Fourier coefficients কী?

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

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

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.