## Orthogonal polyanalytic polynomials and normal matrices

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## 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.## References

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## Additional Information

**Marko Huhtanen**- Affiliation: SCCM program, Computer Science Department, Stanford University, Stanford, California 94305
- Address at time of publication: Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 01239
- Email: Marko.Huhtanen@hut.fi
- Received by editor(s): September 12, 2000
- Received by editor(s) in revised form: March 1, 2001
- Published electronically: February 22, 2002
- Additional Notes: This work was supported by the Academy of Finland and the Alfred Kordelin Foundation
- © Copyright 2002 American Mathematical Society
- Journal: Math. Comp.
**72**(2003), 355-373 - MSC (2000): Primary 42C05; Secondary 15A57
- DOI: https://doi.org/10.1090/S0025-5718-02-01417-5
- MathSciNet review: 1933825