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Schur class

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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 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]

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Schur function

Consider the Carathéodory function of a unique probability measure on the unit circle given by

where implies .[4] Then the association

sets up a one-to-one correspondence between Carathéodory functions and Schur functions given by the inverse formula:

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Schur algorithm

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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 (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

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References

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