Orthogonal polyanalytic polynomials and normal matrices

Huhtanen, Marko

doi:10.1090/S0025-5718-02-01417-5

Orthogonal polyanalytic polynomials and normal matrices
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Math. Comp. 72 (2003), 355-373 Request permission

Abstract:

The Hermitian Lanczos method for Hermitian matrices has a well-known connection with a 3-term recurrence for polynomials orthogonal on a discrete subset of $\mathbb {R}$. This connection disappears for normal matrices with the Arnoldi method. In this paper we consider an iterative method that is more faithful to the normality than the Arnoldi iteration. The approach is based on enlarging the set of polynomials to the set of polyanalytic polynomials. Denoting by $d$ the number of elements computed so far, the arising scheme yields a recurrence of length bounded by $\sqrt {8d}$ for polyanalytic polynomials orthogonal on a discrete subset of $\mathbb {C}$. Like this slowly growing length of the recurrence, the method preserves, at least partially, the properties of the Hermitian Lanczos method. We employ the algorithm in least squares approximation and bivariate Lagrange interpolation.

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