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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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