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This question is about conditions on a mother wavelet that generates a countable familily of child wavelets via scaling and translation, that are both necessary and sufficient for the child wavelets to form a frame in the Hilbert space $L^2(\mathbb{R})$

Here are the precise definitions of these concepts in the context of this question:

Let a mother wavelet be an element $\psi \in L^2({\mathbb{R}})$ with $\|\psi\| = 1$ and a finite admissibility constant $0 \lt C_{\psi} \lt \infty$ that is defined as follows: $$ C_{\psi} = \int_{- \infty}^{\infty} \frac{| \hat{\psi} (\omega) |^2}{|\omega|} d\omega $$ where $\hat{\psi}$ denotes the Fourier transform of $\psi$.

Let $\psi$ be such a mother wavelet and define the countable set of child wavelets $W$ as follows: Let $\sigma \gt 1, \tau \gt 0$ be real numbers and

$$ W := \{ \psi_{j, k}: j, k \in \mathbb{Z}, \psi_{j, k}(t) = \frac{1}{\sigma^{- \frac{j}{2}}} \psi(\frac{t - k \tau \sigma^{-j}}{\sigma^{-j}}) \} $$

Now let a frame in a separable Hilbert space be a countable set of vectors $\{ \phi_j \} $ such that there are constants $a, b \gt 0$ such that for every vector $f$ we have

$$ a \|f\|^2 \le \sum_j | \langle f, \phi_j \rangle |^2 \le b \|f\|^2 $$

My question is: Are there conditions known on the triple $(\psi, \sigma, \tau)$ that are both necessary and sufficient for W, the set of child wavelets, to be a frame in $L^2(\mathbb{R})$?

(AFAIK there are conditions that are necessary, and other conditions that are sufficient, known since Ingrid Daubechies published her results in 1990. But there don't seem to be any conditions that are both necessary and sufficient.)

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No. This question is, I believe, open even for functions of the type $\hat \psi = I_E$, dilations by powers of 2 and translation by integers. See, for example the relatively recent paper of Bownik and Weber http://pages.uoregon.edu/mbownik/papers/16.pdf, where specific examples of $\psi$ of this type are given for which they indicate they don't know how to prove whether it is a frame or not. In the 2004 Proceedings of the AMS paper of Dai, Diao, Gu and Han, they explicitly mention that the characterization of sets $E$ so that $\hat \psi = I_E$ and $\psi$ is a frame wavelet (for dilations by powers of 2 and integer translations) is open. There are currently not sufficient techniques available for dealing with the case that the canonical dual frame is not affine.

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