On the strong summability of double orthogonal series
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.
scientific article; zbMATH DE number 3960224