Convergence Of Double Fourier Series Research Articles

Rate of convergence of double Fourier series of functions of generalized bounded variation

In this paper, the rate of convergence of the double Fourier series of a function 2\(\pi\)-periodic in each variable and of generalized bounded variation is estimated. The result obtained is a generalization of a result of Bojanić and Waterman for a single Fourier series and of a result of F. Móricz and consequently a result of Hardy for a function of two variables.

On weighted β-absolute convergence of double Fourier series

On weighted β-absolute convergence of double Fourier series

English

We extend the results of paper of F. Moricz (2010), where necessary conditions were given for the $L^1$-convergence of double Fourier series. We also give necessary and sufficient conditions for the $L^1$-convergence under appropriate assumptions.

Double Fourier Series

In this paper , we introduce the double Fourier series for functions of two variables , and we study the double Fourier series for even and odd functions , then convergence of double Fourier series is investigated .We discuss also the Parsevals identity for double Fourier series , Parsevals identity for even and odd functions .

Norm convergence of double Fourier series on unbounded Vilenkin groups

We study approximation by rectangular partial sums of double Fourier series on unbounded Vilenkin groups in the spaces C and L1. From these results we obtain criterions of the uniform convergence and L-convergence of double Vilenkin–Fourier series. We also prove that these results are sharp.

Sharp estimates for the rate of convergence of double Fourier series in classical orthogonal polynomials

Sharp estimates are obtained for the convergence rate of “triangular” and “hyperbolic” partial sums of Fourier series in orthogonal (Laguerre, Hermite, Jacobi) polynomials in the classes of differentiable functions of two variables characterized by a generalized modulus of continuity. The proofs are based on the generalized shift operator and generalized modulus of continuity for functions from L2 having generalized partial derivatives in Levi’s sense.

이중 푸리에 급수의 총합가능성과 수렴성에 대한 고전적인 연구들에 관하여

G. H. Hardy laid the foundation of classical studies on double Fourier series at the beginning of the 20th century. In this paper we are concerned not only with Fourier series but more generally with trigonometric series. We consider Norlund means and Cesaro summation method for double Fourier Series. In section 2, we investigate the classical results on the summability and the convergence of double Fourier series from G. H. Hardy to P. Sjolin in the mid-20th century. This study concerns with the <TEX>$L^1(T^2)$</TEX>-convergence of double Fourier series fundamentally. In conclusion, there are the features of the classical results by comparing and reinterpreting the theorems about double Fourier series mutually.

Convergence of double Fourier series and generalized Λ-variation

The paper introduces a new concept of Λ-variation of bivariate functions and investigates its connection with the convergence of double Fourier series.

Convergence of double Fourier series of functions from symmetric spaces

We establish conditions for the convergence of double Fourier series in the trigonometric system of functions belonging to a symmetric space.

Singular Integrals on Product Spaces Related to the Carleson Operator

AbstractWe prove Lp(T2) boundedness, 1 &lt; p ≤ 2, of variable coefficients singular integrals that generalize the double Hilbert transform and present two phases that may be of very rough nature. These operators are involved in problems of a.e. convergence of double Fourier series, likely in the role played by the Hilbert transform in the proofs of a.e. convergence of one dimensional Fourier series. The proof due to C.Fefferman provides a basis for our method.

Convergence of double Fourier series and $W$-classes

The double Fourier series of functions of the generalized bounded variation class {n/ln(n + 1)}*BV are shown to be Pringsheim convergent everywhere. In a certain sense, this result cannot be improved. In general, functions of class A* BV, defined here, have quadrant limits at every point and, for f E A*BV, there exist at most countable sets P and Q such that, for x ¬∈ P and y ¬∈ Q, f is continuous at (x, y). It is shown that the previously studied class ABV contains essentially discontinuous functions unless the sequence A satisfies a strong condition.

Convergence of Double Fourier Series after a Change of Variable

In this paper, we prove that for any compact set \(\Omega \subset C({\mathbb{T}}^2 )\) there exists a homeomorphism τ of the closed interval \({\mathbb{T}} = [ - \pi ,\pi ]\) such that for an arbitrary function f ∈ Ω the Fourier series of the function F(x,y) = f(τ(x),τ(y)) converges uniformly on \(C({\mathbb{T}}^2)\) simultaneously over rectangles, over spheres, and over triangles.

On the convergence of double Fourier series of functions in LP[0, 1]2, where p=(p1, p2)

Pointwise convergence of double trigonometric Fourier series of functions in the Lebesgue space Lp[0, 2π]2 was studied by M.I.Dyachenko. In this paper, we also consider the problems of the convergence of double Fourier series in Pringsheim's sense with respect to the trigonometric as well as the Walsh systems of functions in the Lebesgue space LP[0, 1]2, p=(p1, p2), endowed with a mixed norm, in the particular case when the coefficients of the series in question are monotone with respect to each of the indices.

Uniform convergence of double fourier series for classes of functions with anisotropic smoothness

Uniform convergence of double fourier series for classes of functions with anisotropic smoothness

-convergence almost everywhere of double Fourier series

In this paper we consider -convergence of double Fourier series (-convergence of a double number series implies that it converges in the sense of Pringsheim, over spheres, and so on.) Unextendable classes of Weyl multipliers for -convergence almost everywhere are described. In addition, close to exact sufficient conditions for -convergence almost everywhere in the spaces and are found.

A boundary discontinuous fourier solution for clamped transversely isotropic (pyrolytic graphite) mindlin plates

An analytical solution to the long-standing boundary-value problem of a shear-flexible (moderately-thick) rigidly clamped transversely isotropic rectangular plate, subjected to transverse loading, is presented. A recently developed accurate yet computationally efficient boundary discontinuous Fourier series technique (BDFST) has been utilized to solve the three highly coupled second-order partial differential equations with constant coefficients that result from the Mindlin hypothesis. Numerical results presented (i) testify to the accuracy and computational efficiency of the above-mentioned solution methodology, (ii) help in understanding the nature of convergence of double Fourier series in the presence of edge discontinuities introduced by the fully clamped boundary conditions, (iii) ascertain the limit of applicability of the classical plate theory (CPT), and (iv) provide physical insight into such complex deformation behaviors as the effect of transverse shear deformability and thickness on the deformation of rigidly clamped moderately-thick plates of metallic (isotropic) and pyrolytic graphite (transversely isotropic) constructions.

Convergence of double Fourier series with coefficients of generalized bounded variation

We extend the class of double null sequences of complex numbers that are of bounded variation and prove the almost everywhere pointwise convergence as well as the convergence in the L r(T 2) -metric for 0 < r < 1 of double Fourier series with such coefficients. We study mainly convergence in Pringsheim's sense, either unrestricted or restricted, but regular convergence is also considered.

Variants of the maximal double Hilbert transform

We prove the boundedness on L p ( T 2 ) {L_p}({T^2}) , 1 &gt; p &gt; ∞ 1 &gt; p &gt; \infty , of two variants of the double Hilbert transform and maximal double Hilbert transform. They have an application to a problem of almost everywhere convergence of double Fourier series.

Vector-valued inequalities for Fourier series

Denoting by S ∗ {S^\ast } the maximal partial sum operator of Fourier series, we prove that S ∗ ( f 1 , f 2 , … , f k , … ) = ( S ∗ f 1 , S ∗ f 2 , … , S ∗ f k , … ) {S^\ast }({f_1},{f_2}, \ldots ,{f_k}, \ldots ) = ({S^\ast }{f_1},{S^\ast }{f_2}, \ldots ,{S^\ast }{f_k}, \ldots ) is a bounded operator from L p ( l r ) {L^p}({l^r}) to itself, 1 &gt; p , r &gt; ∞ 1 &gt; p,r &gt; \infty . Thus, we extend the theorem of Carleson and Hunt on pointwise convergence of Fourier series to the case of vector valued functions. We give also an application to the rectangular convergence of double Fourier series.

On the convergence of double Fourier series of functions from Lp, p &gt; 1

It is proved that if a function from Lp, p > 1, satisfies the condition $$\frac{1}{{t \cdot \tau }}\int_0^t {\int_0^\tau {\left| {f(x + u,y + v) - f(x,y)} \right|} dudv = O\left( {\left[ {1n\frac{1}{{(t^2 + \tau ^2 )}}} \right]^{ - 2} } \right),}$$ then the double Fourier series of function f, under summation over a rectangle, converges almost everywhere.