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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

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