Multiple Fourier Integrals Research Articles

On multiple Fourier integrals of piecewise smooth functions with discontinuities of the second kind

It is well known that the Riesz means of eigenfunction expansions of piecewise smooth functions of order s > ( n − 3 ) / 2 converge uniformly on compacts where these functions are smooth. In 2000 L. Brandolini and L. Colzani considered eigenfunction expansions of piecewise smooth functions with discontinuities of the second kind across smooth surfaces. They showed that the Riesz means of these functions of order s > ( n − 3 ) / 2 may diverge even at certain points where these functions are smooth. Here it is argued that this effect depends on the measure of the singularity area, i.e. we consider functions with singularities across more limited areas and prove that the Riesz means of their eigenfunction expansions of order s > ( n − 3 ) / 2 converge uniformly on compacts where these functions are continuous.

On the multiple Fourier integrals of continuous functions from the Sobolev spaces

On the multiple Fourier integrals of continuous functions from the Sobolev spaces

On summability of eigenfunction expansions of piecewise smooth functions

We consider two forms of eigenfunction expansions associated with an arbitrary elliptic differential operator with constant coefficients and order m, that is the multiple Fourier series and integrals. For the multiple Fourier integrals, we prove the convergence of the Riesz means of order s > (N − 3)/2 of piecewise smooth functions of N ≥ 2 variables. The same result is proved in the case of the N ≥ 3 dimensional multiple Fourier series.

Fractional derivatives and the inverse Fourier transform of ℓ1-radial functions

In [Berens, H. and Xu, Y. (1997). ℓ − 1 summability of multiple Fourier integrals and positivity. Math. Proc. Camb. Phil. Soc., 122, 149–172] and [Berens, H. and zu Castell, W. (1998). Hypergeometric functions as a tool for summability of the Fourier integral. Result. Math., 34, 69–84], H. Berens, Y. Xu, and the author proved that the inverse Fourier integral of ℓ1-radial functions, i.e., functions which are radial w.r.t. the ℓ1-norm on ℝ d , can be decomposed into a multi-dimensional integral with B-spline kernel, which is independent of the function, and a transformation on ℝ+ the kernel of which is given by a Meijer G-function. The latter can be seen as an analog of the well-known Fourier–Bessel transform. Here, we associate weights to the underlying transform. There is an analog development where the B-spline has to be replaced by a Dirichlet spline, while on ℝ+ the related transforms can again be defined via G-function kernels. We further show that the underlying transform can be interpreted as a fractional derivative.

On Linear Means of Multiple Fourier Integrals Defined by Special Domains

Weak and strong estimates in weighted L p spaces are obtained for linear means of Fourier integrals defined by a single function with support in a specially organized set.

Approximation of spherical means of multiple Fourier integrals and Fourier series on set of full measure

In this paper we consider the approximation for functions in some subspaces of L2 by spherical means of their Fourier integrals and Fourier series on set of full measure. Two main theorems are obtained.

Decomposability of continuous functions from Nikol'skii classes into multiple fourier integrals

Decomposability of continuous functions from Nikol'skii classes into multiple fourier integrals

Harmonic Analysis on Product Spaces

Rn x R' and commute with the dilations p33 '8(x2, x2) = (61x1, 62x2) for x1 E Rn, x2 E Rm, and 61, 82 > 0, then the corresponding operators are related to many problems in the theory of multiple Fourier integrals (see Zygmund [21]). The maximal operator which commutes with the p31382 is the Jessen, Marcinkiewicz, and Zygmund [14] strong maximal operator Ms defined by 1 Ms(f)(x1, x2) = sup RI A|(Y1, Y2)1 dy1 dy2 (xl, X2)GR JR1

ON THE SUMMABILITY METHOD OF ABEL-POISSON TYPE FOR MULTIPLE FOURIER INTEGRALS

The author examines a class of summability methods for multiple Fourier integrals which contains for certain values of the parameter the Abel-Poisson and Gauss-Weierstrass methods. The properties of the kernels of these methods are studied. A subclass of positive kernels is exhibited. Using the properties established for the kernels, he proves the convergence of the integral means under consideration almost everywhere and in the metric of , as well as the existence of a localization principle. Bibliography: 18 titles.

On Gibbs' phenomenon for Riesz spherical means of multiple Fourier integrals and Fourier series

Вводится понятие явл ения Гиббса для сходя щейся последовательности функций не-скольких переменных и исследу ется наличие этого яв ления в поведении сферическ их сред-них Рисса интегралов и ря дов Фурье функций, име ющих разрывы на дифференц ируемых многообразиях.