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Approximation by Unimodular Functions

Published online by Cambridge University Press:  20 November 2018

Stephen Fisher
Stephen Fisher*
Affiliation:
Northwestern University, Evanston, Illinois
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The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].

In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form

*

where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 257 - 269
Copyright
Copyright © Canadian Mathematical Society 1971

References

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