Which of the following statements is/are TRUE regarding the Fourier series of a periodic function $f(x)$ with period $2L$?
Options:
(A) If $f(x)$ is an even function, all sine coefficients $b_n = 0$
(B) If $f(x)$ is an odd function, all cosine coefficients $a_n = 0$ including $a_0 = 0$
(C) At a point of jump discontinuity $x_0$, the Fourier series converges to $\frac{f(x_0^+) + f(x_0^-)}{2}$
(D) Parseval's identity relates the average of $[f(x)]^2$ over a period to the sum of the squares of the Fourier coefficients
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
Under Dirichlet's conditions for Fourier series on $[-L, L]$:
1. For even $f(x)$, $f(x)\sin(n\pi x/L)$ is odd, so $b_n = \frac{1}{L}\int_{-L}^L f(x)\sin(n\pi x/L) dx = 0$ (A is true).
2. For odd $f(x)$, $f(x)\cos(n\pi x/L)$ is odd, so $a_n = 0$ for all $n \ge 0$ (B is true).
3. By Dirichlet's theorem, at a jump discontinuity $x_0$, the series converges to the midpoint of the jump $\frac{f(x_0^+) + f(x_0^-)}{2}$ (C is true).
4. Parseval's theorem states: $\frac{1}{2L}\int_{-L}^L [f(x)]^2 dx = \frac{a_0^2}{4} + \frac{1}{2}\sum_{n=1}^\infty (a_n^2 + b_n^2)$ (D is true).