Abstract view
Karamata Renewed and Local Limit Results


Published:20061001
Printed: Oct 2006
Abstract
Connections between behaviour of real analytic functions (with no
negative Maclaurin series coefficients and radius of convergence one)
on the open unit interval, and to a lesser extent on arcs of the unit
circle, are explored, beginning with Karamata's approach. We develop
conditions under which the asymptotics of the coefficients are related
to the values of the function near $1$; specifically, $a(n)\sim
f(11/n)/ \alpha n$ (for some positive constant $\alpha$), where
$f(t)=\sum a(n)t^n$. In particular, if $F=\sum c(n) t^n$ where $c(n)
\geq 0$ and $\sum c(n)=1$, then $f$ defined as $(1F)^{1}$ (the
renewal or Green's function for $F$) satisfies this condition if $F'$
does (and a minor additional condition is satisfied). In come cases,
we can show that the absolute sum of the differences of consecutive
Maclaurin coefficients converges. We also investigate situations in
which less precise asymptotics are available.
MSC Classifications: 
30B10, 30E15, 41A60, 60J35, 05A16 show english descriptions
Power series (including lacunary series) Asymptotic representations in the complex domain Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15] Transition functions, generators and resolvents [See also 47D03, 47D07] Asymptotic enumeration
30B10  Power series (including lacunary series) 30E15  Asymptotic representations in the complex domain 41A60  Asymptotic approximations, asymptotic expansions (steepest descent, etc.) [See also 30E15] 60J35  Transition functions, generators and resolvents [See also 47D03, 47D07] 05A16  Asymptotic enumeration
