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Search: MSC category 30E15 ( Asymptotic representations in the complex domain )

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1. CJM 2006 (vol 58 pp. 1026)

Handelman, David
 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)

Sauer, A.
 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 theoryCategories:30D30, 30E10, 30E15