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Fourier coefficients

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The set of functions $\{e_n = e^{inx}: n \in Z\}$ is an orthonormal basis for the space of square-integrable functions of $[−π, π]$. This space is an inner product space: $$\langle f,\, g \rangle \;\stackrel{\mathrm{def}}{=} \; \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\overline{g(x)}\,dx.$$

Then $$f=\sum_{n=-\infty}^\infty \langle f,e_n \rangle \, e_n.$$

Riemann–Lebesgue lemma. $$\lim_{|n|\rightarrow \infty}\hat{f}(n)=0,$$ $$\lim_{n\rightarrow +\infty}a_n=0,$$ $$\lim_{n\rightarrow +\infty}b_n=0.$$

Theorem. The Fourier coefficients $\widehat{f'}(n)$ of the derivative $f'$ can be expressed in terms of the Fourier coefficients $\hat{f}(n)$ of the function $f$: $$\widehat{f'}(n) = in \hat{f}(n).$$

Parseval's theorem. $$\sum_{n=-\infty}^\infty |\hat{f}(n)|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi} |f(x)|^2 \, dx.$$

Plancherel's theorem. If $c_0,\, c_{\pm 1},\, c_{\pm 2},\ldots$ are coefficients and $\sum_{n=-\infty}^\infty |c_n|^2 < \infty$ then there is a unique function $f\in L^2([-\pi,\pi])$ such that $\hat{f}(n) = c_n$ for every $n$.