## 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 $$\mathfrak{T}^d$$.

### MSC:

 42A20 Convergence and absolute convergence of Fourier and trigonometric series

Zbl 1310.42003
Full Text:

### References:

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