arXiv subjects

Aaron Agulnick

Publications and source records attributed to Aaron Agulnick.

2 recordsLinked to original sources

arXiv2026

The Askey-Wilson Function is the Laplace Transform of a Positive Measure

The Askey-Wilson function $ϕ(x, γ)$ is a meromorphic continuation of the Askey-Wilson polynomials in their spectral variable off the lattice $γ= \tilde{a} q^m, m \in \mathbb{Z}_+$. We show that the analytic part of the Askey-Wilson function $ϕ^{an}(x, \tilde{a} q^γ)$ is the Laplace transform of a positive measure of total measure $ϕ^{an}(x, s_0) < \infty$, under certain conditions on the parameters governing the function. We deduce log-convexity of this function, as well as positive-definiteness and an integral formula for a certain intertwining operator associated with the Askey-Wilson operator $L$. This offers a $q$-analogue of several results, in the trigonometric limit due to Sahi-McSwiggen, and in the rational (i.e. Dunkl) limit due to Rösler.

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arXiv2026

Higher-Order Autocorrelations on Finite Abelian Groups

The question of determining a signal from its higher-order autocorrelation data is of practical interest in fields as varied as X-ray crystallography, image processing, and satellite communications. At the heart of the issue is how much of this autocorrelation data one truly needs. We prove two new upper bounds on the order of data needed to determine a signal on a general (i.e. not necessarily cyclic) finite abelian group depending on some knowledge of the vanishing of the signal's Fourier transform. In investigating lower bounds on the required data, we classify signals on $\mathbb{Z}_6$ not determined by their fifth-order data and provide analogous examples on $\mathbb{Z}_{30}$.

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