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Zbl 1043.15020
Barza, Sorina; Persson, Lars-Erik; Popa, Nicolae
A matriceal analogue of Fejér's theory. (English)
[J] Math. Nachr. 260, 14-20 (2003). ISSN 0025-584X; ISSN 1522-2616/e

Let $A=(a_{ij}), i,j\geq 0$ be an infinite matrix with complex entries and let $n\geq 0.$ The matrix $A$ is called of $n$-band type if $a_{ij}=0$ for $\vert i-j\vert >n.$ For $k=0,\pm 1,\pm 2,\cdots ,$ let $A_k=(a'_{ij}),$ where $a'_{ij}=a_{ij}$ if $j-i=k;$ $a_{ij}'=0,$ otherwise. $A_k$ is called the Fourier coefficient of the matrix $A.$ Let $B(\ell_2)$ be the space of bounded linear operators on the one-sided scalar $\ell_2$-space and $A$ be a matrix corresponding to an operator on $B(\ell_2).$ Analogously to Fejér's theory for Fourier series, the authors define Cesaro sum by $$\sigma_n(A)=\sum_{k=-n}^{k=n}A_k\left (1-\frac{\vert k\vert }{n+1}\right ).$$ Then they say that $A$ is a continuous matrix if $\lim_{n\rightarrow \infty }\Vert \sigma _n(A)-A\Vert _{B(\ell_2)}=0.$ In this way the authors extend the classical Banach space of functions $C(T)$ on the torus $T$ to the space of continuous matrices. The space of all continuous matrices is denoted by $C(\ell_2).$ In Section 3, the authors derive some properties and relations between the spaces $B(\ell_2)$ and $C(\ell_2).$ Similar extensions are considered for the space $L^1(T).$ Theorem 4.2, Theorem 4.3 and Theorem 4.10 are the main results.
[Jaspal Singh Aujla (Jalandhar)]

MSC 2000:: *15A57 Other types of matrices
42A16 Fourier coefficients, etc.
42A45 Multipliers, one variable
47B35 Toeplitz operators, etc.
47B37 Operators on sequence spaces, etc.

Keywords: Fourier series; Fejér's theory; Cesaro sums; infinite matrices; Toeplitz matrices; Schur multipliers; Fourier coefficient

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Zentralblatt MATH Berlin [Germany]