open access

https://doi.org/10.14943/83687

https://hdl.handle.net/2115/69291

The Nevanlinna counting functions for Rudin's orthogonal functions

英語

2001-12

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

542

1-7

H∞ and H2 denote the Hardy spaces on the open unit disc: D. Let cf> be a function in H∞ and 11</Jll ∞ = 1. If cf> is an inner function and ¢(0) = 0, then { cpn ; n = 0, 1, 2, · · ·} is orthogonal in H2 . \\7.Rudin asked if the converse is true and C.Sundberg and C.Bishop showed that the converse is not true. Therefore there exists a function c/J such that c/J is not an inner function and { c/Jn } is orthogonal in H2. In this paper, the following is shown : { q'Jn } is orthogonal in H2 if and only if there exists a uniqueprobability measure v0 on [0,1] with 1 E supp v0 such that N4,(z) - S log r/|z| dv0(r) for nearly all z in D where Nc/J is the Nevanlinna counting function of c/J. If 6 is an inner function, then v0 is a Dirac: measure at r = 1.

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