Convolution operations arising from Vandermonde matrices

TL;DR

This paper explores convolution operations involving large Vandermonde matrices, analyzing their behavior, convergence, and dependencies, and provides algorithms and simulations to verify theoretical results and discuss computational challenges.

Contribution

It introduces a comprehensive framework for understanding and computing convolutions of Vandermonde matrices, including criteria for phase or spectral dependence and efficient implementation techniques.

Findings

Convergence of Vandermonde matrix convolutions is almost sure.

Classification of convolutions based on phase distribution or spectra.

Implementation techniques using Fourier-Motzkin elimination are effective.

Abstract

Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution. First additive and multiplicative convolution of Vandermonde matrices and deterministic diagonal matrices are considered. After this, several cases of additive and multiplicative convolution of two independent Vandermonde matrices are considered. It is also shown that the convergence of any combination of Vandermonde matrices is almost sure. We will divide the considered convolutions into two types: those which depend on the phase distribution of the Vandermonde matrices, and those which depend only on the spectra of the matrices. A general criterion is presented to find which type applies for any given convolution. A simulation is presented, verifying the results. Implementations of all considered convolutions are provided and discussed, together with the challenges in making these implementations efficient. The implementation is based on the technique of Fourier-Motzkin elimination, and is quite general as it can be applied to virtually any combination of Vandermonde matrices. Generalizations to related random matrices, such as Toeplitz and Hankel matrices, are also discussed.