Convolution operators preserving the set of totally positive sequences

TL;DR

This paper characterizes convolution operators that preserve total positivity in sequences, showing specific conditions under which the property is maintained and exploring related open problems.

Contribution

It identifies conditions for sequences that preserve total positivity under termwise multiplication and proposes open problems on convolution operators maintaining this property.

Findings

Sequences of the form (a_k a^{-k(k-1)}) are totally positive if a^2 ≥ 3.503.

Characterization of operators preserving total positivity.

Open problems on convolution operators and total positivity.

Abstract

A real sequence $(a_k)_{k=0}^\infty$ is called {\it totally positive} if all minors of the infinite Toeplitz matrix $ \left\| a_{j-i} \right\|_{i, j =0}^\infty$ are nonnegative (here $a_k=0$ for $k<0$). In this paper, which continues our earlier work \cite{kv}, we investigate the set of real sequences $(b_k)_{k=0}^\infty$ with the property that for every totally positive sequence $(a_k)_{k=0}^\infty,$ the sequense of termwise products $(a_k b_k)_{k=0}^\infty$ is also totally positive. In particular, we show that for every totally positive sequence $(a_k)_{k=0}^\infty$ the sequence $\left(a_k a^{-k (k-1)}\right)_{k=0}^\infty$ is totally positive whenever $a^2\geq 3{.}503.$ We also propose several open problems concerning convolution operators that preserve total positivity.