On the strong summability of double orthogonal series
type: Item
instance of: scholarly article
title: On the strong summability of double orthogonal series
zbMATH Open document ID: 0596.42009
published in: Michigan Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The well-known results of P. R. Agnew, L. Csernyák, V. S. Fedulov, K. Tandori and author on the convergence and the order of magnitude of the partial sums \(S_{m,n}(x)\) and the first arithmetic means of the double orthogonal series \(\sum^{\infty}_{i=1}\sum^{\infty}_{k=1}a_{ik}\phi_{ik}(x),\) defined on an arbitrary positive measure spaces (X,F,\(\mu)\), are extended to the case of the strong means. Two typical results of the paper are following: Theorem 1. Under condition \[ \sum^{\infty}_{i=1}\sum^{\infty}_{k=1}a^ 2_{ik}[\log \log (i+3)]^ 2[\log \quad \log (k+3)]^ 2<\infty, \] \(\frac{1}{MN}\sum^{M}_{m=1}\sum^{N}_{n=1}[S_{m,n}(x)-f(x)]^ 2\to 0\) a.e. as \(\min (M,N)\to \infty\) and there exists a function \(F(x)\in L^ 2(X,F,\mu)\) such that \[ \sup_{M,N\geq 1}\{\frac{1}{MN}\sum^{M}_{m=1}\sum^{N}_{n=1}[S_{m,\quad n}(x)- f(x)]^ 2\}^{1/2}\leq F(x) \] a.e. Theorem 2. Under condition \(\sum^{\infty}_{i=1}\sum^{\infty}_{k-1}a^ 2_{ik}<\infty,\) \[ H_{M,N}(x)=\{\frac{1}{MN}\sum^{M}_{m=1}\sum^{N}_{n=1}S^ 2_{m,n\quad}(x)\}^{1/2}/(\log \log (M+3)\log \log (N+3))\to 0 \] a.e. as max(M,N)\(\to \infty\) and there exists a function \(F(x)\in L^ 2(X,F,\mu)\) such that \(\sup_{M,N\geq 1}H_{M,N}(x)\leq F(x)\) a.e. The first of them can be also considered as an extension of a theorem of \textit{S. Borgen} [Math. Ann. 98, 125-150 (1928)] from single orthogonal series to double ones.
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 40F05
zbMATH DE Number: 3960224
zbMATH Keywords: arithmetic means
zbMATH Keywords: double orthogonal series
author: Ferenc Móricz
reviewed by: W. Łenski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003027
OpenAlex ID: W1981980351
DOI: 10.1307/MMJ/1029003027
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description: scientific article; zbMATH DE number 3960224