Sine Fourier series and smooth function.

Question

I was reading through my text on PDEs and came across a theorem (or perhaps Lemma) that states:

"For any smooth function $g_1(y)$ with $g_1(0) = g_1(h) = 0$, it can be expressed as a Fourier sine series, where y is between $0$ and $h$".

Why is this so?

Edit: to add context

Answer 1

This is from Fourier series theory (I guess you should remember it)

1. Every smooth function can be expressed as its Fourier series on in interval say on $(-h, h)$. Remember that Fourier series has only sine terms if function $f$ is odd and it has only cosine terms if function $f$ is even.

2. But on half interval $(0,h)$ we can express the any smooth function using only sines (or cosines). For this we redefine function $f$ as odd function $f(-x)=-f(x)$ for $x\in(-h,0)$.