Asymptotic Properties of Convolution Products of Sequences
Publications de l'Institut Mathématique, _N_S_36 (1984) no. 50, p. 67
Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts
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 :
40A05 40A25
@article{PIM_1984_N_S_36_50_a9,
author = {Edward Omey},
title = {Asymptotic {Properties} of {Convolution} {Products} of {Sequences}},
journal = {Publications de l'Institut Math\'ematique},
pages = {67 },
publisher = {mathdoc},
volume = {_N_S_36},
number = {50},
year = {1984},
language = {en},
url = {http://geodesic.mathdoc.fr/item/PIM_1984_N_S_36_50_a9/}
}
Edward Omey. Asymptotic Properties of Convolution Products of Sequences. Publications de l'Institut Mathématique, _N_S_36 (1984) no. 50, p. 67 . http://geodesic.mathdoc.fr/item/PIM_1984_N_S_36_50_a9/