24 problems

Optimal convergence order conjecture. The order of convergence

Bandeira–et-al. degree-three invariant conjecture. If $L\geq 10$, then the generic orbit in $\mathbf{L}^2(S^2)_L$ can be determined up to a finite list of orbits from invariants of…

Three-term recurrence conjecture. There exist matrices $A_p$, $B_p$, and $C_p$, independent of $t$, such that

Generalized hypergeometric expression conjecture. For a given $\ell\geq 0$, the components of the functions $H(t)$ are expressible in terms of generalized hypergeometric functions…

Hook alternating property conjecture. The coefficients $A_k$ with $k$ in the range $j-i\leq k\leq j+i$ have the hook alternating property.

Matrix-valued product-expansion conjecture. The product admits a unique expansion of this form, with matrix coefficients $A_k$, for all nonnegative $n$.

Hypergeometric-form conjecture. For a given $\ell\geq 0$, the spherical functions corresponding to $(\ell,n)$ have components expressible in terms of generalized hypergeometric fun…

Let $G$ be a semisimple linear group over a local field $F$, let $K<G$ be a maximal compact subgroup, and let $H<G$ be a closed abelian subgroup. Define … For a continuous bi-$K$-i…

Let $X$ be an irreducible symmetric space of compact type, with spectral parameter $\mu\in \Lambda^+\subset i\mathfrak{t}^*$ and spatial parameter $H\in\mathfrak{t}$. Let…

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 $G$ be a semisimple Lie group with maximal compact subgroup $K$, let $\mathfrak{a}^*$ be the Fourier-dual space of the vector group $A$, and let $c(i\lambda)$ denote the Harish…

The new families are families of functions depending on a parameter $\nu$, arising in the study of matrix-valued non-symmetric Jacobi polynomials and related shift operators. Ident…

Optimal uniform regularity conjecture. One has

Spherical-function growth conjecture. If $\lambda,X\in\overline{\mathfrak{a}^+}$, then

The global spherical-function estimate. On flat Riemannian symmetric spaces with complex group $G$, one has

Let $G_0$ be the Cartan motion group of a symmetric space with subexponential growth, and let spherical functions be parametrized by spectral variables such as $\lambda_0$ and eval…

Let $G$ be the ambient group, $K$ a maximal compact subgroup, and $\Pi(G)_{\operatorname{sph}}$ the set of unitary irreducible spherical representations of $G$. For $g\in G$, let…

Let the approximation results established in Theorem and Corollary be understood in the setting of uniform-norm approximation by zonal function networks with activation function…

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

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

Explicit lowest-spherical-function conjecture. For all $n\geq 3$,

The second family is a one-parameter sequence of $2\times2$ matrix-valued orthogonal polynomials obtained from the compact Gelfand pair … and spherical functions of type…

Let $uge 0$ be real, let $mud(q-1)/2$, and let $sinPiq$. Here $phis^{mu+nu}$ and $phiy^{mu}$ denote the spherical functions indexed by $s,yinPiq$, and $M^1(Piq)$ denotes the proba…

For an integer $n\geq 1$, let … where $c>0$, and let $\Psi_{2n+1}^+$ denote the class of strictly positive definite isotropic functions on the $(2n+1)$-sphere. Beatson's conjecture…