What is Fourier coefficients? — Mathematics

Definition 14.1 University Mathematics — Year 2 · Chapter 14 — Fourier Series

The Fourier coefficients of $f$ are

c_n(f) = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(t)\,\eu^{-\iu n t}\,\dd t \qquad (n \in \Z),

and the real-form coefficients $a_n = c_n + c_{-n}$ , $b_n = \iu(c_n - c_{-n})$ , so that the Fourier partial sums are

S_N(f)(t) = \sum_{n=-N}^{N} c_n(f)\,\eu^{\iu nt} = \frac{a_0}{2} + \sum_{n=1}^{N}\bigl(a_n\cos nt + b_n \sin nt\bigr).

On $\mathcal{C}$ , define the Hermitian inner product $\langle f, g\rangle = \frac{1}{2\pi}\int_{-\pi}^{\pi}\conj f\,g$ : the exponentials $e_n(t) = \eu^{\iu nt}$ are orthonormal ( $\langle e_m, e_n\rangle = \delta_{mn}$ , direct computation), and $c_n(f) = \langle e_n, f\rangle$ : Fourier analysis is Hermitian geometry (Chapter 13) in infinite dimension.

Examples

Example 14.3 (Best approximation, measured)

How well do low-degree trigonometric polynomials approximate the sawtooth $f(t) = t$ (on $\intoo{-\pi}{\pi}$ ) in the quadratic mean? By Proposition 14.2 the best degree- $N$ approximation is $S_N(f)$ , with squared error

\norm{f - S_Nf}_2^2 = \norm f_2^2 - \sum_{\abs n\leq N}\abs{c_n}^2 .

Here $\norm f_2^2 = \frac{1}{2\pi}\int_{-\pi}^\pi t^2\dd t = \frac{\pi^2}{3}$ , and from $b_n = \frac{2(-1)^{n+1}}{n}$ (Example 14.12): $\abs{c_n}^2 + \abs{c_{-n}}^2 = \frac{b_n^2}{2} = \frac{2}{n^2}$ . So

\norm{f - S_Nf}_2^2 = \frac{\pi^2}{3} - \sum_{n=1}^{N}\frac{2}{n^2} \qquad\text{: numerically } 1.29,\ 0.79,\ 0.57,\ 0.44

for $N = 1, 2, 3, 4$ — decreasing, but slowly: the tail $\sum_{n>N}\frac2{n^2} \sim \frac2N$ is governed by the slow $\frac1n$ decay of the coefficients, itself the signature of the jump (Exercise 14.6 read backwards). Closing insight: Parseval turns approximation quality into a tail of a numerical series — and predicts, before any picture, that jumps make Fourier series converge reluctantly.

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Examples

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