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}.

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