Generalization of the Hardy-Littlewood theorem on Fourier series

In the theory of one-dimensional trigonometric series, the Hardy-Littlewood theorem on Fourier series with monotone Fourier coefficients is of great importance. Multidimensional versions of this theorem have been extensively studied for the Lebesgue space. Significant differences of the multidimensional variants in comparison with the one-dimensional case are revealed and the strengthening of this theorem is obtained. The Hardy-Littlewood theorem is also generalized for various function spaces and various types of monotonicity of the series coefficients. Some of these generalizations can be seen in works of M.F. Timan, M.I. Dyachenko, E.D. Nursultanov, S. Tikhonov. In this paper, a generalization of the Hardy-Littlewood theorem for double Fourier series of a function in the space L_qϕ(L_q)(0,2π]^2 is obtained.

Bending Analysis of Isotropic Rectangular Kirchhoff Plates Subjected to Non-Uniform Heating Using the Fourier Transform Method

The object of this paper is the bending analysis of isotropic rectangular Kirchhoff plates subjected to non-uniform heating (NUH) using the Fourier transform method. The bottom and top surfaces of the plate are assumed to have different changes in temperature, whereas the change in temperature of the mid-surface is zero. According to classical plate theory, the governing equation of the plate contains second derivatives of the NUH; these derivatives are zero by constant value of the NUH, which leads to its absence in the governing equation. This paper presented an approach by which Fourier sine transform was utilized to describe the NUH, while the double trigonometric series of Navier and the simple trigonometric series of Lévy were utilized to describe the deflection curve. Thus, the NUH appeared in the governing equation, which simplified the analysis. Rectangular plates simply supported along all edges were analyzed, bending moments, twisting moments, and deflections being determined. In addition, rectangular plates simply supported along two opposite edges were analyzed; the other edges having various support conditions (free, simply supported, and fixed).

On uniform convergence of one class of double trigonometric series

We have established sufficient conditions of uniform convergence of the series of the form $\sum\limits_{k,l=1} a_{k,l} \sin kx \sin ly$ in the strip: $-\infty < x < +\infty$, $\delta \leqslant y \leqslant 2\pi - \delta$ ($\delta$ is a fixed number, $0 < \delta < \pi$).

Refinements of Wilker–Huygens-Type Inequalities via Trigonometric Series

The study of even functions is important from the symmetry theory point of view because their graphs are symmetrical to the Oy axis; therefore, it is essential to analyse the properties of even functions for x greater than 0. Since the functions involved in Wilker–Huygens-type inequalities are even, in our approach, we use cosine polynomials expansion method in order to provide new refinements of the above-mentioned inequalities.

Fourier transforms on weighted amalgam type spaces

We introduce weighted amalgam-type spaces and analyze their relations with some known spaces. Integrability results for the Fourier transform of a function with the derivative from one of those spaces are proved. The obtained results are applied to the integrability of trigonometric series with the sequence of coefficients of bounded variation.

Bending Analysis of Isotropic Rectangular Kirchhoff Plates Subjected to a Thermal Gradient Using the Fourier Transform Method

The object of this paper is the bending analysis of isotropic rectangular Kirchhoff plates subjected to a thermal gradient (TG) using the Fourier transform method. The bottom and top surfaces of the plate are assumed to have different changes in temperature, whereas the change in temperature of the mid-surface is zero. According to classical plate theory, the governing equation of the plate contains second derivatives of the TG; these derivatives are zero by constant value of the TG, which leads to the absence of the TG in the governing equation. This paper presented an approach by which Fourier sine transform was utilized to describe the TG, while the double trigonometric series of Navier and the simple trigonometric series of L&eacute;vy were utilized to describe the deflection. Thus, the TG appeared in the governing equation, which simplified the analysis. Rectangular plates simply supported along all edges were analyzed, bending moments, twisting moments, and deflections being determined. In addition, rectangular plates simply supported along two opposite edges were analyzed, the other edges having various support conditions (free, simply supported, and fixed).