On Characterizations of Convex and Approximately Subadditive Sequences

TL;DR

This paper characterizes convex and approximately subadditive sequences, showing their properties, local polynomial interpolation, stability, and their interrelation under minimal assumptions.

Contribution

It provides new characterizations of convex sequences, establishes stability results for approximately subadditive sequences, and links convex and subadditive sequences with minimal assumptions.

Findings

Convex sequences can be locally interpolated by quadratic polynomials.

Approximately subadditive sequences can be decomposed into subadditive and bounded sequences.

Stability results for approximately subadditive sequences are established.

Abstract

A sequence $\Big(u_n\Big)_{n=0}^{\infty}$ is said to be convex if it satisfies the following inequality $$ 2u_n\leq u_{n-1}+u_{n+1}\qquad \mbox{for all}\qquad n\in\mathbb{N}. $$ We present several characterizations of convex sequences and demonstrate that such sequences can be locally interpolated by quadratic polynomials. Furthermore, the converse assertion of this statement is also established. On the other hand, a sequence $\Big(u_n\Big)_{n=1}^{\infty}$ is called approximately subadditive if for a fixed $\epsilon>0$ and any partition $n_1,\cdots,n_k$ of $n\in\mathbb{N}$; the following discrete functional inequality holds true $$ u_n\leq u_{n_1}+\cdots+ u_{n_k}+\varepsilon. $$ We show Ulam's type stability result for such sequences. We prove that an approximately subadditive sequence can be expressed as the algebraic summation of an ordinary subadditive and a non-negative sequence bounded above by $\varepsilon.$ A proposition portraying the linkage between the convex and subadditive sequences under minimal assumption is also included.