New estimates for the Fourier transform in the space L2(Rn)

Mohamed El Hamma,R Daher,Ch Khalil,N Djellab

doi:10.5269/bspm.50604

Abstract

In this paper, we prove new estimates are presented for the integral \int_{|t|>N}|\widehat(f)(t)|^{2}dt, where \widehat(f) stands for the Fourier transform of f and N ≥ 1, in the space L2(Rn) characterized by the generalized modulus of continuity of the kth order constructed with the help of the generalized spherical mean operator.

Highlights

Stands for the Fourier transform of f and N ≥ 1, in the space L2(Rn) characterized by the generalized modulus of continuity of the kth order constructed with the help of the generalized spherical mean operator
From formula (1.2), we conclude that the Fourier transform of ∆khDrf (x) is k
Since the serie ir−1mi(f ), r = 1, 2, .., converge f ∈ L2r

Summary

Introduction

|t|≥N stands for the Fourier transform of f and N ≥ 1, in the space L2(Rn) characterized by the generalized modulus of continuity of the kth order constructed with the help of the generalized spherical mean operator. Stands for the Fourier transform of f and N ≥ 1, in the space L2(Rn) characterized by the generalized modulus of continuity of the kth order constructed with the help of the generalized spherical mean operator. Consider in L2(Rn) the spherical mean operator (see [3]) The finite differences of the first and higher orders are defined by ∆hf (x) = Mhf (x) − f (x) = (Mh − I)f (x)

Results

Conclusion