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

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1. CJM Online first

Speissegger, Patrick
Quasianalytic Ilyashenko algebras
I construct a quasianalytic field $\mathcal{F}$ of germs at $+\infty$ of real functions with logarithmic generalized power series as asymptotic expansions, such that $\mathcal{F}$ is closed under differentiation and $\log$-composition; in particular, $\mathcal{F}$ is a Hardy field. Moreover, the field $\mathcal{F} \circ (-\log)$ of germs at $0^+$ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.

Keywords:generalized series expansion, quasianalyticity, transition map
Categories:41A60, 30E15, 37D99, 03C99

2. 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

3. 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 theory
Categories:30D30, 30E10, 30E15

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