Convergence Of Trigonometric Fourier Series Research Articles

Convergence of Trigonometric Fourier Series of Functions with a Constraint on the Fractality of Their Graphs

For a function f continuous on a closed interval, its modulus of fractality ν(f, e) is defined as the function that maps any e > 0 to the smallest number of squares of size e that cover the graph of f. The following condition for the uniform convergence of the Fourier series of f is obtained in terms of the modulus of fractality and the modulus of continuity ω(f, δ): if $$\begin{array}{*{20}{c}} {\omega (f,\pi /n)\ln \left( {\frac{{v(f,\pi /n)}}{n}} \right) \to 0}&{\text{as}}&{n \to + \infty ,} \end{array}$$ then the Fourier series of f converges uniformly. This condition refines the known Dini–Lipschitz test. In addition, for the growth order of the partial sums Sn(f, x) of a continuous function f, we derive an estimate that is uniform in x ∈ [0, 2π]: $${S_n}(f,x) = o\left( {\ln \left( {\frac{{v(f,\pi /n)}}{n}} \right)} \right).$$. The optimality of this estimate is shown.

On the generalized absolute convergence of Fourier series

Abstract In the present paper the sufficient conditions are obtained for the generalized r-absolute convergence ( 0 < r < 2 {0<r<2} ) of the single Fourier trigonometric series in terms of the modulus of δ-variation of a function. It is proved that these conditions are unimprovable in a certain sense. The classical results of Berstein, Szasz, Zygmund and others, related to the absolute convergence of single trigonometric Fourier series, were previously generalized by [L. Gogoladze and R. Meskhia, On the absolute convergence of trigonometric Fourier series, Proc. A. Razmadze Math. Inst. 141 2006, 29–40].

Absolute convergence of eigenfunction expansions of integral operators with kernels admitting discontinuities of derivatives on diagonals

In this paper, for expansions in eigenfunctions and associated functions, an analog of the known A. Zygmund theorem on the absolute convergence of trigonometric Fourier series is proved.

Absolute and uniform convergence of eigenfunction expansions of integral operators with kernels admitting derivative discontinuities on the diagonals

An analog of Szasz’s theorem on the absolute convergence of trigonometric Fourier series is established for expansions in the eigen and associated functions of integral operators some of whose kernels involve derivatives with discontinuities on the diagonals.

Absolute convergence of expansions in eigen- and adjoint functions of an integral operator with a variable limit of integration

We establish an analogue of Zygmund's criterion for the absolute convergence of trigonometric Fourier series for expansions in eigen- and adjoint functions of the integral operator .

On uniform and Lp-convergence of eigenfunction expansions for indefinite eigenvalue problems

We show that the classical results of L.Fejer and M. Riesz concerning norm convergence of trigonometric Fourier series can be carried over to eigenfunction expansions arising from regular indefinite boundary eigenvalue problems.

ON INTEGRAL NORMS FOR POLYNOMIALS

In this paper various transformations of trigonometrical polynomials are introduced; these are then used to deduce asymptotic formulas for the behavior of sequences of integrals of the moduli of the polynomials. As a consequence a definitive form is found for a relationship (in both directions) between the summability and absolute convergence of trigonometric Fourier series which has been noted earlier by the author.Bibliography: 26 titles.

Almost Everywhere Convergence of Fourier Series on the Ring of Integers of a Local Field

It is shown that the partial sums of the Fourier series of $L^p (\mathfrak{D})$-functions $(p > 1)$ converge almost everywhere (a.e.), where $\mathfrak{D}$ is the ring of integers in a local field K. This includes the case where K is a p-adic number field as well as the case where $\mathfrak{D}$ is the Walsh–Paley or dyadic group $2^\omega $. The techniques are essentially those used by Carleson [2] in establishing the a.e. convergence of trigonometric Fourier series for $L^2 ( - \pi ,\pi )$-functions as modified by Hunt [4] to obtain this same result for $L^p ( - \pi ,\pi )$-functions, $p > 1$. The necessary modifications for the local field setting are made in the context of the Sally’Taibleson [7] development of harmonic analysis on local fields and by use of Taibleson’s multiplier theorem [11]. These same results for $2^\omega $ have already been obtained by Billiard $(L^2 (2^\omega ))$ [1] and by Sjolin $(L^p (2^\omega ))$, $p > 1$) [8]. Many advantages (in particular the non-Archimidean nature of th...