Schur class
In complex analysis, the Schur class is the set of holomorphic functions defined on the open unit disk and satisfying that solve the Schur problem: Given complex numbers , find a function
which is analytic and bounded by 1 on the unit disk.[1] The method of solving this problem as well as similar problems (e.g. solving Toeplitz systems and Nevanlinna-Pick interpolation) is known as the Schur algorithm (also called Coefficient stripping or Layer stripping). One of the algorithm's most important properties is that it generates n + 1 orthogonal polynomials which can be used as orthonormal basis functions to expand any nth-order polynomial.[2] It is closely related to the Levinson algorithm though Schur algorithm is numerically more stable and better suited to parallel processing.[3]
Schur function[edit]
Consider the Carathéodory function of a unique probability measure on the unit circle given by
where implies .[4] Then the association
sets up a one-to-one correspondence between Carathéodory functions and Schur functions given by the inverse formula:
Schur algorithm[edit]
Schur's algorithm is an iterative construction based on Möbius transformations that maps one Schur function to another.[4][5] The algorithm defines an infinite sequence of Schur functions and Schur parameters (also called Verblunsky coefficient or reflection coefficient) via the recursion:[6]
which stops if . One can invert the transformation as
or, equivalently, as continued fraction expansion of the Schur function
by repeatedly using the fact that
See also[edit]
References[edit]
- ^ Schur, J. (1918), "Über die Potenzreihen, die im Innern des Einheitkreises beschränkten sind. I, II", Journal für die reine und angewandte Mathematik, Operator Theory: Advances and Applications, 147: 205–232, I. Schur Methods in Operator Theory and Signal Processing in: Operator Theory: Advances and Applications, vol. 18, Birkhäuser, Basel, 1986 (English translation), doi:10.1007/978-3-0348-5483-2, ISBN 978-3-0348-5484-9
- ^ Chung, Jin-Gyun; Parhi, Keshab K. (1996). Pipelined Lattice and Wave Digital Recursive Filters. The Kluwer International Series in Engineering and Computer Science. Boston, MA: Springer US. p. 79. doi:10.1007/978-1-4613-1307-6. ISBN 978-1-4612-8560-1. ISSN 0893-3405.
- ^ Hayes, Monson H. (1996). Statistical digital signal processing and modeling. John Wiley & Son. p. 242. ISBN 978-0-471-59431-4. OCLC 34243409.
- ^ Jump up to: a b Simon, Barry (2005), Orthogonal polynomials on the unit circle. Part 1. Classical theory, American Mathematical Society Colloquium Publications, vol. 54, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3446-6, MR 2105088
- ^ Conway, John B. (1978). Functions of One Complex Variable I (Graduate Texts in Mathematics 11). Springer-Verlag. p. 127. ISBN 978-0-387-90328-6.
- ^ Simon, Barry (2010), Szegő's theorem and its descendants: spectral theory for L² perturbations of orthogonal polynomials, Princeton University Press, ISBN 978-0-691-14704-8