# Sampling expansions associated with quaternion difference equations

## TL;DR

This paper develops sampling expansions in quaternion function spaces derived from difference equations and explores the spectral properties of quaternion matrices, linking them to quaternion polynomials.

## Contribution

It introduces a novel quaternion difference equation framework and connects quaternion matrices with polynomial eigenvalue problems, expanding quaternion analysis tools.

## Key findings

- Derived a complete quaternion sequence from difference equations.
- Established sampling expansions in quaternion function spaces.
- Linked quaternion matrices to characteristic polynomials and eigenvalues.

## Abstract

Starting with a quaternion difference equation with boundary conditions, a parameterized sequence which is complete in finite dimensional quaternion Hilbert space is derived. By employing the parameterized sequence as the kernel of discrete transform, we form a quaternion function space whose elements have sampling expansions. Moreover, through formulating boundary-value problems, we make a connection between a class of tridiagonal quaternion matrices and polynomials with quaternion coefficients. We show that for a tridiagonal symmetric quaternion matrix, one can always associate a quaternion characteristic polynomial whose roots are eigenvalues of the matrix. Several examples are given to illustrate the results.