Functions consistent with real numbers, and global extrema of functions in exponential Takagi class

TL;DR

This paper investigates the global extrema and extreme points of a class of functions called the exponential Takagi class, which generalizes the classic Takagi function, analyzing their properties based on polynomial and series behavior.

Contribution

It introduces a detailed study of the global extrema of exponential Takagi functions for various parameters, expanding understanding of their extremal structure and properties.

Findings

Characterization of global extrema for different parameter values

Identification of sets of extreme points for the functions

Analysis based on properties of consistent and anti-consistent polynomials

Abstract

The functions of the Takagi exponential class are similar in construction to the continuous, nowhere differentiable Takagi function described in 1901. They have one real parameter $v\in (-1;1)$ and at points $x\in{\mathbb R}$ are defined by the series $T_v(x) = \sum_{n=0}^\infty v^n T_0(2^nx)$, where $T_0(x)$ is the distance between $x$ and the nearest integer point. If $v=1/2$ then $T_v$ coincides with Takagi's function. In this paper, for different values of the parameter $v$, we study the global extremes of the functions $T_v$, as well as the sets of extreme points. All functions of $T_v$ have a period of $1$, so they are investigated only on the segment $[0;1]$. This study is based on the properties of consistent and anti-consistent polynomials and series, which the first half of the work is devoted to.