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Schur class

From Wikipedia, the free encyclopedia

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

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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

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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

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References

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  1. ^ 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
  2. ^ 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.
  3. ^ Hayes, Monson H. (1996). Statistical digital signal processing and modeling. John Wiley & Son. p. 242. ISBN 978-0-471-59431-4. OCLC 34243409.
  4. ^ 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
  5. ^ 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.
  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