On Characterizations of Convex and Approximately Subadditive Sequences
Angshuman Robin Goswami
TL;DR
The paper addresses convex and approximately subadditive sequences in discrete settings, showing that convex sequences admit several characterizations and a local quadratic interpolation via $P_2^{n}(x)$ with $u_{n-1}-2u_n+u_{n+1}\ge 0$, while approximately subadditive sequences satisfy a robust Ulam-type stability. It proves that convexity is equivalent to a monotone slope representation and to local quadratic interpolation, providing both forward and converse implications. For approximately subadditive sequences, it establishes that such a sequence can be decomposed as $u_n=v_n+w_n$ with $v_n$ subadditive and $0\le w_n\le \varepsilon$, and gives a constructive method for obtaining $v_n$ from partitions. The results link convexity and subadditivity under minimal assumptions and yield discrete functional inequalities and stability insights with potential applications in numerical analysis and discrete modelling.
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 $ε>0$ 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.