Schur class - Alchetron, The Free Social Encyclopedia

In mathematics, the Schur class consists of the Schur functions: the holomorphic functions from the open unit disk to the closed unit disk. These functions were studied by Schur (1918).

The Schur parameters γ_j of a Schur function f₀ are defined recursively by

γ j = f j ( 0 ) z f j + 1 = f j ( z ) − γ j 1 − γ j ¯ f j ( z ) .

The Schur parameters γ_j all have absolute value at most 1.

This gives a continued fraction expansion of the Schur function f₀ by repeatedly using the fact that

f j ( z ) = γ j + 1 − | γ j | 2 γ j ¯ + 1 z f j + 1 ( z )

which gives

f 0 ( z ) = γ 0 + 1 − | γ 0 | 2 γ 0 ¯ + 1 z γ 1 + z ( 1 − | γ 1 | 2 ) γ 1 ¯ + 1 z γ 2 + ⋯ .

References

Schur class Wikipedia

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