On Wiener’s Tauberian theorems and convolution for oscillatory integral operators

Luis Pinheiro De Castro,Nguyen Minh Tuan,Rita Correia Guerra

doi:10.3906/mat-1801-90

Abstract

The main aim of this work is to obtain Paley--Wiener and Wiener's Tauberian results associated with an oscillatory integral operator, which depends on cosine and sine kernels, as well as to introduce a consequent new convolution. Additionally, a new Young-type inequality for the obtained convolution is proven, and a new Wiener-type algebra is also associated with this convolution.

Highlights

We will build a new convolution associated with an integral operator of oscillatory nature which exhibits very distinctive properties
This gives rise to a nonclassic notion of translation which interconnects well with the kernel of the integral operator, being a key ingredient in our construction of a Wiener–Tauberian-type theorem
This event was seen as a clear identification of the Wiener’s and Tauber’s exceptionally original ideas, and caused a great attention, giving rise to a significant number of studies related to the theory of oscillatory integrals and their convolutions

Summary

Introduction

We will build a new convolution associated with an integral operator of oscillatory nature which exhibits very distinctive properties. This event was seen as a clear identification of the Wiener’s and Tauber’s exceptionally original ideas, and caused a great attention, giving rise to a significant number of studies related to the theory of oscillatory integrals and their convolutions (see [6,7,8] and the interesting comprehensive analysis therein). As it is well known, the above issues are concerned with certain oscillatory integrals and convolutions. B(x0, δ) := {x ∈ Rn : |x − x0| < δ} denotes the open ball centered at x0 ∈ Rn with radius δ , and S stands for the Schwartz space

Auxiliary results

Convolution and Young-type inequality

Paley–Wiener theorems

Wiener’s Tauberian theorems