61 problems

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 $p$ be a prime, let ${{\mathbb Q}}_p$ be the field of $p$-adic numbers, and let $\Omega(|x|_p)$ denote the characteristic function of the unit disc. A function $\phi$ satisfyin…

General surjectivity conjecture. Some much more general surjectivity result should hold for affine systems generated by $\psi$, allowing synthesis surjectively onto $L^p$, Hardy, o…

Norm-arcwise connectedness conjecture. The set $\mathcal{W}(D,T)$ is norm-arcwise-connected.

The preceding construction associates generalized filter systems to unitary-matrix-valued functions on ${\mathbb T}$ whose dimensions vary with $x\in{\mathbb T}$, and an analogue o…

Let a divergence-free three-dimensional vector wavelet basis consist of wavelet functions $\boldsymbol{\psi}_\lambda(\boldsymbol{x})$, and let $r$-regularity require rapid decrease…

Consider an orthonormal solenoidal wavelet basis on $\mathbb{R}^3$, obtained by the integral helical pull-up of a scalar basis. A wavelet is $r$-regular when it decreases rapidly a…

Let $\psi_\lambda$ be scalar wavelets used to construct an orthonormal solenoidal wavelet basis by helical pull-up, and let $r$-regularity mean rapid decrease of the wavelet functi…

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 $A,P\in\mathrm{GL}(d,\mathbb{R})$ and $\psi\in L^2(\mathbb{R}^d)$. Consider the wavelet system … Assume that this system is an orthonormal basis, or more generally a Parseval f…

Ionascu-type interpolation conjecture. There exists a measurable, nonnegative function $h$ such that $f\leq h\leq\max\{g_1,g_2\}$ a.e. and

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 $psi\in L^{2}(\mathbb{R})$ and define … Suppose that the wavelet system $\{\psi_{j,k}:j,k\in\mathbb{Z}\}$ is orthonormal. The discrete Calderón condition conjecture. The condit…

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 $u$…

For the orthonormal Fourier basis and the orthonormal Haar wavelet basis, the local-coherence estimate used in the paper involves the numerical factor $2700$, through the bound…

Let $n\in{\mathbb N}$, $-n<\varrho<0$, $s\in{\mathbb R}$, and $0<p<\infty$, $0<q\leq\infty$. Let ${\varrho}$-$B^s_{p,q}({\mathbb R}^n)$ denote either…

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 $Y_\ell^0$ be the degree-$\ell$, order-zero spherical harmonic, let $\eta_a$ denote the wavelet at scale $a$, and let $\kappa_0$ and $\widetilde\eta$ be the functions appearing…

Seip's conjecture. The quantity

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 $\widehat{\beta}_\psi$ denote the empirical mean of a wavelet basis function, and let $\beta_\psi^p$ denote its wavelet mean under the density $p$. Consider density estimators…

Wavelet characterization conjecture. The Bergman spaces $A^p_\alpha$ should admit a wavelet characterization throughout this entire range, without the restriction requiring…

Let $W_{\mathrm{Aff}}^{\psi}$ denote the affine Wigner distribution of a function $\psi\in L^{2}(\mathbb{R}_{+})$. The affine positivity conjecture. The only functions…