# Which Fourier series are “legal”?

Let's consider a continuous function $f(x)$ and real numbers $\lambda_n=(\alpha+\beta n)\pi$ where both $\alpha$ and $\beta$ are integers. In any interval $I$, is it true that $$f(x)=\sum_{n\geq 0}a_n\sin(\lambda_n x) \quad\text{and}\quad a_n=\frac{\int_I f(x)\sin (\lambda_n x)\,\mathrm{d}x}{\int_I \sin^2(\lambda_n x)\,\mathrm{d}x}?$$

If not, can I modify the hypothesis so that it becomes true?

• you missed to write question :-) – devraj Jul 11 '15 at 4:59
• Definitely not true in general. To begin with, even the ordinary Fourier sine series may not be pointwise convergent everywhere if you only assume $f$ continuous. Secondly, for your formula for $a_n$ you need the functions $\sin(\lambda_n x)$ to be orthogonal w.r.t. the $L^2$ inner product on $I$, which puts restrictions on the length of the interval. – Hans Lundmark Jul 11 '15 at 18:50
• @HansLundmark Thanks for your comment. I am not a mathematician and so I dont know too much about this. The theorem that I was looking for is that from Sturm-Liouville. – Julio Jul 13 '15 at 22:50