location:  Publications → journals
Search results

Search: MSC category 30B10 ( Power series (including lacunary series) )

 Expand all        Collapse all Results 1 - 2 of 2

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 2003 (vol 55 pp. 1019)

Handelman, David
 More Eventual Positivity for Analytic Functions Eventual positivity problems for real convergent Maclaurin series lead to density questions for sets of harmonic functions. These are solved for large classes of series, and in so doing, asymptotic estimates are obtained for the values of the series near the radius of convergence and for the coefficients of convolution powers. Categories:30B10, 30D15, 30C50, 13A99, 41A58, 42A16
 top of page | contact us | privacy | site map |