Summation Of Multiple Fourier Series Research Articles

Rectangular Summation of Multiple Fourier Series and Multi-parametric Capacity

We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichlet space functions. As a by-product of the methods of proof, the results also hold in the vector-valued setting.

Summation of Multiple Fourier Series in Matrix Weighted -Spaces

This paper is concerned with rectangular summation of multiple Fourier series in matrix weighted -spaces. We introduce a product Muckenhoupt condition for matrix weights and prove that rectangular Fourier partial sums converge in the corresponding matrix weighted space , , if and only if the weight satisfies the product Muckenhoupt condition. The same result is shown to hold true for other summation methods such as Cesàro and summation with the Jackson kernel.

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 Cesaro method of negative order, the condition of continuity in Λ-variation, is essential in contrast to the one-dimensional case.

Summation of multiple Fourier series in multiplicative systems

Summation of multiple Fourier series in multiplicative systems

A radial multiplier and a related Kakeya maximal function

In this paper we state some results for a maximal function and a Fourier multiplier that are connected with the Bochner-Riesz spherical summation of multiple Fourier series (see Fefferman [3], [5] ). Our purpose will be to get sharp estimates for the norm of these operators in dimension two. Proofs will appear elsewhere [2]. Let N > be a real number. By a rectangle of eccentricity N we mean a rectangle R such that Length of the bigger side of R __ .. Length of the smaller side of R We will define the direction of R as the direction of its bigger side. Given a locally integrable function ƒ we consider the maximal function

On the summation of multiple Fourier series

On the summation of multiple Fourier series

On the Summation of Multiple Fourier Series, IV

On the Summation of Multiple Fourier Series, IV

Riesz Summation of Multiple Fourier Series by Spherical Means

Riesz Summation of Multiple Fourier Series by Spherical Means

On the Summation of Multiple Fourier Series. I

On the Summation of Multiple Fourier Series. I

On the Summation of Multiple Fourier Series. II

On the Summation of Multiple Fourier Series. II

On absolute convergence of multiple Fourier series

Introduction. The results of this paper are extensions of corresponding results for simple Fourier series, given by one of the authors (cf. [5 ]) (1). The main problem was to study the relationship between the mean modulus of a function f(x) and series of the type E Icn I , /3 >0, where the cn are the Fourier coefficients of f(x). We obtain here analoguous results, employing spherical means of a function of several variables. These means were first used by Bochner [1 ] in the study of summation of multiple Fourier series. A particular result is: if a,,, . . .nn are the Fourier coefficients of f(xi, x,x) and f satisfies a Lipschitz condition of degree a, then |iI a., ... .jI 2KI/(K +2a), while the series may be divergent for /3= 2 K/(K+2a). For some previous results concerning the absolute convergence of double Fourier series cf. [3]. 1. Notations. We denote by capital letters vectors in the K-dimensional space, so that X = (x1, X2, * xi), N = (n1, n2, * ni); I NJ = ( ',n2)1I2 is the norm of N; NX =En'x is the scalar product of N and X. The xi, *, are real variables, the ni, * * *, n are integers. f(x1, * * * , XK) =f(X) is a realvalued integrable function of period 27r in each variable. The formal Fourier series of f(X) is

On the summation of multiple Fourier series. III

On the summation of multiple Fourier series. III

Summation of Multiple Fourier Series by Spherical Means

Summation of Multiple Fourier Series by Spherical Means

Summation of multiple Fourier series by spherical means

Summation of multiple Fourier series by spherical means

Summation of Multiple Fourier Series by Spherical Means

Summation of Multiple Fourier Series by Spherical Means