ON THE ORDER OF SUMMABILITY OF THE FOURIER INVERSION FORMULA

In this article we show that the order of the point value, in the sense of Lojasiewicz, of a tempered distribution and the order of summability of the pointwise Fourier inversion formula are closely related. Assuming that the order of the point values and certain order of growth at infinity are given for a tempered distribution, we estimate the order of summability of the Fourier inversion formula. For Fourier series, and in other cases, it is shown that if the distribution has a distributional point value of order k, then its Fourier series is e.v. Cesaro summable to the distributional point value of order k＋1. Conversely, we also show that if the pointwise Fourier inversion formula is e.v. Cesaro summable of order k, then the distribution is the （k ＋ 1）-th derivative of a locally integrable function, and the distribution has a distributional point value of order k ＋ 2. We also establish connections between orders of summability and local behavior for other Fourier inversion problems.     

On the absolute summability of Fourier series of almost-periodic besicovitch functions

For almost-periodic Besicovitch functions whose spectrum has a limit point only at infinity, we establish criteria for the absolute Cesàro summability of their Fourier series of order greater than –1.     

Distributional versions of littlewood’s Tauberian theorem

We provide several general versions of Littlewood’s Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We employ two types of Tauberian hypotheses; the first kind involves distributional boundedness, while the second type imposes a one-sided assumption on the Cesàro behavior of the distribution. We apply these Tauberian results to deduce a number of Tauberian theorems for power series and Stieltjes integrals where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.     

On the absolute Cesàro summability of Fourier series of functions of Lebesgue classL v and some related problems in the theory of Fourier constants

Summary The author studies certain aspects of a problem on Fourier constants which emerges out of his attempt at proving a recent result ofTsuchikura on the absolute Cesàro summability of Fourier series by means of a technique that exploits properties of Fourier costants. He proves,inter alia, the Fourier-power series analogue of aconjecture on Fourier constants, which he has raised in this paper.     

Summability of multiple Fourier series of functions of bounded generalized variation

The paper is devoted to summability problems for multiple Fourier series of functions of bounded generalized variation. We find necessary and sufficient conditions on a sequence Λ = {λ _n } that guarantee the convergence of the Cesàro means of multiple Fourier series of functions of bounded Λ-variation.     

Pointwise convergence of strong summability on sphere

In this paper, strong summability of Cesàro means (of critical order) of Fourier-Laplace series on unit sphere is discussed. The Pointwise convergence conditions are established. The results of this paper are analogous to those of single and multiple Fourier series.     

Strong universal consistency of smooth kernel regression estimates

The paper deals with kernel estimates of Nadaraya-Watson type for a regression function with square integrable response variable. For usual bandwidth sequences and smooth nonnegative kernels, e.g., Gaussian and quartic kernels, strongL ₂-consistency is shown without any further condition on the underlying distribution. The proof uses a Tauberian theorem for Cesàro summability.     

(<Emphasis Type="Italic">C</Emphasis>, 1)-Summation of fourier series over rearranged Vilenkin system

The Cesàro means of a function f with respect to the Vilenkin system in the Kaczmarz rearrangement are considered. We prove that these means of a summable function converge to it almost everywhere.     

On the Cesàro summability of the ultraspherical series

Summary The author has obtained theorems for Cesàro summability of the ultraspherical series which are analogous to those of Izumi and Sunouchi (2) in Fourier series.     

Continuity in Λ-variation and summation of multiple Fourier series by Cesàro methods

We construct new examples of functions of bounded Λ-variation not continuous in Λ-variation. Using these examples, we show that, in the problem of the summability of multiple Fourier series by the Cesàro method of negative order, the condition of continuity in Λ-variation, is essential in contrast to the one-dimensional case.     

On Cesàro summability of vector valued multiplier spaces and operator valued series

In this paper, we introduce and study vector valued multiplier spaces with the help of the sequence of continuous linear operators between normed spaces and Cesàro convergence. Also, we obtain a new version of the Orlicz–Pettis Theorem by means of Cesàro summability.     

Power series with radius of convergence zero — Overconvergence,elongations, summability

We deal with overconvergence phenomena of power series with radius of convergence zero. Among others it is shown that the partial sums of such a series can be elongated to become Cesàro summable on a set S ⊂ {z: |z| > 0} if and only if the considered power series is overconvergent.     

Fourier Series and Approximation on Hexagonal and Triangular Domains

Several problems on Fourier series and trigonometric approximation on regular hexagonal and triangular domains are studied. The results include Abel and Cesàro summability of Fourier series, degree of approximation, and best approximation by trigonometric functions with both direct and inverse theorems. One of the objectives of this study is to demonstrate that Fourier series on spectral sets enjoy a rich structure that permits an extensive theory for Fourier series and approximation.     

Fourier Inversion of Distributions Supported by a Hypersurface

Let μ _Σ be the natural measure on R ^N (N≥3) supported by a compact oriented analytic hypersurface Σ, ψ a smooth function on R ^N and P(D) a differential operator in N variables of order m. We determine a sufficient condition on the number λ such that the Fourier integral of the distribution P(D)ψ μ _Σ be summable by Cesàro means of order λ to zero in a point outside the hypersurface. This condition depends on m and on the position of the point with respect to the caustic of the hypersurface.     

A remark about the statistical Cesàro summability and the Orlicz-Pettis theorem

We prove a version of the Orlicz-Pettìs theorem within the frame of the Statistical Cesàro summability.     

Summability of Fourier-Laplace Series with the Method of Lacunary Arithmetical Means at Lebesgue Points

Let ∑ _{n −1} be the unit sphere in the n-dimensional Euclidean space ℝ ⁿ . For a funcion ƒ∈L(∑ _{n −1}) denote by σ^δ _N (ƒ) the Cesàro means of order δ of the Fourier-Laplace series of ƒ. The special value of δ is known as the critical index. In the case when n is even, this paper proves the existence of the ‘rare’ sequence {n _k } such that the summability takes place at each Lebesgue point satisfying some antipole conditions. Received June 28, 1999, Revised August 11, 1999, Accepted February 16, 2000     

Rate of pointwise approximation of functions by the Cesàro (C, β)-means of their Fourier series

e consider the interrelation between the smoothness of a function at a fixed point with the behavior at this point of the Cesàro (C, β)-means of the Fourier series of this function.     

Application of Banach limits to the study of sets of integers

The Kamae and Mendes France version of the Van der Corput equidistribution theorem is extended further to summability methods different from Cesàro summability and groups different from the circle. The theorem is shown to follow naturally from consideration of Banach limits and spectral theory.