29 problems

Let $\mathcal{S}$ denote the class of normalized univalent functions in the unit disk, and let $\mathcal{S}\ast\mathcal{S}$ be the class of convolutions $f\ast g$ with…

The Student family of probability laws is infinitely divisible but is not closed under convolution. Consider instead families consisting of mixtures of Student distributions. Closu…

Schinzel–Schmidt minimizer conjecture. Among all probability density functions supported on $[0,\frac12]$, this function minimizes $\|f\ast f\|_\infty$.

Let $f$ be a probability density function supported on $[-\frac14,\frac14]$. The norms $\|f\ast f\|_\infty$ and $\|f\ast f\|_2$ are defined using the autoconvolution $f\ast f$. Aut…

Let $X_1$ and $X_2$ be independent and identically distributed random variables, and let ${\mathcal{C}}(X)$ denote the Poincaré-type constant defined in the paper. Consider the nor…

Lusztig's tensor-category conjecture. One should have $K\circ K'\in\mathcal{C}_{\mathcal{U}}$, and the assignment $(K,K')\mapsto K\circ K'$ should define a tensor-category structur…

A probability density function (pdf) supported on $[0,\frac12]$ is nonnegative and integrates to $1$. For such a function, write $f\ast f$ for its autoconvolution and…

Let $f$ be a probability density function supported on $[-\frac14,\frac14]]$. The convolution $f\ast f$ has supremum norm $\|f\ast f\|_\infty$ and $L^2$ norm $\|f\ast f\|_2$. Convo…

Let ${\mathbf G}$ be a reductive group over a finite field, let $\rho$ be a good representation of ${\mathbf G}$, and let $\psi$ be a non-trivial additive character of the finite f…

Let ${\mathbf G}$ be a reductive group over a finite field, let $\rho$ be a good representation, and let $\psi$ be a non-trivial additive character of the finite field. Let…

Let $\mu$ and $\nu$ be compactly supported probability measures, let $c\in[0,\infty]$, and let $\mu\oplus_c\nu$ denote their $c$-convolution. Let $f\in C^{4}(\mathbb{R})$ satisfy…

Zartash–Robeva conjecture. The convolution $f*g$ is log-supermodular.

Hecke convolution equivalence conjecture. The functor $\mathrm{Conv}^{\mathrm{Hecke}}$ is an equivalence of categories between…

Pólya–Schoenberg conjecture. The class $\mathcal{K}$ is preserved under convolution: if $f,g\in\mathcal{K}$, then $f*g\in\mathcal{K}$.

Direct-sum convolution conjecture. If

Saeki's $L^p$-conjecture. The space $L^p(G)$ is closed under convolution if and only if $G$ is compact.

Let $G$ be a locally compact group and let $p$ satisfy $1<p<\infty$. The Lebesgue space $L^p(G)$ consists of measurable functions on $G$ whose $p$th powers are integrable, with con…

Let $C_4$ be the smallest constant such that … for the function class specified in Theorem 4 of the source. The compactly supported formulation concerns functions…

Let $\scalebox{1.15}{\textsf{\upshape A}}^{\left(n\right)}$ and $\scalebox{1.15}{\textsf{\upshape C}}^{\left(n'\right)}$ be a polyadic associative algebra and a coassociative coalg…

Let $G$ be a locally compact group, and let $L^p(G)$ denote the space of measurable functions on $G$ whose $p$th powers are integrable, with convolution defined by … Classical…

Diaconis–Perlman unique crossing conjecture. If $\eta\prec\theta$, but $\theta$ is not a permutation of $\eta$, then $F_\eta(x)-F_\theta(x)$ changes signs exactly once, from $-$ to…

Let $F$ be a distribution, let $F^{*n}$ denote its $n$-fold convolution, and let $p_n\ge0$ satisfy … The corresponding compound distribution is $\sum_{n=0}^{\infty}p_nF^{*n}$. Let…

Generalized Fuglede conjecture. The measure $\mu$ is spectral if and only if there exist $\nu$ and $Q$ as above such that

Let $f_\beta=h_\beta+\overline{g_\beta}$ and $F_a=H_a+\overline{G_a}$ be harmonic mappings, where $f_\beta$ has dilatation $\omega(z)=e^{i\theta}z^n$ with $\theta\in\mathbb{R}$ and…

Let $\bmu$ be a parabolic character, and let $\sHg^c$ and $\sWg^c$ be the associated twisted equivariant perverse-sheaf categories. Convolution is the product operation on these ca…