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January 3, 2011

Harmonic Analysis Lecture Notes 1

Filed under: 2011 Harmonic Analysis Course — xuhmath @ 11:28 pm

Introduction

The posts “Harmonic Analysis Lecture Notes {n}”, where {1 \leq n \leq 100}, 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

Definition An lca group is a locally compact Hausdorff topological group ({(g,h) \mapsto gh^{-1}} is continuous) that is abelian.

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

Definition A character of a lca group {G} is a (group) homomorphism from {G} into {U(1) = \{ z : |z| = 1\}}. The set of characters of {G} is called the dual group and is denoted {\hat G}. This is indeed a group under pointwise multiplication

\displaystyle  	(\chi \cdot \varphi)(g) = \chi(g) \varphi(g).

Furthermore, with the topology of uniform convergent on compact sets, then {\hat G} is an lca group.

Theorem (Pontryagin duality) {(\hat G)\hat\; } is naturally isomorphic to {G} via {g \mapsto (\chi \mapsto \chi(g))}.

Example

  1. If {G = {\mathbb R}^n} then {\hat G = \{ x \mapsto e^{-2\pi i x \cdot \xi} : \xi \in {\mathbb R}^n\}}, which is naturally isomorphic to {{\mathbb R}^n} (public health warning: they may be isomorphic but do not treat them as the same)
  2. If {G = {\mathbb R}^n / {\mathbb Z}^n} then {\hat G = \{ x + {\mathbb Z}^n \mapsto e^{-2\pi i m \cdot x} : m \in {\mathbb Z}^n\}}, which is naturally isomorphic to {{\mathbb Z}^n}.
  3. If {G = {\mathbb Z}^n} then {\hat G = \{ m \mapsto e^{-2\pi i x \cdot m} : x \in {\mathbb R}^n / {\mathbb Z}^n\}}, which is naturally isomorphic to {{\mathbb R}^n / {\mathbb Z}^n}.
  4. {G = ( 0, \infty) \subseteq {\mathbb R}} with multiplication {\ldots} Mellin transform.

Remark

  1. We see that {{\mathbb R}^n} is self-dual while {{\mathbb R}^n / {\mathbb Z}^n} and {{\mathbb Z}^n} 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 discrete{\iff}finite, we see that the dual of a finite group is a finite group, in fact it is itself.)
  2. In all these examples, there is a Haar measure (a non-zero regular Borel measure {\mu} that is finite on compact sets and {\mu(E) = \mu(g^{-1} (E))} for all Borel sets {E} and all {g \in G}). In fact, all lca groups have a Haar measure.

Theorem (Duality for finite cyclic groups) The characters of {{\mathbb Z} / N {\mathbb Z}} are {\{ n + N {\mathbb Z} \mapsto e^{-2\pi i \frac{k}{N} \cdot n} : k \in {\mathbb Z} / N{\mathbb Z}\}}.

Proof: If {\chi} is a group homomorphism of {G} into {U(1)}, then {\chi (1 + N{\mathbb Z})^N = \chi (N + N{\mathbb Z}) = 1} (recall here we are writing the group operation as multiplication). Thus {\chi(1 + N {\mathbb Z}) = e^{-2\pi i \frac{k}{N}}} for some {k \in {\mathbb Z} / N {\mathbb Z}}. Immediately we see that {\chi} belongs to the set above. Conversely, we can just check that they are all characters. \Box

Theorem {\widehat{ G \times H } \cong \hat G \times \hat H} in the natural way {(\chi, \varphi)(g,h) = \chi(g) \varphi(h)}.

Remark From 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

  1. As an {\ell^p} space ({\ell^1 \equiv} measures);
  2. As the group algebra {{\mathbb C} [G] = \{ \sum_{g \in G} c(g) g: c(g) \in {\mathbb C}\}} where multiplication of algebra elements is

    \displaystyle  	\left(\sum_{g \in G} c(g) g\right) \cdot \left( \sum_{h \in G} d(h) h \right) = \sum_{g, h \in G} c(g) d(h) \cdot (gh) = \sum_{g' \in G} \underbrace{\left(\sum_{g \in G} c(g) d(g' g^{-1} )\right)}_{(c * d)(g')} \cdot (g').

    where {(c*d)(g')} is called the convolutions of the functions {c} and {d}. Observe that we can extend characters {\chi} of {G} to to be characters of {{\mathbb C} [G]} by linearity {\chi(\sum c(g) g) = \sum c(g) \chi(g)}. The result is an algebra homomorphism from {{\mathbb C}[G]} into {{\mathbb C}}. This is also really the Fourier transform: putting this all in terms of functions

    \displaystyle  		(f : G \rightarrow {\mathbb C}) \stackrel{\widehat{ \:}}{\longrightarrow} ( \hat f : \hat G \rightarrow {\mathbb C})

    via {\hat f(\chi) = \chi(\sum f(g) g) = \sum f(g) \chi(g)}. Or, in the case of {{\mathbb Z}/ N{\mathbb Z}}, then

    \displaystyle  		\hat f(k + N{\mathbb Z}) = \sum_{n=0}^{N-1} e^{-2\pi i \frac{k}{N} \cdot n} f(n + N {\mathbb Z}).

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