Chebyshev–Hankel matrices and the splitting approach for centrosymmetric Toeplitz-plus-Hankel matrices

Abstract

Centrosymmetric Toeplitz-plus-Hankel matrices are investigated on the basis of their “splitting property”, which is their similarity to the direct sum of two special Toeplitz-plus-Hankel matrices. These matrices can be considered as Hankel matrices (moment matrices) in bases of Chebyshev polynomials and are called Chebyshev–Hankel matrices. Chebyshev–Hankel matrices have similar properties like Hankel matrices. This concerns inversion formulas and fast algorithms. A superfast algorithm for solving Chebyshev–Hankel and centrosymmetric Toeplitz-plus-Hankel systems is presented that is based on real trigonometric transforms. The main tool of investigation is the interpretation of Chebyshev–Hankel matrices as matrices of restricted multiplication operators with respect to Chebyshev bases.