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Fourier Notes

This page collects some useful results from Fourier Analysis.

Fourier Series

Definition 1. Let $f\in L^1(0,1)$ be a Lebesgue integrable function on $[0, 1]$ and define the Fourier coefficients by $$\hat{f}(n) = \int_0^1 f(x) e^{-2\pi i nx} \,dx \qquad n\in\ZZ$$

Theorem 2. Suppose that $f$ is continuous on $\TT$ and $\hat{f}(n)$ is absolutely summable. Then $$f(x) = \sum_{n=-\infty}^\infty \hat{f}(n) e^{2\pi inx}$$ where the series converges uniformly on $[0,1]$.

Theorem 3. (Parseval's Theorem) Let $f \in L^2(0,1)$. Then its Fourier coefficients are in $\ell^2(\ZZ)$ and $$\int_0^1 |f(x)|^2 dx = \sum_{n=-\infty}^\infty |\hat{f}(n)|^2$$ Thus the map $f' \in L^2(0,1) \rightarrow {f}\in\ell^2(\ZZ)$ is unitary.

Fourier Transforms

Definition 4. Let $f\in L^1(\RR)$ and define the Fourier transform $$(\F f)(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi i x\xi} \,dx$$ Likewise for integrable $g(\xi)$ define the inverse Fourier transform $$(\F^{-1}g)(x) = \int_{-\infty}^\infty g(x) e^{2\pi i x\xi} \, d\xi$$

Theorem 5. (Fourier Inversion Theorem) If $f \in L^1(\RR)$ and also $\F f \in L^1(\RR)$ then $(\F^{-1}\F f)(x) = f(x)$ for almost every $x\in\RR$.

Corollary 6. If, in addition, $f$ is continuous, then $(\F^{-1}\F f)(x) = f(x)$ for all $x\in\RR$.

Theorem 7. (Plancherel Theorem) If $f\in L^1(\RR)\cap L^2(\RR)$ then $\F f\in L^2(\RR)$ and $$\int_{-\infty}^\infty |(\F f)(\xi)|^2\,d\xi = \int_{-\infty}^\infty |f(x)|^2 \,dx$$

Thus the map $f\rightarrow\F f$ is an isometry from $f\in L^1(\RR)\cap L^2(\RR)$ into $L^2(\RR)$, which extends to a unitary $L^2(\RR)\rightarrow L^2(\RR)$. This unitary is called the Plancherel transform.