8 problems

Convergence conjecture. The series converges when

Let $f\in L^2_{\operatorname{imp}}$, let $(A_k)$ be pairwise disjoint Borel subsets of $\mathbb R_+$, and define $f_k=f\mathbf 1_{A_k}$. Let $(S_n)$ be the arrival times of a Poiss…

Let $N$ be a Poisson random measure and let $L^2_{\operatorname{imp}}$ be the space of bounded, square-integrable functions for which the relevant improper integrals exist. For a f…

Let $b^2 \notin \mathbb Q$ and suppose that $b^2$ is not a Liouville number: there exist $m,q_0\in\mathbb N$ such that, for every $(p,q)\in\mathbb Z\times\mathbb Z_{\geq q_0}$, … L…

Let $\Omega=\{x+iy:x>0,\ y\in\mathbb{R}^*\}$, and for $z\in\Omega$ let $\{c_n(z)\}_{n\in\mathbb{N}}$ be the sequence defined from the sets $\Delta_n(z)$ and the associated sequence…

Convergence conjecture. This condition is sufficient for the series in those cases to converge.

Let $q$ be the nome in the series expansions … where the functions $\mathcal{P}^{(\ell)}_n(z)$ and coefficients $\mathcal{E}^{(\ell)}_n$ arise in the constructed solutions of the n…

Let $f$ be a nonanalytic, possibly periodic function, and let $\hat{f}(t)$ denote the associated series. Piecewise-constant series conjecture. There exist such functions $f$ for wh…