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1. CJM 2006 (vol 58 pp. 1026)
Karamata Renewed and Local Limit Results 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, $a(n)\sim
f(1-1/n)/ \alpha n$ (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 $(1-F)^{-1}$ (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.
Categories:30B10, 30E15, 41A60, 60J35, 05A16 |
2. CJM 1999 (vol 51 pp. 117)
Meromorphic functions with prescribed asymptotic behaviour, zeros and poles and applications in complex approximation |
Meromorphic functions with prescribed asymptotic behaviour, zeros and poles and applications in complex approximation We construct meromorphic functions with asymptotic power series
expansion in $z^{-1}$ at $\infty$ on an Arakelyan set $A$ having
prescribed zeros and poles outside $A$. We use our results to prove
approximation theorems where the approximating function fulfills
interpolation restrictions outside the set of approximation.
Keywords:asymptotic expansions, approximation theory Categories:30D30, 30E10, 30E15 |