26 problems

Let $d\geq 1$ and let $f\in L^2(\mathbb{R}^d)$ be nonzero. A sequence of pure translates of $f$ is a sequence of the form $(f(\cdot-\lambda_n))_{n\geq 1}$ for…

Let $A$ be a unital C-algebra, let $\mathcal H$ be a Hilbert $A$-module, and let $\{x_j:j\in\mathbb J\}$ be a standard frame with two standard alternate dual frames…

Generalized monomial Carleson-frame conjecture. The sequence

Let $H$ be a Hilbert space, let $T_1,T_2\in B(H)$ commute, and suppose that $\\{T_1^iT_2^j\varphi\\}_{i,j\geq 0}$ is an overcomplete frame for $H$. Let $R_1$ and $R_2$ be the assoc…

Let $H$ be a Hilbert space, let $T_1,T_2\in B(H)$ commute, and suppose that $\\{T_1^iT_2^j\varphi\\}_{i,j\geq 0}$ is a frame for $H$ satisfying the conditions of Lemma. The iterati…

Strong-frame conjecture. Every $\mathfrak s$-frame for $\mathcal S$ induced by a pair of sequences as in Theorem 2 of KNS is a strong frame.

Let $\varphi(t)=2^{1/4}e^{-\pi t^2}$ be the Gaussian atom, and let $\alpha,\beta>0$. For the rectangular lattice $\alpha\mathbb{Z}\times\beta\mathbb{Z}$, write…

Let $(\Omega,\mu)$ and $(\Delta,\nu)$ be measure spaces, let $\mathcal{H}$ be a Hilbert space, and let $\{\tau_\alpha\}_{\alpha\in\Omega}$ and $\{\omega_\beta\}_{\beta\in\Delta}$ b…

Let $H$ be the separable Hilbert space under consideration. A subset of $H$ is $ell^1$-bounded if its coefficient sums are uniformly bounded with respect to some Riesz basis of $H$…

Let $\mathcal{H}$ be a Hilbert space and let $E(\mathcal{H})$ denote the class of operators referred to in the source. Let $T\in E(\mathcal{H})$. The Hilbert space $\mathcal{H}$ ca…

Let $H$ be a separable Hilbert space, let $X=(x_j)_{j=1}^{\infty}$ and $F=(f_j)_{j=1}^{\infty}$ be sequences in $H$, and let $m=(m_j)_{j=1}^{\infty}$ be the symbol of the unconditi…

Schauder–Badea conjecture. The series converges unconditionally for every $x \in H$ if and only if there exist scalar sequences $(c_n)_{n \in \mathbb{N}}$ and…

The unilateral frame number conjecture.

Fourier completeness criterion. The Fourier frequencies $\{e_n\}_{n\in\mathbb{N}_0^d}$ are total in $L^2(\mu)$ if and only if $v$ is non-constant, namely, not proportional to…

Normal-reductive extension conjecture. Theorem remains true if $A$ is a normal reductive operator.

Normalized-frame conjecture. The system above is not a frame for $\mathcal{H}$.

Atomicity conjecture. If $R(\Phi)<\infty$, then $(X,\mu)$ is atomic.

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…

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…

Complete boundedness conjecture. The example was conjectured to fail to induce a completely bounded operator-valued measure.

Let $n$ be the dimension of the complex Hilbert space and let $m$ be the number of vectors in a frame. A frame is phase retrievable when its coefficient magnitudes determine every…

Let $m^*_{\mathbb{C}}(n)$ denote a critical cardinal such that phase retrievable frames are generic for every $m\geq m^*_{\mathbb{C}}(n)$, while no phase retrievable frame of $m$ v…

Let $D$ and $T$ denote the dyadic dilation and integer translation operators, respectively. Let $\{D^jT_k\psi_1\}_{j,k\in\mathbb Z}$ be a wavelet frame for $L^2(\mathbb R)$ with up…

Let $f_1,\ldots,f_n\in L^2(\mathbb{R})$ and let $\Gamma_1,\ldots,\Gamma_n\subset\mathbb{R}$. For $f\in L^2(\mathbb{R})$ and $\Gamma\subset\mathbb{R}$, write … where…

Casazza–Han–Larson framing conjecture. This framing should fail to induce a completely bounded operator-valued measure.