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On the integrability of double Walsh series with special coefficients. (English) Zbl 0714.42017
We study double Walsh series (1) \(\sum \sum a_{jk}w_ j(x)w_ k(y)\), where \(\{a_{jk}:j,k=0,1,...\}\) is a sequence of real numbers such that \((2)\quad a_{jk}\to 0\quad as\quad j+k\to \infty\) and \((3)\quad \sum \sum | \Delta_{11}a_{jk}| <\infty,\) where \(\Delta_{11}a_{jk}=a_{jk}-a_{j+1,k}-a_{j,k+1}+a_{j+1,k+1}.\) We prove that under conditions (2), (3) series (1) converges to a finite limit \(f(x,y)\) for all \(0<x,y<1,\) but f is not Lebesgue integrable in general. Our Theorem 2 shows that conditions (2), (3) are sufficient for the integrability of f in the sense of the improper Riemann integral, and in addition, series (1) is the Walsh-Fourier series of f in the same sense: for all \(j,k\geq 0,\) we have \[ \int^{1}_{\delta}\int^{1}_{\epsilon}f(x,y)w_ j(x)w_ k(y)dx dy\to a_{jk}\quad as\quad \delta,\epsilon \to 0. \]
Reviewer: F.Móricz

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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