On approximately Convex and Affine Sequences

TL;DR

This paper investigates the structure of approximately convex and affine sequences, showing they can be decomposed into sums of convex or affine sequences and bounded sequences, with precise bounds depending on the approximation parameter.

Contribution

It introduces a novel decomposition for approximately convex and affine sequences into sums of standard convex or affine sequences and bounded sequences, with explicit bounds.

Findings

Approximately convex sequences decompose into convex plus bounded sequences.

Approximately affine sequences decompose into affine plus bounded sequences.

Explicit bounds on the bounded components are provided based on .

Abstract

In this paper, our primary objective is to study a possible decomposition of an approximately convex sequence. For a given $\varepsilon>0$; a sequence $\big<u_n\big>_{n=0}^{\infty}$ is said to be $\varepsilon$-convex, if for any $i,j\in\mathbb{N}$ with $i<j$ there exists an $n\in]i,j]\cap \mathbb{N}$ such that the following discrete functional inequality holds \begin{equation*} { u_i-u_{i-1}-\dfrac{\varepsilon}{n-i}\leq u_j-u_{j-1}. } \end{equation*} We show that such a sequence can be represented as the algebraic summation of a convex and a controlled sequence which is bounded in between $\left[-\dfrac{\varepsilon}{2}, \dfrac{\varepsilon}{2}\right].$ On the other hand, if for any $i,j\in\mathbb{N}$ with $i<j$, if a sequence $\big<u_n\big>_{n=0}^{\infty}$ satisfies the following form of inequality \begin{equation*} { \left|\big(u_i-u_{i-1}\big)-\big(u_j-u_{j-1}\big)\right|\leq\dfrac{\varepsilon}{n-i}\quad \quad\mbox{for some} \quad n\in]i,j]\cap\mathbb{N}; } \end{equation*} then we term it as $\varepsilon$-affine sequence. Such a sequence can be decomposed as the algebraic summation of an affine and a bounded sequence whose supremum norm doesn't exceed $\varepsilon.$