Some integrability theorems for multiple trigonometric series. (English) Zbl 1250.42028

Lee, Tuo-Yeong

Math. Bohem. 136, No. 3, 269-286 (2011).

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.

Reviewer: Ferenc Moricz (Szeged)

MSC:

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)

Keywords:

multiple cosine series; sine series; regular convergence; integrability theorem