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Cím:On the integrability of double Walsh series with special coefficients.
Szerző:M\'oricz, F.; Schipp, F.; Wade, W.R.
Forrás:Mich. Math. J. 37, No.2, 191-201 (1990).
Nyelv:English
Absztrakt:We study double Walsh series (1) $\sum \sum a\sb{jk}w\sb j(x)w\sb k(y)$, where $\{a\sb{jk}:j,k=0,1,...\}$ is a sequence of real numbers such that $(2)\quad a\sb{jk}\to 0\quad as\quad j+k\to \infty$ and $(3)\quad \sum \sum \vert \Delta\sb{11}a\sb{jk}\vert <\infty,$ where $\Delta\sb{11}a\sb{jk}=a\sb{jk}-a\sb{j+1,k}-a\sb{j,k+1}+a\sb{j+1,k+1}.$ 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 $j,k\ge 0,$ we have $$ \int\sp{1}\sb{\delta}\int\sp{1}\sb{\epsilon}f(x,y)w\sb j(x)w\sb k(y)dx dy\to a\sb{jk}\quad as\quad \delta,\epsilon \to 0.$$} \RV{F.M\'oricz 
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