** Introduction **

The posts “Harmonic Analysis Lecture Notes ”, where , are a transcription and editing of lecture notes from Professor Rowan Killip’s course on Harmonic Analysis for the Winter 2011 and Spring 2011 quarters at UCLA. In particular, some discussion topics have been from one post to an adjacent post when appropriate. Longer proofs that were discussed over the course of two (or more) lectures are all lumped together in one post.

From the course webpage, “**References:** We will not be following any single source. The lectures are most strongly influenced by the following:

- Thomas H. Wolff,
*Lectures on harmonic analysis*. American Mathematical Society, Providence, RI, 2003. - Elias M. Stein,
*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*. Princeton University Press, Princeton, NJ, 1993.

Other good references include:

- Loukas Grafakos, Classical Fourier analysis. Graduate Texts in Mathematics, 249. Springer, New York, NY, 2008.
- Loukas Grafakos, Modern Fourier analysis. Graduate Texts in Mathematics, 250. Springer, New York, NY, 2008.
- Yitzhak Katznelson,
*An introduction to harmonic analysis*. Dover Publications, Inc., New York, NY, 1976. - Elias M. Stein and Guido Weiss,
*Introduction to Fourier analysis on Euclidean spaces*. Princeton University Press, Princeton, NJ, 1971.”

** Algebraic properties of the Fourier transform **

DefinitionAnlca groupis a locally compact Hausdorff topological group ( is continuous) that is abelian.

For the remainder of this discussion, we will be writing the group operation multiplicatively.

DefinitionAcharacterof a lca group is a (group) homomorphism from into . The set of characters of is called thedual groupand is denoted . This is indeed a group under pointwise multiplication

Furthermore, with the topology of uniform convergent on compact sets, then is an lca group.

Theorem (Pontryagin duality)is naturally isomorphic to via .

Example

- If then , which is naturally isomorphic to (public health warning: they may be isomorphic but do not treat them as the same)
- If then , which is naturally isomorphic to .
- If then , which is naturally isomorphic to .
- with multiplication Mellin transform.

Remark

- We see that is self-dual while and are dual to one another. There is a general theorem that the dual of a compact group is discrete and vice versa. (Note that since compact and discretefinite, we see that the dual of a finite group is a finite group, in fact it is itself.)
- In all these examples, there is a Haar measure (a non-zero regular Borel measure that is finite on compact sets and for all Borel sets and all ). In fact, all lca groups have a Haar measure.

Theorem (Duality for finite cyclic groups)The characters of are .

*Proof:* If is a group homomorphism of into , then (recall here we are writing the group operation as multiplication). Thus for some . Immediately we see that belongs to the set above. Conversely, we can just check that they are all characters.

Theoremin the natural way .

RemarkFrom these two results it follows that the dual of a finite abelian group is isomorphic to itself.

** The Fourier Transform **

The Fourier transform is about *functions*. There are several ways we can think about functions on a finite group. For example

- As an space ( measures);
- As the group algebra where multiplication of algebra elements is
where is called the convolutions of the functions and . Observe that we can extend characters of to to be characters of by linearity . The result is an algebra homomorphism from into . This is also really the Fourier transform: putting this all in terms of

*functions*via . Or, in the case of , then

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