generalizing fourier transform, part 1
TRANSCRIPT
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