On approximately monotone and approximately Holder functions

Goswami, Angshuman Robin and Páles, Zsolt (2020) On approximately monotone and approximately Holder functions. PERIODICA MATHEMATICA HUNGARICA, 81 (1). pp. 65-87. ISSN 0031-5303

Preview

Text 1909.06243.pdf Download (287kB) | Preview

Abstract

A real valued function f defined on a real open interval I is called Phi-monotone if, for all x, y is an element of I with x <= y it satisfiesf(x) <= f(y) + Phi(y - x),where Phi : [0, l(I)[-> R+ is a given nonnegative error function, where l(I) denotes the length of the interval I. If f and - f are simultaneously Phi-monotone, then f is said to be a Phi-Holder function. In the main results of the paper, we describe structural properties of these function classes, determine the error function which is the most optimal one. We show that optimal error functions for Phi-monotonicity and Phi-Holder property must be subadditive and absolutely subadditive, respectively. Then we offer a precise formula for the lower and upper Phi-monotone and Phi-Holder envelopes. We also introduce a generalization of the classical notion of total variation and we prove an extension of the Jordan Decomposition Theorem known for functions of bounded total variations.

Actions (login required)

Edit Item