Schur class


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 on the unit disk. The method of solving this problem as well as similar problems is known as the Schur algorithm. 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. 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 on the unit circle given by
where implies. Then the association
sets up a one-to-one correspondence between Carathéodory functions and Schur functions given by the inverse formula:

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 and Schur parameters via the recursion:
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