Sine coefficients, sine series and the odd periodic extension

Let $L>0$. A function $F:\mathbb{R}\to \mathbb{R}$ is $L$-periodic when $F(x+L)=F(x)$ for all $x\in \mathbb{R}$, and odd when $F(-x)=-F(x)$ for all $x$.

(a) Sine coefficients. Let $F:[0,L]\to \mathbb{R}$ be continuous (Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point); a function on a larger domain is used through its restriction to $[0,L]$, with no periodicity required for the coefficient definition. In particular both $L$-periodic and odd $2L$-periodic functions are admissible. Then ${\int }_0^L\mathrm{\mid }F\mathrm{\mid }<\mathrm{\infty }$ (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion). For every integer $n\ge 1$ the $n$-th sine coefficient of $F$ is

$b_n(F):=\frac{2}{L}{\int }_0^{L}F(x)\sin \text{\hspace{0.17em}⁣}\left(\frac{n\pi x}{L}\right)dx\in \mathbb{R},$

an ordinary Riemann integral of a continuous function on a compact interval (The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$, Sine and cosine defined by their real power series). The sine series of $F$ is the series of functions

${\sum }_{n\ge 1}{b}_n(F)\sin \text{\hspace{0.17em}⁣}\left(\frac{n\pi x}{L}\right),$

whose $N$-th partial sum is $S_N^{\mathrm{s}}F(x):={\sum }_{n\le N}{b}_n(F)\sin (n\pi x\mathrm{/}L)$.

(b) Odd periodic extension. Let $F:[0,L]\to \mathbb{R}$ be continuous with $F(0)=F(L)=0$. The odd $2L$-periodic extension of $F$ is the function on $\mathbb{R}$ defined by

$F^{\mathrm{o}\mathrm{d}\mathrm{d}}(x):=\{\begin{array}{ll}F(x),& 0\le x\le L,\\ -F(-x),& -L\le x\le 0,\end{array}{F}^{\mathrm{o}\mathrm{d}\mathrm{d}}(x+2L):=F^{\mathrm{o}\mathrm{d}\mathrm{d}}(x)\mathrm{.}$

Equivalently, $F^{\mathrm{o}\mathrm{d}\mathrm{d}}$ is the unique $2L$-periodic odd function whose restriction to $[0,L]$ is $F$ (the two definitions agree at $0$ and $L$ because both values vanish; the extension is well defined and continuous).

(c) Given a specified wave speed $c>0$, the angular frequency of mode $n$ is ${\omega }_n:=cn\pi \mathrm{/}L$; its fundamental ($n=1$) value is $c\pi \mathrm{/}L$.

• All coefficients are ordinary integrals of continuous functions on compact intervals; no choice principle is needed, and no convergence of the sine series is asserted by the definition. Sine coefficients are real because $F$ and the sine functions are real-valued.

• The normalization in $b_n$ is the coefficient normalization for the orthogonal family $\{\sin (n\pi x\mathrm{/}L){\}}_{n\ge 1}$ with squared norm $L\mathrm{/}2$: ${\int }_0^L\sin (n\pi x\mathrm{/}L)\sin (k\pi x\mathrm{/}L)dx=(L\mathrm{/}2){\delta }_{nk}$, the factor $(2\mathrm{/}L)$ being the inverse of the squared norm $L\mathrm{/}2$; the orthogonality is proved in the companion lemma, not assumed here.

• A function that is odd and $2L$-periodic automatically satisfies $F(0)=F(L)=0$ by oddness and periodicity; the extension in (b) is a construction, not a claim that $F$ already has these symmetries.