17 problems

Let $n$ be a positive integer, let $g\in L^{2}(\mathbb{R}^{n})$ be nonzero, and for $x,\omega\in\mathbb{R}^{n}$ let $T_x$ and $M_\omega$ denote the translation and modulation opera…

Let a Daubechies filter have length at least $4$, and let its associated basic wavelet packets be the functions obtained from the corresponding low-pass/high-pass filter pair. For…

Let $\psi$ be a dyadic orthonormal wavelet, let $f:\mathbb{R}\to\mathbb{R}$ tile $obreakmathbb{R}$ by translations and dilations, and let $mu$ denote Lebesgue measure. Define a me…

Let $f:\mathbb{R}\to\mathbb{R}$ be a nonnegative function that tiles $obreakmathbb{R}$ by translations and dilations, meaning … A wavelet set is a measurable set…

Larson's conjecture. The support of the Fourier transform of a dyadic, orthonormal wavelet contains a wavelet set.

Let $D>0$ be a constant, and let $\mathbf{a}=(a_n)_{n=1}^{\infty}$ be a sequence such that the hyperbolic distance between $a_n$ and $a_{n+1}$ is at most $D$ for every…

Let $C$ be the phase space for the universal quasiperiodic factorization, with $u=U\circ\Phi$ and $\Phi:S^1\to C$. An appropriate wavelet basis is used to decompose the signal $u$…

Orthonormal wavelet implies wavelet-set conjecture. For each pair $(B,\Gamma^*)$ such that there exists a $(B,\Gamma^*)$ orthonormal wavelet, there exists an $(A,\Gamma)$ wavelet s…

Let $I=\{i:c_i>0\}$ be the support of the mask of a dyadic subdivision scheme with non-negative coefficients. Convergence conjecture. The subdivision scheme converges unless the gr…

Let $\phi\in\mathcal{S}(\mathbb{R})\setminus\{0\}$, and let a finite wavelet system generated by $\phi$ be a finite collection of wavelet dilates and translates of $\phi$. Finite-w…

Let $A$ be an invertible matrix, let $\Gamma$ be a lattice in $\mathbb{R}^n$, let $B=A^T$, and let $\Psi=\\{\psi_1,\dots,\psi_L\\}$ generate the wavelet system…

Optimality conjecture. The local-set-based wavelet basis $\operatorname{W}_{\rm LSPC}$ minimizes, among all orthonormal bases $\operatorname{F}$, the worst-case number of nonzero c…

Let $\varphi$ be a refinable function, and suppose that its local Hölder exponent is constant and equal to $\alpha$. Constant local Hölder exponent conjecture. Then … The paper stu…

Let $n\ge 2$ and let $\Lambda_n=\{\lambda_0,\ldots,\lambda_{n-1}\}$ be an arbitrary set of real numbers. Let $\psi^{\Lambda_n}$ be the generalized Daubechies wavelet associated wit…

Let $M_\sigma$ denote the wavelet transform kernel at positive scale $\sigma$, and let $f$ be a locally integrable function of sufficiently rapid decay. For a sequence of positive…

Let $Q$ be the unit cube in $\mathbb{R}^n$ for $n\geq 2$. For $1\leq p<\infty$, $0<q<\infty$, and $s>\sigma_{p,q}$, let $F_{p,q}^{s,\operatorname{rinf}}(Q)$ denote the reinforced f…

Let $G$ be a unimodular exponential solvable Lie group, and let $\tau$ be a monomial representation of $G$. A vector $\eta$ is admissible if the associated wavelet transform…