Very slowly varying functions

J. Marshall Ash, P. Erdős, L. A. Rubel

Research output: Contribution to journalArticle

14 Citations (Scopus)

Abstract

A real-valued function f of a real variable is said to be φ{symbol}-slowly varying (φ{symbol}-s.v.) if limx→∞φ{symbol}(x) [f(x+α)-f(x)]=0 for each α. It is said to be uniformly φ{symbol}-slowly varying (u. φ{symbol}-s.v.) if limx→∞ supα ∈ Iφ{symbol}(x) |f(x+α)-f(x)|=0 for every bounded interval I. It is supposed throughout that φ{symbol} is positive and increasing. It is proved that if φ{symbol} increases rapidly enough, then every φ{symbol}-s.v. function f must be u. φ{symbol}-s.v. and must tend to a limit at ∞. Regardless of the rate of increase of φ{symbol}, a measurable function f must be u. φ{symbol}-s.v. if it is φ{symbol}-s.v. Examples of pairs (φ{symbol},f) are given that illustrate the necessity for the requirements on φ{symbol} and f in these results.

Original languageEnglish
Pages (from-to)1-9
Number of pages9
JournalAequationes Mathematicae
Volume10
Issue number1
DOIs
Publication statusPublished - Feb 1974

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Slowly Varying Function
Real variables
Measurable function

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  • Mathematics(all)

Cite this

Very slowly varying functions. / Ash, J. Marshall; Erdős, P.; Rubel, L. A.

In: Aequationes Mathematicae, Vol. 10, No. 1, 02.1974, p. 1-9.

Research output: Contribution to journalArticle

Ash, J. Marshall ; Erdős, P. ; Rubel, L. A. / Very slowly varying functions. In: Aequationes Mathematicae. 1974 ; Vol. 10, No. 1. pp. 1-9.
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