Are Fourier Analysis and Harmonic Analysis the same subject?
I believe that they are not the same.
Maybe there is big difference between those subjects but I need to know what is the main difference between those subjects and what is the main intersection?
What is the common between those subjects?
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asked Aug 22, 2013 at 6:52
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$\begingroup$ Basic Fourier analysis is the beginning of harmonic analysis. General harmonic analysis is done on topological groups. "Fourier analysis" can also mean doing harmonic analysis afforded by the very special structure on $\mathbb{R}^n$. $\endgroup$– MichaelAug 22, 2013 at 7:02
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1$\begingroup$ @Michael: unfortunately the line is very blurred, and that distinction of terminology you described, while used often, is inconsistently applied by different people. For example, an expert in Fourier analysis wrote a book called Harmonic Analysis a large part of which strongly dependent on the special properties of $\mathbb{R}^n$. $\endgroup$– Willie WongAug 22, 2013 at 16:15
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$\begingroup$ @Michael If basic Fourier analysis is the beginning of harmonic analysis, then harmonic analysis is the beginning of what? $\endgroup$– OokerSep 26, 2017 at 13:13
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2 Answers
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Your question will maneuver elegantly deep into a discussion to restrict the term Fourier Analysis to refer to the process of expanding functions on a locally compact abelian group $G$ as a sum of the characters of the group, while the generalization, when the group $G$ is not assumed to be abelian, should be referred to as Harmonic Analysis. Refer to a great dialogue between Dick Palais and Emerton (see MathOverflow here>>>) in their answer sections.
answered Aug 22, 2013 at 7:41
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I think there is great ambiguity about "Fourier analysis" and "harmonic analysis". To know what anyone means by either term one must know the context, typically, one must know the next noun or phrase in "harmonic/Fourier analysis on X".
It is true that some patterns of use use one term as a special case, the other as more general, but this can be reversed, also. If I had to "bet", it might be that "harmonic" is more general than "Fourier" 60 percent of the time? Maybe? Luckily, we do not have to "commit".
Sometimes, yes, "Fourier..." is meant to restrict to _abelian_groups_, ... but sometimes "Fourier analysis" is done without admitting that anything is a group. "Harmonic..." may refer to structured analysis on a group, or homogeneous space, but, again, by observation, classical "harmonic analysis" did analysis on chunks of Euclidean spaces.
If anything, using either of these terms suggests that any groups (or chunks thereof) have more structure than just topological groups (although, historically, originally people had hopes that not much more would be necessary to develop a satisfactory theory). Especially for non-compact, non-abelian groups, often one would want a Lie group structure, or totally-disconnected group structure, or combination of the two.
answered Aug 22, 2013 at 16:38