Generalization of Fourier Transform from Swartz Class to Lp Space

part 1. 연세대학교 수학과 설진석

고등교육재단

Basic Terminologies

Complete SpaceRiemann Integrable Space is not Complete

Lebesgue Integrable Space does!

Banach SpaceComplete and Norm

Space

Hilbert SpaceComplete and Inner-Product Space

Lp Space

Lp is Banach Space L2 is also Hilbert Space

Swartz Class

Smooth, Moderate Decreasing

Fourier Transform on Swartz Class

Well-Defined

Need to define Fourier Transform on general Lp space!

Main Idea

#1 Prove

Swartz Class is dense in Lp

#2 Define

using completion

Lemmas

Dominant Convergence Theorem

shortly, DCT

Mollifier

or Approximation to the Identity or Good Kernel

then

continue on next part