Continuous functions with complicated local structure that defined in terms of alternating Cantor series representation of numbers

TL;DR

This paper studies a class of continuous functions with complex local structures defined via alternating Cantor series, exploring their differential, integral, and self-affine properties, and solving related functional equations.

Contribution

It introduces a new class of continuous functions based on alternating Cantor series and investigates their properties and associated functional equations.

Findings

Functions have unique solutions to specific functional equations.

Functions exhibit complex local structures and self-affinity.

Differential and integral properties are characterized for these functions.

Abstract

The article is devoted to one infinite parametric class of continuous functions with complicated local structure. In the article differential, integral, self-affine and other properties of functions, that their argument is represented by the alternating Cantor series are investigated. The functional equations systems, that investigating functions are unique solution of these systems in the class of determined and bounded on $[0;1]$ functions are indicated. The investigation was represented in seminar on fractal analysis of Institute of Mathematics of the National Academy of Sciences of Ukraine on, October 16, 2014 (http://www.imath.kiev.ua/events/index.php?seminarId=21&archiv=1). The following investigations in the list of references in the article one can to find by the corresponding links: [6] (http://fmi.npu.edu.ua/images/files/publications/naukchasopys1/NZ2013_15.pdf ) [7] (http://fmi.npu.edu.ua/images/files/publications/naukchasopys1/NZ2013_14.pdf) [8] (http://fmi.npu.edu.ua/images/files/publications/naukchasopys1/NZ2012_13_2.pdf) [9](http://www.ekmair.ukma.edu.ua/bitstream/handle/123456789/7409/Serbeniuk_Funktsii_oznacheni_systemamy.pdf)