Posted by: cnpde | September 16, 2012

## Fourier series 1

Fourier analysis is a power tool for PDE problem on $\mathbb{R}^n$ or $\mathbb{T}^n$. For $\mathbb{R}^n$, we usually use Fourier transform

$\hat f(\xi)=\int_{\mathbb{R}^n} e^{-ix\xi}f(x)dx$;

while in the case of $\mathbb{T}^n$, we calculate Fourier coefficients $\hat f(k)$.

These two cases are much similar, especially for the algebraic properties. However, the difference between $L^p$ and $l^p$ and other difference between $\mathbb{R}^n$ and $\mathbb{T}^n$ will show they are two different things.

In this set of posts, I will discuss something about Fourier series. The main topic may be focused on:

1) Decay of Fourier coefficients $\hat f(\xi)$ and smoothness of function $f(x)$.

2)  Pointwise convergence.

3) Bochner-Riesz summation.

4) Lacunary series.

Fourier transform is mainly used to measure oscillation. We can look functions $\cos \sin$ as signals. The higher frequency, the more oscillation. By calculating $\hat f(k)$, we ‘take out’ the wave with frequency k. The larger $|\hat f(k)|$, the larger the amplitude, and the heavier of this frequency weighs in the function $f$.

By Riemann-Lebesgue lemma, we know for $L^1$ function $f$, $|\hat f(k)|$ approaches to 0 as $|m|\rightarrow 0$, and this rate of decay is arbitarily slow.

For smooth functions, we have the decay estimate

$|\hat f(m)|\leq C_{s,n}\frac{\sup\limits_{|\alpha|=s}|\partial^\alpha f(m)|}{|m|^s}$.

Conversely, if the Fourier coefficients decay as

$|\hat f(m)|\leq C(1+|m|)^{-s-n}$ for all $m\in \mathbb{Z}^n$, then $f$ has partial derivative of orer $|\alpha|\leq [[s]]$, where $[[s]]$ is the largest integer strictly less than s.

For bonded vatiational function, $|\hat f(m)|\leq \frac{Var (f)}{2\pi m}$.