16 problems

Ismail's reproducing kernel Hilbert space conjecture. The operators with kernels defined by these functions should have an underlying reproducing kernel Hilbert space structure.

Let $\Theta_N$ be the space of theta functions equipped with the metric specified in the paper, and let $K_v$ denote its reproducing kernel at $v$. Sklyanin's reproducing-kernel co…

Coefficient ratio conjecture. Then

Let $X$ be the compact Hausdorff space equipped with the group action and relation from the paper, and let $K_x$ denote the reproducing kernel associated with $x\in X$. Write…

Let $d$ be the dimension, let $Omega\subset[0,\infty)^2$ be a simple low-pass spectral shape, and let $N$ be a positive real bandwidth. Define … and let…

Let $\Omega$ be a bounded symmetric domain of rank $r$, with invariants $p$, $b$, $d$, and $q_\Omega$, and let $\Gamma_\Omega$ denote the associated gamma function. Let $m$ and…

Polyharmonic-spline eta-norm conjecture. The $\eta$-norm is given by

Let $D$ be a bounded planar domain with analytic boundary and connectivity $n>1$. For $z e w$ in $D$, let $GD(w,z)$ be the Green function and define the logarithmic capacity by … L…

Let $K_{[t_0,T]}$ be the reproducing kernel of the LQ state space, let $K_{[t_0,T]}(t_0,t_0)^{\ominus}$ denote its pseudoinverse for the associated seminorm, and let…

Let $D$ be a planar regular region whose boundary consists of $n$ analytic Jordan curves. Let $\hat{R}(z,\bar{w})$ be its conjugate Hardy $H^{2}$ kernel and $B(z,\bar{w})$ its Berg…

Let a Fock space be a Hilbert space of analytic functions with its associated system of reproducing kernels. A system of reproducing kernels is complete and minimal if its closed l…

Let $\mathbb T$ be the one-dimensional torus, let $K_1$ be the kernel of $\mathcal H_r^{\mathrm{Kor}}(\mathbb T)$, and let $d_{\mathbb T}$ denote the torus distance. Korobov-kernel…

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

A-F-M conjecture. The multiplication operator $M_z$ on $\mathcal H(K_1+K_2)$ is subnormal.

Let $K$ be a generalized Mercer kernel, and let $K^{\ast(2m-1)}$ denote its $(2m-1)$-fold kernel convolution. A Green function can compute the kernel $K^{\ast(2m-1)}$. This conject…

Let $\mathbb{M}$ be a general $C^\infty$ compact Riemannian manifold, let $\kappa_m$ be the reproducing kernel considered in the paper, and let $s_f$ denote the $\kappa_m$-interpol…