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Some integrability theorems for multiple trigonometric series. (English) Zbl 1250.42028
Among others, the following two theorems are proved.
Theorem 2.4. Let \(\{a_k\: k=1,2,\ldots \}\) be a sequence of real numbers such that \(\sum | a_k| < \infty \). Then \[ \int ^{\pi }_0 | {1\over x} \sum ^{\infty }_{k=1} a_k (1-\cos 2^k x)| dx <\infty \] if and only if \[ \sum ^{\infty }_{k=0} \Big \{\Big (\sum ^{\infty }_{j=k+1} a_j\Big )2 + \sum ^{\infty }_{j=k+1} a^2_j\Big \}^{1/2} < \infty . \]
Theorem 2.5. Let \(\{b_k\: k=1,2,\ldots \}\) be a null sequence of real numbers such that \(b_k - b_{k+1} = 0\) for \(k\in \mathbb {N} \backslash \{2^r: r\in \mathbb {N}\}\) and \(\sum ^{\infty }_{k=1} | b_k - b_{k+1}| <\infty \). Then \(\sum ^{\infty }_{k=1} b_k \sin kx\) is a Fourier series if and only if \[ \sum ^{\infty }_{k=0} \Big \{\sum ^{\infty }_{j=k+1} (b_{2^j} - b_{2^j+1})2 + \Big (\sum ^{\infty }_{j=k+1} (b_{2^j} - b_{2^j+1})\Big )^2\Big \}^{1/2} < \infty . \] Higher-dimensional analogues of Theorems 2.4 and 2.5 are also proved.

42B05 Fourier series and coefficients in several variables
40B05 Multiple sequences and series (should also be assigned at least one other classification number in this section)
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