General: If $f \in C^1$ is a periodic function defined over some multi-dimensional space, then it should be possible to express $f$ as a FINITE fourier series.
- is this true of any periodic basis?
- is there a way to determine the number of terms in this finite series?
Specific: I have a function that is smooth and continuous and is defined on the unit hypersphere. I want to know:
- is it possible to represent this function as a FINITE series in the hyperspherical harmonic basis?
- how many terms will it take?
If $f$ is a $C^1$ function, $2\pi$-periodic, defined on the real line, then $$\widehat{f'}(n)=\frac 1{2\pi}\int_0^{2\pi}f'(x)e^{-2i nx}dx=-\frac 1{2\pi}\int_0^{2\pi}f(x)e^{-2inx}(-2in)dx=\frac{in}{\pi}\widehat f(n),$$ so $\{n\widehat{f}(n)\}\in\ell^2(\mathbb Z)\}$. But it doesn't imply that only finitely many terms are nonzero. For example, let $\{\alpha_n\}_{n=-\infty}^{+\infty}$ a bounded sequence and $f(x)=\sum_{n=-\infty}^{+\infty}\alpha_n e^{-|n|}e^{inx}$. $f$ is $C^1$ and all the terms are not $0$, so there are a lot of function, which are $C^1$ (an in fact $C^{\infty}$ as the example shows), but all the Fourier coefficients are different from $0$.