R}^{m}\right) \) and all \(t=\left( t_{1},\ldots ,t_{m}\right) \in \mathbb {R}^{m}\) we have
$$\begin{aligned}{} & {} \lim \limits _{n\rightarrow \infty }\frac{1}{\left( 2\pi \right) ^{m}}\int _{ \left[ -\pi ,\pi \right] ^{m}}K_{n}\left( \theta _{1},\ldots ,\theta _{m}\right) f\left( t_{1}-\theta _{1},\ldots ,t_{m}-\theta _{m}\right) \textrm{d}\theta _{1}\cdot \cdot \cdot \textrm{d}\theta _{m}\\{} & {} \qquad =f\left( t\right) . \end{aligned}$$ -
(iii)
\(\int _{T_{m}}\varphi ^{2}\left( t_{1},\ldots ,t_{m}\right) \textrm{d}t_{1}\cdot \cdot \cdot \textrm{d}t_{m}=1\).
Proof
Let us define \(\varphi _{n}:\left\{ 0,1,\ldots ,n\right\} ^{m}\rightarrow \mathbb {R}\) by \(\varphi _{n}\left( k_{1},\ldots ,k_{m}\right) =\frac{1}{\sqrt{ n^{m}}}\varphi \left( \frac{k_{1}}{n},\ldots ,\frac{k_{m}}{n}\right) \). The condition \(\lim \nolimits _{n\rightarrow \infty }\sum \nolimits _{k_{1}+\cdot \cdot \cdot +k_{m}\le n}\left[ \varphi _{n}\left( k_{1},\ldots ,k_{m}\right) \right] ^{2}=1\) is equivalent to
that is \(\int _{\left[ 0,1\right] ^{m}}\varphi ^{2}\left( t_{1},\ldots ,t_{m}\right) \textrm{d}t_{1}\cdot \cdot \cdot \textrm{d}t_{m}=1\). As in the proof of Proposition 4 we verify all the conditions from Theorem 5(iii).\(\square \)
Data Availability
No datasets were generated or analyzed during the current study.
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Popa, D. The Convergence of Some Positive Linear Operators on the Space of Multivariate Continuous Periodic Functions. Mediterr. J. Math. 21, 152 (2024). https://doi.org/10.1007/s00009-024-02689-y
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DOI: https://doi.org/10.1007/s00009-024-02689-y
Keywords
- Korovkin approximation theorem
- continuous \(2\pi \)-periodic functions
- test set of convergence
- singular integral
- approximation of multivariate periodic functions
- rectangular method of summation
- triangular method of summation