📂Fourier Analysis

The Limit of Fourier Coefficients is Zero

Theorem

The Fourier Coefficients $a_{n}, b_{n}$ and Complex Fourier Coefficients $c_{\pm n}$ are the limit $n \rightarrow \infty$

$$\begin{align*} \lim \limits_{n \rightarrow \infty} a_{n} &= 0 \\ \lim \limits_{n \rightarrow \infty} b_{n} &= 0 \\ \lim \limits_{n \rightarrow \infty} c_{\pm n} &= 0 \end{align*}$$

Proof

By the Bessel’s Inequality, we know that the sum of the Fourier coefficients converges.

$$\dfrac{1}{4}|a_{0}|^2 +\dfrac{1}{2}\sum\limits_{n=1}^{\infty} \left(|a_{n}|^2 + |b_{n}|^2 \right) =\sum \limits_{-\infty}^{\infty} | c_{n} |^2 \le \dfrac{1}{2L}\int_{-L}^{L} | f(t)|^2 dt$$

Therefore, $|a_{n}|^2,\ |b_{n}|^2,\ |c_{\pm n}|^2$ is the $n$th term of a converging series. If a series converges, the limit of the sequence is 0.

$$\lim \limits_{n \rightarrow \infty} |a_{n}|^2=0$$

$$\lim \limits_{n \rightarrow \infty} |b_{n}|^2=0$$

$$\lim \limits_{n \rightarrow \infty} |c_{\pm n}|^2=0$$

Therefore,

$$\lim \limits_{n \rightarrow \infty} a_{n}=0$$

$$\lim \limits_{n \rightarrow \infty} b_{n}=0$$

$$\lim \limits_{n \rightarrow \infty} c_{\pm n}=0$$

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