0x024 Fourier Analysis

Fourier Series

Definition (fourier coefficient) If \(f\) is an integrable function given on an interval \([a,b]\) of length \(L\)(i.e \(L=b-a\)), then \(n\)-th Fourier coefficient is defined by

\[\hat{f}(n) = \frac{1}{L} \int_a^b f(x)e^{-2\pi inx/L} dx, n \in \mathbb{R}\]

Definition (fourier series) Fourier series is given by

\[\sum_{n=-\infty}^{\infty} \hat{f}(n) e^{2\pi i nx/L}\]

the N-th partial sum of fourier series is

\[S_N(f)(x) = \sum_{n=-N}^{N} \hat{f}(n) e^{2\pi i nx/L}\]

we want to know whether \(S_N(f) \to f\) when \(N \to \infty\)

Uniqueness of Fourier Series

We are interested in the statement: if \(f,g\) have same fourier coefficients, then \(f=g\). It is not true as two Riemann integral function differs only at one point have the same fourier series.

Convolution

Mean-square Convergence

Reference

[1] Stein, Elias M., and Rami Shakarchi. Fourier analysis: an introduction. Vol. 1. Princeton University Press, 2011.