ASYMPTOTIC PROPERTIES OF CONVOLUTION PRODUCTS OF SEQUENCES

E. Omey

Abstract: Suppose three sequences $\{a_n\}_{\bold N}$, $\{b_n\}_{\boldN}$ and $\{c_n\}_{\bold N}$ are related by the equation $c_n=\sum^n_{k=0}a_{n-k}b_k$. In this paper we examine the asymptotic behavior of $c_n/a_n$ under various conditions on $\{a_n\}_{\bold N}$ and $\{b_n\}_{\bold N}$. If $\sum^\infty_{k=0}|b_k|<\infty$ we discuss conditions under which $c_n/a_n\to\sum^n_{k=0}b_k$ and give sharp rate of convergence results. From our results we obtain asymptotic expansions of the form $$ c_n = a_n \sum^\infty_{k=0} b_k + (a_n - a_{n-1}) \sum^\infty_{k=1} k b_k + O (|a_n - a_{n-1}|/n). $$

Classification (MSC2000): 40A05, 40A25

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