Take a positive integer $\alpha$. I am looking for an orthonormal basis $(\phi_n)$ for $L^2(\mathbf R)$ with roughly speaking the following properties:
- each $\phi_n$ is compactly supported and $C^\alpha$
- consider $g \in C_0^\infty(\mathbf R)$ and the $L^2$ expansion: $$g = \sum_{n = 0}^\infty \langle g, \phi_n\rangle \phi_n$$ and consider the partial sum: $$g_N = \sum_{n = 0}^N \langle g, \phi_n\rangle \phi_n$$ I want the $g_N$s to be supported within a common compact interval, ie. some neighborhood of the support of $g$.
- nice (geometric $\mathcal O(2^{-n})$ or better) decay on $\langle g, \phi_n\rangle$ as $n \to \infty$ would be grand.
Wavelets are the first candidate but I think they do not work. Specifically, if we let $\psi$ be a $C^\alpha$ mother wavelet generating an orthonormal basis $\{\psi_{j, k} : j, k \in \mathbf Z\}$ ($\psi_{j, k}(x) = 2^{j/2} \psi(2^j x - k)$). You can get excellent (arbitrarily good increasing the smoothness of $\psi$) geometric decay on the Fourier coefficients by e.g. Theorem 5 in https://math.unm.edu/~crisp/courses/wavelets/fall07/Sobolev.pdf.
I believe property $(2)$ does not hold for wavelets. Take such an orthonormal basis $\{\psi_{j, k} : j, k \in \mathbf Z\}$ and expand: $$g = \sum_{j, k} \langle g, \psi_{j, k}\rangle \psi_{j, k}$$ We can truncate using say: $$g_n = \sum_{|j| \le n} \sum_k \langle g, \psi_{j, k}\rangle \psi_{j, k}$$ Note that for each $j$ only finitely many $\langle g, \psi_{j, k}\rangle$ are non-zero so this is a finite sum. If we take $g$ to have support exactly $[0, 1]$, note that we may have $\langle g, \psi_{-10^5, 0}\rangle$ non-zero, at which point the support of $g_{10^5}$ may contain points near $2^{10^5}$ or something silly, though the actual values it takes near there will be tiny or be taken for a very short time. Unfortunately I want to say, for example, that truncations $g_n$ will (eventually) be supported in $[-1, 2]$ say, or so on, any compact interval will do. Is this actually possible? It must be linear combinations of the basis, I cannot multiply by a cutoff function or similar.
If I've missed any fact about wavelet theory and my concern is actually entirely invalid, please let me know as well!