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Weighted integrability and $$L^ 1$$-convergence of multiple trigonometric series. (English) Zbl 0821.42007
The author considers the double trigonometric series $$(*)$$ $$\sum_ j \sum_ k c_{jk} e^{i(jx + ky)}$$, where $$j$$ and $$k$$ run through $$\mathbb{Z}$$; and assumes that $$c_{jk} \to 0$$ and $$| j | + | k | \to \infty$$ and $$\sum_ j \sum_ k \theta (| j | + 1) \vartheta (| k | + 1) | \Delta_{11} c_{jk} | < \infty$$, where $$\theta (t)$$ and $$\vartheta (t)$$ are positive, nondecreasing functions on $$[1, \infty)$$, and $$\Delta_{11} c_{jk} = c_{jk} - c_{j + 1,k} - c_{j,k + 1} + c_{j + 1, k + 1}$$. He proves that then the rectangular partial sums $$s_{mn} (x,y)$$ of the series $$(*)$$ converge to some function $$f(x,y)$$ for $$x \neq 0$$ and $$y \neq 0$$, $$f(x,y) \varphi (x) \psi (y) \in L^ 1 (T^ 2)$$, and $\iint_{T^ 2} \bigl | s_{mn} (x,y) - f(x,y) \bigr | \varphi (x) \psi (y) dx dy \to 0 \quad \text{as} \quad \min (m,n) \to \infty,$ where $$T^ 2 = \{(x,y) \in R^ 2 : - \pi \leq x,y < \pi\}$$, and $$(\varphi, \theta)$$ and $$(\psi, \vartheta)$$ are of type I functions (in the sense as defined in the paper).
Extension of these results to double series of orthogonal functions is also considered.
Reviewer: F.Móricz (Szeged)

##### MSC:
 42B05 Fourier series and coefficients in several variables
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