# Math Help - Fourier vs. Harmonic Analysis

1. ## Fourier vs. Harmonic Analysis

I was going to secretly ask this question in a PM to CaptainBlank for he did research in this area. But then I assumed I should publically post it.

What is the distinction between Fourier and Harmonic Analysis? I understand it as that Fourier Analysis is a more specific area than Harmonic Analysis. Thus, in Fourier we study primarily Fourier series, its properties, its theorems about convergence, the Seperation of Variables techinque for solving partial differencial equations analytically and so on ... which are not so easy to show. However, in Harmonic Analysis we do this in a more general manner, i.e. systems of orthogonal functions.

But I do not thnk there is a distintion, I believe that that is just a more modern term for Fourier Analysis.

2. Originally Posted by ThePerfectHacker
I was going to secretly ask this question in a PM to CaptainBlank for he did research in this area. But then I assumed I should publically post it.

What is the distinction between Fourier and Harmonic Analysis? I understand it as that Fourier Analysis is a more specific area than Harmonic Analysis. Thus, in Fourier we study primarily Fourier series, its properties, its theorems about convergence, the Seperation of Variables techinque for solving partial differencial equations analytically and so on ... which are not so easy to show. However, in Harmonic Analysis we do this in a more general manner, i.e. systems of orthogonal functions.

But I do not thnk there is a distintion, I believe that that is just a more modern term for Fourier Analysis.
Informally (and this is my usage I don't know if there is an official line on this):

Fourier Analysis tends to denote the field of study connected with Fourier
transform and series of functions over $\bold{R}^n$.

Harmonic Analysis is the study of the Fourier and related transforms as a
linear operator on function spaces over more general manifolds (for arguments
sake; an example might be the $\bold{L}^p$ space over a Lie group).

RonL