14 problems

Gröchenig's conjecture. For every $n\in\mathbb{N}$,

Let $Wf(z)$ be the Wigner distribution of some function $f\in L^2(\mathbb{R})$, with bounded nodal set $\mathcal{N}(Wf)$, and suppose that $Wf$ is centered at $z=0$. For a Hermite…

Asymptotic bound conjecture. A stronger property is conjectured to hold: for any $n\geq 1$,

The frame set conjecture for Hermite functions. The frame set for Hermite functions of even orders is

Let $h_n(x)$ and $F_n(x)=h_n(x)/h_{n+1}(x)$ be the ratios considered above, and let $g_{\alpha,\beta}(x)$ denote the comparison function introduced in the paper. Hermite-ratio uppe…

Lyubarskii–Nes conjecture. Besides this obstruction, there are no other obstructions preventing the resulting Gabor system from being a frame.

Let $f$ and $g$ be finite linear combinations of Hermite functions, and call their Wigner distribution zero-free if it has no zeros. Nonexistence conjecture. No zero-free Wigner di…

Rotational invariance conjecture. The following are equivalent:

Let $\{\mathbf{T}_n\}_{0\le n<N}$ be the minimal Hermite-type basis of $\mathbb{R}^N$, and let $\psi_n$ denote the corresponding Hermite functions. Convergence conjecture. The vect…

Let $H u$ denote the Hermite function of order $u$, and let $u<0$. Hermite-function quotient monotonicity conjecture. The function … is decreasing. This property would imply the…

Let $I_N$ be the index set used for the discrete Fourier transform, and let ${\boldsymbol\phi}_n\in\mathbb R^N$ be the constructed eigenvectors. Define the number of zero crossings…

Let $N=4L+1$, let $I_N$ be the index set used for the discrete Fourier transform, and let ${\boldsymbol\phi}_n$ be the orthonormal eigenvectors constructed in the paper. Set … Let…

Higher-dimensional Hermite coefficient decay conjecture. A similar result should hold for functions on ${\mathbb R}^n$ in higher dimensions.

Let $\psi \in L^2(\mathbb{R}^n)$ satisfy the hypotheses of Theorem 4.7, namely Gaussian bounds with $a=b=\tanh 2t<1$ in Hardy's theorem. Write $(\psi,\Phi_\alpha)$ for the Fourier–…