Contents

Idea

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.

Fourier series

Definition

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

$f: \mathbb{T} \to \mathbb{C} \; \text{and} \; \| f \|_{L_1} := \int_{\mathbb{T}} |f(t)| d t \lt \infty$

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

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

Order of Magnitude of Fourier Coefficients

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

Polynomial Asymptotics

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.

It shows that the Fourier coefficients decay faster than any polynomial for smooth functions. In the context of spectral methods it is sometimes stated that expansion coefficients have exponential decay, this fact is commonly called exponentially convergence or spectral accuracy. One should be careful however, because this claim is true for the Fourier coefficients of analytic functions, but not for functions that are only smooth.

References

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