Sampling expansions associated with quaternion difference equations

Abstract

Starting with a quaternion difference equation with boundary conditions, a parameterized sequence that is complete in the finite-dimensional quaternion Hilbert space is derived. By employing the parameterized sequence as the kernel of the discrete transform, we construct a quaternion function space whose elements have sampling expansions. It is shown that the sample points of the sampling expansions are the eigenvalues of an associated quaternion difference operator. For the operator that has multiple standard eigenvalues, we provide a method to find the ‘missing sample points’. Through formulating the boundary-value problems, we make a connection between a class of tridiagonal quaternion matrices and the polynomials with quaternion coefficients. We prove that for a tridiagonal symmetric quaternion matrix, one can always associate a quaternion characteristic polynomial whose roots are the eigenvalues of the matrix. By representing colour image pixels as quaternions, an application of the proposed sampling theorem in colour image encryption is discussed. Throughout the paper, examples are given to illustrate the results.