Let $f(x)$ be a piecewise continuous periodic function of period $2L$ defined on $[-L, L]$. Which of the following statements regarding its Fourier series is/are TRUE?
Options:
(A) If $f(x)$ is an even function, then all sine coefficients $b_n = 0$
(B) If $f(x)$ is an odd function, then 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 total average signal power to the sum of squares of its Fourier coefficients
Answer Key:
A, B, C, D
Step-by-Step Solution & Explanation
1. **Even Functions (Option A)**: The product of an even function and $\sin(n\pi x/L)$ is odd; integrating over $[-L, L]$ yields $b_n = 0$ (True).
2. **Odd Functions (Option B)**: The product of an odd function and $\cos(n\pi x/L)$ is odd; integrating yields $a_n = 0$ for all $n \ge 0$ (True).
3. **Dirichlet Conditions (Option C)**: At any point of finite jump discontinuity, the Fourier series converges to the arithmetic mean of the left- and right-hand limits (True).
4. **Parseval's Theorem (Option D)**: $\frac{1}{L} \int_{-L}^L [f(x)]^2 \, dx = \frac{a_0^2}{2} + \sum_{n=1}^\infty (a_n^2 + b_n^2)$ establishes energy conservation between space and frequency domains (True).