Harmonic analysis (Rev #2, changes)

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**Harmonic analysis** is the study of certain abstractions of Fourier series and transformations, that is the study of the representations of periodic functions as sums of basis functions.

This page will collect definitions, theorems and references that are scattered and heterogeneous throughout the literature.

The main reference for this page is the classic textbook of Katznelson, see references.

In one dimension we define the **torus** $\mathbb{T} := \mathbb{R} / 2 \pi \mathbb{Z}$, and define nth Fourier coefficient of a function $f \in L^1(\mathbb{T})$, that is

$$f:\mathbb{T}\to \u2102\phantom{\rule{thickmathspace}{0ex}}\text{and}\phantom{\rule{thickmathspace}{0ex}}\Vert f{\Vert}_{{L}_{1}}:={\int}_{\mathbb{T}}|f(t)|dt<\mathrm{\infty}$$ f: \mathbb{T} \to \mathbb{C} \; \text{and} \; \| f \|_{L_1} := \int_{\mathbb{T}} |f(t)| d t \lt \infty

as

$\hat f(n) := \frac{1}{2 \pi} \int_{\mathbb{T}} f(t) e^{- i n t} d t$

The Fourier series $S[f]$ of $f$ is defined to be

$S[f] := \sum_{- \infty}^{\infty} \hat f(n) e^{i n t}$

With additional assumptions about the differentiability of a function it is possible to prove asymptotic properties of the Fourier coefficients:

For a function $f$ that is k-times differentiable with $f^{(k)}$ absolutely continuous, we have

$| \hat f(n)| \leq \min_{1 \le j \le k} \frac{ \|f^{(j)} \|_{L_1} }{|n^j|}$

This is theorem 4.4 in Katznelson.

Theorems like this are used to prove that algorithms using spectral methods have **exponentially fast convergence** when approximating smooth functions.

- Yitzhak Katznelson:
*An introduction to harmonic analysis. 3rd ed.*(Cambridge University Press, 2004, ZMATH)

category: mathematical methods