Uniform Approximation by Finite Wavelet Sum

Question

Suppose $\psi:\mathbb{R}\rightarrow\mathbb{C}$ is a rapidly decreasing, bounded function with zero integral. For $j,k\in\mathbb{Z}$ define a function $\psi_{jk}(x):=\psi(2^{j}x-k)$. Suppose is the functions $\left\{\psi_{jk}\right\}_{j,k\in\mathbb{Z}}$ form an orthonormal basis of $L^{2}(\mathbb{R})$ (i.e. $\psi$ is a one-dimensional mother wavelet). Note here that we are not assuming that $\psi$ arises from a multiresolutional analysis (MRA).

Question: If $f:\mathbb{R}\rightarrow\mathbb{C}$ is a smooth, compactly supported function, can $f$ be uniformly approximated by a finite linear combination of the functions $\psi_{jk}$?

I have looked through [Y. Meyer, Wavelets and Operators] and [E. Hernandez and G. Weiss, A First Course in Wavelets] but did not find this result. Furthermore, those references seem to consider only wavelets arising from an MRA.

I believe that I can show that, for $f$ as above, the wavelet projections $$P_{J}f:=\sum_{j\leq J}\sum_{k\in\mathbb{Z}}\langle{f,\psi_{jk}}\rangle\psi_{jk}$$ converge to $f$ in the $L^{\infty}$ norm as $J\rightarrow\infty$. So if we could show that the series defining $P_{J}f$ converges uniformly, then we would be done.

Answer 1

We assume the following result:

Proposition: If $f\in C_{0}^{\infty}(\mathbb{R})$, then $\left\|f-P_{J}f\right\|_{\infty}\rightarrow 0$ as $J\rightarrow\infty$.

Write $a_{jk}:=\langle{f,\psi_{jk}}\rangle$. Observe that by Holder's inequality and translation/dilation invariance, $$\left|a_{jk}\right|\leq 2^{j/2}\left\|f\right\|_{p}\left\|\psi_{jk}\right\|_{p'}=2^{j/2-j/p'}\left\|f\right\|_{p}\left\|\psi\right\|_{p'}=2^{-j/2+j/p}\left\|f\right\|_{p}\left\|\psi\right\|_{p'}$$ for $1\leq p\leq\infty$.

I now claim that the series $\sum_{k\in\mathbb{Z}}\psi(x-k)$ converges uniformly. By periodicity, we may assume that $0\leq x\leq 1$. Since $\psi$ is rapidly decreasing, we have that $$\sum_{k\in\mathbb{Z}}\left|\psi(x-k)\right|\leq C_{N}\sum_{k\in\mathbb{Z}}(1+\left|x-k\right|)^{-N}$$ for any $N>0$. The RHS above is equal to $$C_{N}\sum_{k=0,1}(1+\left|x-k\right|)^{-N}+\sum_{k\neq 0,1}(1+\left|x-k\right|)^{-N}\leq C_{N}\left[2+\sum_{k\in\mathbb{Z}}(1+\left|k\right|)^{-N}\right]$$ Taking $N>1$ gives a convergent series on the RHS.

Combining this estimate with our bound on the wavelet coefficients, we see that $$\left|P_{J}f\right|\leq\sum_{j\leq J}C_{N}2^{j/p}\left\|f\right\|_{p}\left\|\psi\right\|_{p'}\left(1+\sum_{k\in\mathbb{Z}}(1+\left|k\right|)^{-N}\right)<\infty,$$ since there only finitely many positive scale terms. Clearly, the bound on the RHS is independent of $x\in\mathbb{R}$.

Note that we don't use the full hypothesis that $f\in C_{0}^{\infty}(\mathbb{R})$ in the above proof. We just need some integrability condition. I don't think this strong of hypothesis is necessary either in the assumed proposition. Something weaker like Lipschitz continuous would suffice.