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 of a ship's skin panel loaded along the axis of symmetry

Задача изгиба прямоугольной панели обшивки от действия распределенной по оси симметрии поперечной нагрузки не имеет точного решения в конечном виде в виду сложности краевых условий и вида нагрузки. Использование другими авторами различных приближенных методов оставляет открытым вопрос о точности полученных результатов. Целью исследования является получение точного решения с помощью гиперболо-тригонометрических рядов по двум координатам. Для этого используется метод бесконечной суперпозиции указанных рядов, которые в отдельности удовлетворят лишь части граничных условий. Порождаемые ими невязки взаимно компенсируются в ходе итерационного процесса и стремятся к нулю. Частное решения представлено двойным рядом Фурье. Точное решение достигается увеличением количества членов в рядах и числа итераций. При достижении заданной точности процесс прекращается. Получены численные результаты для прогибов и изгибающих моментов для квадратной пластины при различной длине загруженной части оси пластины. Представлены 3D-формы изогнутой поверхности пластины и эпюры изгибающих моментов. The problem of bending a rectangular skin panel from the action of a transverse load distributed along the axis of symmetry does not have an exact solution in the final form due to the complexity of the boundary conditions and the type of load. The use of various approximate methods by other authors leaves open the question of the accuracy of the results obtained. The aim of the study is to obtain an exact solution using hyperbolo-trigonometric series in two coordinates. To do this, we use the method of infinite superposition of these series, which individually satisfy only part of the boundary conditions. The residuals generated by them are mutually compensated during the iterative process and tend to zero. The quotient of the solution is represented by a double Fourier series. The exact solution is achieved by increasing the number of terms in the series and the number of iterations. When the specified accuracy is reached, the process stops. Numerical results are obtained for deflections and bending moments for a square plate with different lengths of the loaded part of the plate axis. 3D shapes of the curved surface of the plate and diagrams of bending moments are presented.

Improved double Fourier series on a sphere and its application to a semi-implicit semi-Lagrangian shallow water model

The computational cost of a spectral model using spherical harmonics (SH) increases significantly at high resolution because the transform method with SH requires O(N3) operations, where N is the truncation wavenumber. One way to solve this problem is to use double Fourier series (DFS) instead of SH, which requires O(N2 log N) operations. This paper proposes a new DFS method that improves the numerical stability of the model compared with the conventional DFS methods by adopting the following two improvements: a new expansion method that employs the least-squares method (or the Galerkin method) to calculate the expansion coefficients in order to minimize the error caused by wavenumber truncation, and new basis functions that satisfy the continuity of both scalar and vector variables at the poles. In the semi-implicit semi-Lagrangian shallow water model using the new DFS method, the Williamson test cases 2 and 5 and the Galewsky test case give stable results without the appearance of high-wavenumber noise near the poles, even without using horizontal diffusion. The new DFS model is faster than the SH model, especially at high resolutions, and gives almost the same results.

Estimates of Functions with Transformed Double Fourier Series

The paper provides a detailed study of inequalities of complete moduli of smoothness of functions with transformed Fourier series by moduli of smoothness of initial functions. Upper and lower estimates of the norms and best approximations of the functions with transformed Fourier series by the best approximations of initial functions are also obtained.

Trigonometric approximation of functions $f(x,y)$ of generalized Lipschitz class by double Hausdorff matrix summability method

In this paper, we establish a new estimate for the degree of approximation of functions $f(x,y)$ f ( x , y ) belonging to the generalized Lipschitz class $Lip ((\xi _{1}, \xi _{2} );r )$ L i p ( ( ξ 1 , ξ 2 ) ; r ) , $r \geq 1$ r ≥ 1 , by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from $Lip ((\alpha ,\beta );r )$ L i p ( ( α , β ) ; r ) and $Lip(\alpha ,\beta )$ L i p ( α , β ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and $(C, \gamma , \delta )$ ( C , γ , δ ) means.

Nonlocal Problem for a Three-dimensional Elliptic Equation with Singular Coefficients in a Rectangular Parallelepiped

The nonlocal problem for an elliptic equation with two singular coefficients in a rectangular parallelepiped is studied. The uniqueness of the solution of the problem is proved with the use of the method of energy integrals. The spectral Fourier method based on the separation of variables is used to prove the existence of solutions. The solution of the problem is constructed as double Fourier series in terms of a sum of trigonometric and Bessel functions. Under some conditions on parameters and given functions the uniform convergence of the constructed series and its derivatives up to the second order inclusive is proved

Approximation of a Function in Hölder Class Using Double Karamata (Kλ,μ) Method

In this paper, we establish a new theorem on the best approximation of a function of two variables belonging to H ̈older class by double Karamata (Kλ,μ) means of its double Fourier series.