Schur class

In complex analysis, the Schur class is the set of holomorphic functions $f (z)$ defined on the open unit disk $𝔻 = {z \in ℂ : | z | < 1}$ and satisfying $| f (z) | \leq 1$ .

Schur problem (or Carathéodory-Fejér interpolation)

Given complex numbers $c_{0}, c_{1}, \dots, c_{n}$ , find a function $f (z) = \sum_{j = 0}^{n} c_{j} z^{j} + \sum_{j = n + 1}^{\infty} f_{j} z^{j},$ which is analytic and bounded by $1$ on the unit disk. 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 $n$ th-order polynomial. It is closely related to the Levinson algorithm though Schur algorithm is numerically more stable and better suited to parallel processing.

Schur function

Consider the Carathéodory function of a unique probability measure $d μ$ on the unit circle $𝕋 = {z \in ℂ : | z | = 1}$ given by $F (z) = \int \frac{e^{i θ} + z}{e^{i θ} - z} d μ (θ),$ where $\int d μ (θ) = 1$ implies $F (0) = 1$ . Then the association $F (z) = \frac{1 + z f (z)}{1 - z f (z)},$ sets up a one-to-one correspondence between Carathéodory functions and Schur functions $f (z)$ given by the inverse formula: $f (z) = z^{- 1} (\frac{F (z) - 1}{F (z) + 1}) .$

Schur algorithm

Schur's algorithm is an iterative construction based on Möbius transformations that maps one Schur function to another. The algorithm defines an infinite sequence of Schur functions $f \equiv f_{0}, f_{1}, \dots, f_{n}, \dots$ and Schur parameters $γ_{0}, γ_{1}, \dots, γ_{n}, \dots$ (also called Verblunsky coefficient or reflection coefficient) via the recursion: $f_{j + 1} = \frac{1}{z} \frac{f_{j} (z) - γ_{j}}{1 - \overline{γ_{j}} f_{j} (z)}, f_{j} (0) \equiv γ_{j} \in 𝔻,$ which stops if $f_{j} (z) \equiv e^{i θ} = γ_{j} \in 𝕋$ . One can invert the transformation as $f (z) \equiv f_{0} (z) = \frac{γ_{0} + z f_{1} (z)}{1 + \overline{γ_{0}} z f_{1} (z)}$ or, equivalently, as continued fraction expansion of the Schur function $f_{0} (z) = γ_{0} + \frac{1 - | γ_{0} |^{2}}{\overline{γ_{0}} + \frac{1}{z γ_{1} + \frac{z (1 - | γ_{1} |^{2})}{\overline{γ_{1}} + \frac{1}{z γ_{2} + \dots}}}}$ by repeatedly using the fact that $f_{j} (z) = γ_{j} + \frac{1 - | γ_{j} |^{2}}{\overline{γ_{j}} + \frac{1}{z f_{j + 1} (z)}} .$

References

↑ 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, vol. 147, pp. 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 {{citation}}: ISBN / Date incompatibility (help)
↑ 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.
1 2 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

[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, vol. 147, pp. 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 {{citation}}: ISBN / Date incompatibility (help)

[Chung_Parhi_1996_p.-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.

[SimonP12005-4] 1 2 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-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.

[Simon2011-6] 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

Schur class

Schur problem (or Carathéodory-Fejér interpolation)

Schur function

Schur algorithm

See also

References