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Behavior of Lacunary Series at the Natural Boundary by O. Costin; M. Huang is a Mathematics article available to read on EtoBox.

What is Behavior of Lacunary Series at the Natural Boundary about?

We develop a local theory of lacunary Dirichlet series of the form ∞ k=1 c k exp(-zg(k)), Re(z) > 0 as z approaches the boundary iR, under the assumption g → ∞ and further assumptions on c k . These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with c k = 1, the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at z = 0. When sufficient analyticity information on g exists, we obtain Borel summable expansions at points on the boundary, giving exact local description. Borel summability of the expansions provides propertypreserving extensions beyond the barrier. The singular behavior has remarkable universality and self-similarity features. If The Bötcher map at infinity of polynomial iterations of the form x n+1 = λP (x n ), |λ| < λ 0 (P ), turns out to have uniformly convergent Fourier expansions in terms of simple lacunary series. For the quadratic map P (x) = xx 2 , λ 0 = 1, and the Julia set is the graph of this Fourier expansion in the main cardioid of the Mandelbrot set.

Who reads Behavior of Lacunary Series at the Natural Boundary?

Author

O. Costin; M. Huang

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0001-8708)

Published

2009

Language

EN

Field

Mathematics (Physical Sciences)