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Generalized Variation and Functions of Slow Growth

Published online by Cambridge University Press:  20 November 2018

Robert D. Berman
Robert D. Berman*
Affiliation:
Wayne State University, Detroit, Michigan
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Many of the basic results of HP theory on the disk Δ = {|z| < 1} are proved using the Cauchy-Stieltjes representation

1.1

and the Poisson-Stieltjes representation

1.2

Here, μ:R → C is a complex-valued function of a real variable that is of bounded variation on [0, 2π] such that μ(t + 2π) = μ(t) + μ(2t) — μ(0), t ∊ R,

is the Cauchy kernel, and

is the Poisson kernel. It is therefore natural to generalize these representations in such a way that some of the basic properties and results carry over. Such a generalization occurs when the assumption that μ is of bounded variation on [0, 2μ] is replaced by the requirement that it is measurable and bounded on [0, 2μ] (cf. [9]). The integrals in (1.1) and (1.2) are then defined by a formal integration by parts. After some preliminaries in Section 2, we catalogue a variety of results which remain valid in Section 3.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 1 , 01 February 1988 , pp. 55 - 85
Copyright
Copyright © Canadian Mathematical Society 1988

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