Schur class
In complex analysis, the Schur class is the set of holomorphic functions [math]\displaystyle{ f(z) }[/math] defined on the open unit disk [math]\displaystyle{ \mathbb{D} = \{ z\in \mathbb{C} : |z|\lt 1\} }[/math] and satisfying [math]\displaystyle{ |f(z)| \leq 1 }[/math] that solve the Schur problem: Given complex numbers [math]\displaystyle{ c_0,c_1,\dotsc,c_n }[/math], find a function
- [math]\displaystyle{ f(z) = \sum_{j=0}^{n} c_j z^j + \sum_{j=n+1}^{n}f_j z^j }[/math]
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
Consider the Carathéodory function of a unique probability measure [math]\displaystyle{ d\mu }[/math] on the unit circle [math]\displaystyle{ \mathbb{T} =\{z\in\mathbb{C} :|z|=1\} }[/math] given by
- [math]\displaystyle{ F(z) = \int \frac{e^{i\theta} + z}{e^{i\theta} - z} d\mu(\theta) }[/math]
where [math]\displaystyle{ \int d\mu(\theta) = 1 }[/math] implies [math]\displaystyle{ F(0)=1 }[/math].[4] Then the association
- [math]\displaystyle{ F(z) = \frac{1+zf(z)}{1-zf(z)} }[/math]
sets up a one-to-one correspondence between Carathéodory functions and Schur functions [math]\displaystyle{ f(z) }[/math] given by the inverse formula:
- [math]\displaystyle{ f(z) = z^{-1}\left( \frac{F(z)-1}{F(z)+1} \right) }[/math]
Schur algorithm
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 [math]\displaystyle{ f\equiv f_0,f_1,\dotsc,f_n,\dotsc }[/math] and Schur parameters [math]\displaystyle{ \gamma_0,\gamma_1,\dotsc,\gamma_n,\dotsc }[/math] (also called Verblunsky coefficient or reflection coefficient) via the recursion:[6]
- [math]\displaystyle{ f_{j+1}=\frac{1}{z}\frac{f_j(z)-\gamma_j}{1-\overline{\gamma_j}f_j(z)}, \quad f_j(0)\equiv \gamma_j \in \mathbb{D}, }[/math]
which stops if [math]\displaystyle{ f_j(z)\equiv e^{i\theta} = \gamma_j \in \mathbb{T} }[/math]. One can invert the transformation as
- [math]\displaystyle{ f(z)\equiv f_0 (z) = \frac{\gamma_0 + zf_1(z)}{1 + \overline{\gamma_0} z f_1(z) } }[/math]
or, equivalently, as continued fraction expansion of the Schur function
- [math]\displaystyle{ f_0(z)=\gamma_0+\frac{1-|\gamma_0|^2}{\overline {\gamma_0}+\frac{1}{z \gamma_1+\frac{z(1-|\gamma_1|^2)}{\overline {\gamma_1}+\frac{1}{z\gamma_2+\cdots}}}} }[/math]
by repeatedly using the fact that
- [math]\displaystyle{ f_j(z)=\gamma_j+\frac{1-|\gamma_j|^2}{\overline {\gamma_j}+\frac{1}{zf_{j+1}(z)}}. }[/math]
See also
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 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.
- ↑ Hayes, Monson H. (1996). Statistical digital signal processing and modeling. John Wiley & Son. p. 242. ISBN 978-0-471-59431-4. OCLC 34243409. https://www.worldcat.org/oclc/34243409.
- ↑ 4.0 4.1 Simon, Barry (2005), Orthogonal polynomials on the unit circle. Part 1. Classical theory, American Mathematical Society Colloquium Publications, 54, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3446-6, https://books.google.com/books?id=d94r7kOSnKcC
- ↑ 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, https://books.google.com/books?id=e9R5gcz_x0YC
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