42 research outputs found
Polynomial interpolation and Gaussian quadrature for matrix valued functions
The techniques for polynomial interpolation and Gaussian quadrature are
generalized to matrix-valued functions. It is shown how the zeros and
rootvectors of matrix orthonormal polynomials can be used to get a quadrature
formula with the highest degree of precision
Orthogonal matrix polynomials and applications
AbstractOrthogonal matrix polynomials, on the real line or on the unit circle, have properties which are natural generalizations of properties of scalar orthogonal polynomials, appropriately modified for the matrix calculus. We show that orthogonal matrix polynomials, both on the real line and on the unit circle, appear at various places and we describe some of them. The spectral theory of doubly infinite Jacobi matrices can be described using orthogonal 2×2 matrix polynomials on the real line. Scalar orthogonal polynomials with a Sobolev inner product containing a finite number of derivatives can be studied using matrix orthogonal polynomials on the real line. Orthogonal matrix polynomials on the unit circle are related to unitary block Hessenberg matrices and are very useful in multivariate time series analysis and multichannel signal processing. Finally we show how orthogonal matrix polynomials can be used for Gaussian quadrature of matrix-valued functions. Parallel algorithms for this purpose have been implemented (using PVM) and some examples are worked out
A connection between orthogonal polynomials on the unit circle and matrix orthogonal polynomials on the real line
Szego's procedure to connect orthogonal polynomials on the unit circle and
orthogonal polynomials on [-1,1] is generalized to nonsymmetric measures. It
generates the so-called semi-orthogonal functions on the linear space of
Laurent polynomials L, and leads to a new orthogonality structure in the module
LxL. This structure can be interpreted in terms of a 2x2 matrix measure on
[-1,1], and semi-orthogonal functions provide the corresponding sequence of
orthogonal matrix polynomials. This gives a connection between orthogonal
polynomials on the unit circle and certain classes of matrix orthogonal
polynomials on [-1,1]. As an application, the strong asymptotics of these
matrix orthogonal polynomials is derived, obtaining an explicit expression for
the corresponding Szego's matrix function.Comment: 28 page