On approximately Convex and Affine Sequences

In this paper, our primary objective is to study a possible decomposition of an approximately convex sequence. For a given ε>0; a sequence <uₙ>ₙ₌₀∞ is said to be ε-convex, if for any i,j with $i<j$ there exists an n∈]i,j]∩ N such that the following discrete functional inequality holds {equation*} { u_i-uᵢ₋₁-ε/n-i≤ u_j-uⱼ₋₁. } {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 [-ε/2, ε/2]. On the other hand, if for any i,j with $i<j$, if a sequence <uₙ>ₙ₌₀∞ satisfies the following form of inequality {equation*} { |(u_i-uᵢ₋₁)-(u_j-uⱼ₋₁)|≤ε/n-i some n∈]i,j]; } {equation*} then we term it as ε-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 ε.