On \(\Lambda \)-convergence almost everywhere of multiple trigonometric Fourier series. (English) Zbl 1450.42001

Let \(\Lambda=\{\lambda_{ \nu}\}_{\nu=1}^{\infty}\) be a nonincreasing sequence of positive numbers. Set \[ \Omega_{\Lambda}=\{\mathfrak{n}=(n^1, n^2, \dots, n^d)\in\mathbb{N}^d:\, \frac{1}{1+\lambda_{n^i}}\leq \frac{n^i}{n^j}\leq 1+\lambda_{n^j}, \,\, 1\leq i, j\leq d\}. \] We will say that a multiple Fourier series of a function \(f\in L(\mathfrak{T}^d)\) is \(\Lambda\)-convergent at a point \(\mathfrak{x}\in\mathfrak{T}^d\) if there exists a limit \[ \lim_{\mathfrak{n} \in \Omega_{\Lambda},\,\min\{n^j: 1\leq j \leq d\}\to \infty } S_{\mathfrak{n}}(f, \mathfrak{x}), \] where \(S_{\mathfrak{n}}(f,\mathfrak{x})\) are the rectangular partial sums of the trigonometric Fourier series of \(f\).
It is clear that \(\Lambda\)-convergence is a generalization of the classical \(\lambda\)-convergence (\(\lambda>1\)) and that it is stronger than convergence over cubes. Earlier the author [Izv. Sarat. Univ. (N.S.), Ser. Mat. Mekh. Inform. 14, No. 4, Part 2, 497–505 (2014; Zbl 1310.42003)] proved that if a sequence \(\Lambda=\{\lambda_{\nu}\}_{\nu=1}^{\infty}\) satisfies the condition \(\ln^2\lambda_{\nu}=o(\ln \nu)\) as \(\nu \to \infty,\) then there exists a function \(F\in C(\mathfrak{T}^2)\) such that its Fourier series is \(\Lambda\)-divergent almost everywhere on \(\mathfrak{T}^2\).
In the present paper it is established that if \(d\geq 2\) and \(\lambda_{\nu}=O (1/\nu)\) then the Fourier series of any function from the class \[ L(\ln+ L)^d(\ln^+ \ln^+ \ln^+ L)(\mathfrak{T}^d) \] is \(\Lambda\)-convergent almost everywhere on \(\mathfrak{T}^d\).


42A20 Convergence and absolute convergence of Fourier and trigonometric series


Zbl 1310.42003
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