Showing papers on "Uniform boundedness published in 1970"

Holomorphic functions of polynomial growth on bounded domains

Abstract: In [4] R. Narasimhan proved the following theorem concerning holomorphie functions of polynomial growth: Suppose g is a holomorphic function defined on some open neighborhood of the closure of a bounded open subset gt of Cn. If ] is a holomorphic function of polynomial growth on t such that ] gh for some holomorphic function h on t, then h has polynomial growth on t. It is natural to raise the following question:

On Measures Determined by Continuous Functions that are not of Bounded Variation

Abstract: In [1] it was shown that a continuous function of bounded variation on the real line determined a Method II outer measure for which the Borel sets were measurable and the measure of an open interval was equal to the total variation of f over the interval. The monotone property of measures implied that if an open interval I on which f was not of bounded variation contained subintervals on which f was of finite but arbitrarily large total variation then the measure of I was infinite. Since there are continuous functions that are not of bounded variation over any interval (e.g. the Weierstrasse nondifferentiable function) the general case was not resolved.