In complex analysis, the Schur class is the set of holomorphic functions
defined on the open unit disk
that solve the Schur problem: Given complex numbers
, find a function 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] Consider the Carathéodory function of a unique probability measure
on the unit circle
∫ d μ ( θ ) = 1
[4] Then the association sets up a one-to-one correspondence between Carathéodory functions and Schur functions
given by the inverse formula: 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
and Schur parameters
γ
(also called Verblunsky coefficient or reflection coefficient) via the recursion:[6] 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