Repeated integrals of increasing functions

TL;DR

This paper characterizes the possible endpoint integral values of increasing functions and constructs smooth functions with prescribed integral properties, addressing a problem in comonotone approximation.

Contribution

It provides a complete characterization of integral triplets for increasing functions and constructs smooth functions matching these integral conditions.

Findings

Characterization of possible integral values for increasing functions.

Construction of smooth functions with prescribed integral and derivative conditions.

Proof of conjecture for derivatives at endpoints for functions with increasing derivatives.

Abstract

Motivated by a problem on comonotone approximation of $C^n$ functions by entire functions, for increasing functions $f\colon[0,1]\to[0,1]$, we characterize the possible values of $(a,b,c)$, where $a=I(f)(1)$, $b=I^2(f)(1)$, $c=I^3(f)(1)$ ($I$ is the integral operator $I(f)(x)=\int_0^xf(t)\,dt$), as those which satisfy the conditions $0\leq a\leq 1$, $a^2/2\leq b\leq a/2$, $2b^2\leq 3ac$, $a^2 + 4b^2 + 6c\leq 6ac +2ab+2b$, and $0\leq c\leq a/6$. Our main theorem states that if $a,b,c$ are real numbers for which the inequalities are strict, then there is a function $f$ satisfying $a=I(f)(1)$, $b=I^2(f)(1)$, $c=I^3(f)(1)$ which is $C^\infty$ with $f(0)=0$, $f(1)=1$, $Df(x)>0$ for $0<x<1$, and whose derivatives $D^jf(0)$ and $D^jf(1)$, $j\geq 1$, are arbitrary as long as they are consistent with the increasing nature of $f$. The construction of $f$ proceeds by starting with a continuous parametrization $s\mapsto \rho_s\in C^\infty([0,1])$ defined on an open subset of $\mathbb{R}^4$, and composing with successive continuous transversals through the open set to fix the values of $I^j(\rho_s)(1)$ for $j=0,1,2,3$. Addressing the aforementioned problem on comonotone approximation, we examine the set $V_n\subseteq\mathbb{R}^{2(n+1)}$ of possible values $D^jf(0)$, $D^jf(1)$, $j=0,\dots,n$, of the derivatives of a $C^n$ function at the endpoints when $D^nf$ is increasing but not constant. We make a conjecture about the nature of this set and prove our conjecture for $n\leq 3$ as a consequence of the theorem mentioned above.