Fourier analysis is a power tool for PDE problem on or . For , we usually use Fourier transform

;

while in the case of , we calculate Fourier coefficients .

These two cases are much similar, especially for the algebraic properties. However, the difference between and and other difference between and 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 and smoothness of function .

2) Pointwise convergence.

3) Bochner-Riesz summation.

4) Lacunary series.

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

By Riemann-Lebesgue lemma, we know for function , approaches to 0 as , and this rate of decay is arbitarily slow.

For smooth functions, we have the decay estimate

.

Conversely, if the Fourier coefficients decay as

for all , then has partial derivative of orer , where is the largest integer strictly less than s.

For bonded vatiational function, .

### Like this:

Like Loading...

## Leave a Reply