19 problems

Let $PW^p$ denote the Paley–Wiener space in exponent $p$, and let Theorem denote the trace theorem stated earlier in the paper. Trace-theorem extension conjecture. Theorem extends…

Strong divergence conjecture. There exists an $f\in\mathcal{PW}^{1}_{\pi}$ such that

Let $PW=\{f\in L^2(\mathbb{R}):\operatorname{supp}\widehat{f}\subseteq[-\pi,\pi]\}$, and fix $s>0$. Consider all $f\in PW$ with $\|f\|_2=1$ and all measurable sets…

Let $\mathscr{C}_p$ denote the optimization constant considered in the paper for $p>2$. Brevig's conjecture. The function … is decreasing for $p>2$. The conjecture concerns the glo…

Donoho–Stark conjecture. Among all functions $f$ in the Paley–Wiener space and all measurable subsets $\Omega$ of the real line with fixed Lebesgue measure $|\Omega|=\ell$, the con…

Let $n$ be an integer and let an admissible solution mean a solution of the convolution equation defined earlier in the paper satisfying the stated admissibility conditions. Parity…

Let $\varphi$ be a solution of the Paley–Wiener extremal problem, and let $\widehat{\varphi}$ denote its Fourier transform. Fourier-transform shape conjecture. The function…

Let $\varphi_p$ be the extremal function for the Paley–Wiener extremal problem, and let $\widehat{\varphi}_p$ denote its Fourier transform. Fourier-transform regularity conjecture.…

Let $1<p<\infty$, let $\varphi_p$ be the extremal function, and let $\mathscr{Z}(\varphi_p)$ be its zero set. Write $\operatorname{dist}(x,\mathscr{Z}(\varphi_p)\cup\{0\})$ for the…

Let $\varphi_p$ be the unique solution of the extremal problem, let $\mathscr{Z}(\varphi_p)$ denote its zero set, and let $PW^p$ be the Paley–Wiener space. A set is a uniqueness se…

Assume the extremal problem has a unique solution $\varphi_p$, and let $t_1=t_1(p)$ be its first positive zero. First-zero asymptotic conjecture. … and … The source presents these…

Assume the extremal problem has a unique solution $\varphi_p$ for each $p>0$, and let $t_n=t_n(p)$, $n\geq1$, be the positive zeros of $\varphi_p$. Zero-spacing conjecture. … for a…

Assume the extremal problem has a unique solution $\varphi_p$ for each $p>0$, and let $t_n=t_n(p)$, $n\geq1$, be the positive zeros of $\varphi_p$. Zero-monotonicity conjecture. Th…

Let $0<p<\infty$, and let a solution mean a function solving the extremal problem defined earlier in the paper. Uniform-discreteness conjecture. The zero set of any solution of the…

Let $0<p<1$, and let a solution mean a function solving the extremal problem defined earlier in the paper. Evenness conjecture. Any solution of the extremal problem is even. The cl…

For $0<p<\infty$, consider the extremal problem defining the Paley–Wiener extremal function, and call a function attaining the extremum a solution. Uniqueness conjecture. The extre…

Fix $0<p<\infty$, and let $\mathscr{K}_p$ be the best constant in Korevaar's inequality for point evaluation on the imaginary axis, namely…

Let $\mathscr{C}_p$ denote the point-evaluation constant associated with the Paley–Wiener space $PW^p$. Monotonicity conjecture. The function … is strictly decreasing on…

Let $X$ be a configuration sampled from the sine process, with law denoted by ${{\mathbb P}_{\mathscr S}}$, and let $p\in X$ be a particle. A set is hereditarily complete if every…