Schur class

In complex analysis, the Schur class is the set of holomorphic functions $f(z)$ defined on the open unit disk $\mathbb {D} =\{z\in \mathbb {C} :|z|<1\}$ and satisfying $|f(z)|\leq 1$ that solve the Schur problem: Given complex numbers $c_{0},c_{1},\dotsc ,c_{n}$ , find a function

f(z)=\sum _{j=0}^{n}c_{j}z^{j}+\sum _{j=n+1}^{n}f_{j}z^{j}

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 $n$ th-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 $d\mu$ on the unit circle $\mathbb {T} =\{z\in \mathbb {C} :|z|=1\}$ given by

F(z)=\int {\frac {e^{i\theta }+z}{e^{i\theta }-z}}d\mu (\theta )

where $\int d\mu (\theta )=1$ implies $F(0)=1$ .^[4] Then the association

F(z)={\frac {1+zf(z)}{1-zf(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}\left({\frac {F(z)-1}{F(z)+1}}\right)

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 $f\equiv f_{0},f_{1},\dotsc ,f_{n},\dotsc$ and Schur parameters $\gamma _{0},\gamma _{1},\dotsc ,\gamma _{n},\dotsc$ (also called Verblunsky coefficient or reflection coefficient) via the recursion:^[6]

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} ,

which stops if $f_{j}(z)\equiv e^{i\theta }=\gamma _{j}\in \mathbb {T}$ . One can invert the transformation as

f(z)\equiv f_{0}(z)={\frac {\gamma _{0}+zf_{1}(z)}{1+{\overline {\gamma _{0}}}zf_{1}(z)}}

or, equivalently, as continued fraction expansion of the Schur function

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 }}}}}}}}

by repeatedly using the fact that

f_{j}(z)=\gamma _{j}+{\frac {1-|\gamma _{j}|^{2}}{{\overline {\gamma _{j}}}+{\frac {1}{zf_{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, 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.
^ ^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

[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

[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] 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

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Schur function

Schur algorithm

See also

References