On spherical convergence of numerical and functional series (Q1358330)

Chandrasekharan and Minakshisundaram proved that the double Fourier series of the function \(f(x,y):= f(x)g(y)\), where \(f\) and \(g\) are periodic functions of bounded variation, is spherically convergent everywhere. In this case, the order of magnitude of the corresponding Fourier coefficients \(\widehat f(n)\) and \(\widehat g(n)\) is \(O(1/n)\). Now, the present author proves the following Theorem 1. Let \(\{a_n\}\) and \(\{b_n\}\) be sequences of numbers such that the series \(\sum a_n\) and \(\sum b_n\) converge to \(a\) and \(b\), respectively, and \(a_n,b_n= O(1/n)\). Then the double series \(\sum\sum a_mb_n\) is spherically convergent to \(ab\), that is, the limit \[ \lim_{R\to\infty} \sum_{m^2+ n^2\leq R^2} a_mb_n= ab \] exists. This result is exact in the following sense. Theorem 2. Let \(\{c_n\geq 0\}\) be an arbitrary sequence of numbers such that \(\lim c_n=\infty\). Then there exists a sequence \(\{a_n\}\) of numbers such that the series \(\sum a_n\) converges, \(|na_n|\leq c_n\) for all \(n\geq 1\), and the double series \(\sum\sum a_ma_n\) is spherically divergent. The author deduces interesting corollaries for functional series with respect to bounded orthonormal systems, in particular, for trigonometric Fourier series.