Schur class explained

f(z) defined on the open unit disk D = \{ z\in C : |z|< 1 \} and satisfying |f(z)| \leq 1 that solve the Schur problem: Given complex numbers c _{0,c} _{1,...c,c} _{n} , find a functionf(z) = \sum ^{n} _{j =0} c _{j} z ^{j} + \sum ^{n} _{j = n +1} f _{j} z ^{j} which is analytic and bounded by 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 orthogonal polynomials which can be used as orthonormal basis functions to expand any th-order polynomial.^[1] It is closely related to the Levinson algorithm though Schur algorithm is numerically more stable and better suited to parallel processing.^[2]

Schur function

Consider the Carathéodory function of a unique probability measure

d\mu on the unit circle T = \{z\in C :|z| =1 \} given by F(z) = \int (e ^{i\theta} + z)/(e ^{i\theta} - z) d\mu(\theta) where \int d\mu(\theta) = 1 implies F( 0 ) =1 . Then the associationF(z) = (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( (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.^[3] The algorithm defines an infinite sequence of Schur functions

f\equiv f _{0,f} _{1,...c,f} _{n,...c} and Schur parameters \gamma _{0,\gamma} _{1,...c,\gamma} _{n,...c} (also called Verblunsky coefficient or reflection coefficient) via the recursion:f _{j+1} = (1)/(z) (f _{j(z)-\gamma} _{j})/(1- \overline{\gamma _{j}) f _{j(z)},} f _{j(0)\equiv} \gamma _{j} \in D , which stops if f _{j(z)\equiv} e ^{i\theta} = \gamma _{j} \in T . One can invert the transformation asf(z)\equiv f _{0} (z) = (\gamma _{0} + zf _{1(z)})/(1+ \overline{\gamma _{0}) z f _{1(z)} } or, equivalently, as continued fraction expansion of the Schur functionf _{0(z)=\gamma} + _{0+ (1- |\gamma ^{2} _{0 |})/(\overline {\gamma _{0})} (1)/(z \gamma _{1+ (z( 1- |\gamma ^{2 )} _{1 |})/(\overline {\gamma _{1})} + (1)/(z\gamma _{2+ …})) }}by repeatedly using the fact thatf _{j(z)=\gamma} + _{j + (1- |\gamma ^{2} _{j|})/(\overline {\gamma _{j})} (1)/(zf _{j+1} (z)) }.

See also

• Orthogonal polynomials on the unit circle
• Szegő polynomial

Notes and References

1. Book: Chung . Jin-Gyun . Parhi . Keshab K. . The Kluwer International Series in Engineering and Computer Science . Pipelined Lattice and Wave Digital Recursive Filters . Springer US . Boston, MA . 1996 . 978-1-4612-8560-1 . 0893-3405 . 10.1007/978-1-4613-1307-6 . 79.
2. Book: Hayes, Monson H.. Statistical digital signal processing and modeling . 242. 1996 . John Wiley & Son . 978-0-471-59431-4. 34243409.
3. Book: Conway, John B. . Functions of One Complex Variable I (Graduate Texts in Mathematics 11) . Graduate Texts in Mathematics . Springer-Verlag . 1978 . 978-0-387-90328-6. 127.