31 problems

Let $G$ be the finite reflection group associated with the Dunkl operators, let $k\geq 0$ be a multiplicity function, and let $K_G$ be the generalized exponential kernel defined by…

Let $K$ denote the Dunkl kernel associated with the multiplicity function $k$. For each $y\in\mathbb R^N$, consider the function $\xi\mapsto K(iy,\xi)$ on $\mathbb R^N$. Dunkl-kern…

Let $d\geq 1$, set $\lambda=\frac{d-1}{2}$, let $C_n^\lambda$ denote the Gegenbauer polynomial, and let $\mathbb{N}_0=\{0,1,2,\ldots\}$. For $\delta\geq\lambda+1$, any…

Let $d\geq 1$, set $\lambda=\frac{d-1}{2}$, and let $({\theta}-t)_+=\max\{{\theta}-t,0\}$. For $\delta\geq\lambda+1$ and any ${\theta}\in(0,\pi)$, define … Beatson–Castell–Xu conje…

Let $S^{n-1}$ be the unit sphere, and let a positive definite function on $S^{n-1}$ be understood in the usual sense for functions on the sphere. Damase's continuity approximation…

For $n>2$, consider the homogeneous space … Call a homogeneous space $G/H$ pathological if it admits a positive definite function that is not a pointwise limit of continuous positi…

Let $n>2$, let $f:[-1,1]\to\mathbb{R}$ satisfy $f(1)=1$, and define $\widehat f:SO(n)\to\mathbb{R}$ by $\widehat f(\gamma)=f(\gamma_{11})$. Assume that $\widehat f$ defines a posit…

Let $SO(n)$ be the special orthogonal group, and let $SO(n-1)$ be the subgroup fixing the first coordinate. A class-one representation means an irreducible representation admitting…

Let $K_r$ denote the weighted Kubo–Mori mean and $L$ the logarithmic mean, and define … A function is positive definite if, for every finite set of real numbers $t_1,\ldots,t_n$, t…

Let $U(d)$ be the unitary group, let $H(d)$ be the set of complex Hadamard matrices in $U(d)$, and let $B(d)$ be the optimal positive-definite-functions upper bound for mutually un…

Let $U(d)$ be the unitary group, let $H(d)$ denote the set of complex Hadamard matrices in $U(d)$, and let $B_{\leq k}(d)$ be the infimum of $h(\bm I)$ over real-valued positive de…

Let $\mathcal{S}_n$ be the symmetric group and let $c:\mathcal{S}_n\to\mathbb{C}$ satisfy … for every complex-valued function $x$ on $\mathcal{S}_n$. For $A=[a_{i,j}]\in{\cal H}_n$…

Let $\mathbb{F}$ be the free group, let $\mathbb{B}_r$ be the radius-$r$ ball, and let $\mathrm{NSPD}(r,d)$ denote the class of normalized strictly positive definite $d\times d$ ma…

Optimality conjecture. This constant is the best among the admissible ones. The conjecture asserts that the positive-definite-function approach gives the optimal constant for this…

Beatson–zu Castell–Xu conjecture. A necessary and sufficient condition for positivity for every dimension is that all Taylor coefficients $c_k$ in this power expansion are positive…

Let $\alpha,\beta\in(0,\infty)\setminus\mathbb{N}$, and let $F_n^{(\alpha,\beta),\delta}(t)$ denote the Jacobi-polynomial integral defined earlier in the paper. For $t\in(0,\pi]$ a…

Positivity conjecture. $F_n^{\lambda,\delta}(t)>0$ for every $t\in(0,\pi]$ if $\delta\geq\lambda+1$.

Let $D\subset\mathbb{R}^{n}$ be the domain in the spatial Turán problem, and let $a_{\mathbb{R}^{n}}(D)$ denote the supremum of $\widehat{h}(0)$ over admissible functions $h$ suppo…

Even-moment asymptotic conjecture. There is a constant $c(k)>0$ such that, as $j\to\infty$,

Higher-derivative representation conjecture. The second part of the lemma holds for all $k\geq 1$.

For $0<t<\pi$ and an integer $\mu\geq 2$, let … Here $(t-\theta)_+=\max\{t-\theta,0\}$, and the corresponding sphere is $\mathbb{S}^{2\mu-1}$. Strict positive definiteness conjectu…

Let $S^d$ be the $d$-dimensional sphere, and let $K$ be a continuous, positive definite, zonal kernel on $S^d$. Write its radial part as $K_r^d:[-1,1]\to\mathbb{R}$, so that…

Let $Y=\{y_k\}_{k=1}^{n+2}$ be a configuration of $n+2$ points in $\mathbb{R}^{n+1}$, and let ${\mathcal S}_Y(\Omega_n)$ denote its Schoenberg matrix for the Schoenberg kernel…

Let $G$ be a semisimple Lie group with a unitary, possibly projective, discrete-series representation $\pi$ on a Hilbert space $H$. Assume that the restriction $\pi|_\Gamma$ to a l…

Let $G$ be a semisimple Lie group and let $bGamma$ be a lattice in $G$. Let $K_{G,\Gamma}$ be the convex set of continuous positive definite functions $\phi$ on $G$ satisfying … ……