1,850 research outputs found
Behavior of lacunary series at the natural boundary
We develop a local theory of lacunary Dirichlet series of the form
as approaches the
boundary i\RR, under the assumption and further assumptions on
. 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 , the case we primarily focus
on, we obtain blow up rates in measure along the imaginary line and asymptotic
information at . When sufficient analyticity information on exists, we
obtain Borel summable expansions at points on the boundary, giving exact local
description. Borel summability of the expansions provides property-preserving
extensions beyond the barrier. The singular behavior has remarkable
universality and self-similarity features. If , , or
, n\in\NN, behavior near the boundary is roughly of the standard
form where if x=p/q\in\QQ and zero otherwise.
The B\"otcher map at infinity of polynomial iterations of the form
, , turns out to have uniformly
convergent Fourier expansions in terms of simple lacunary series. For the
quadratic map , , and the Julia set is the graph of
this Fourier expansion in the main cardioid of the Mandelbrot set
Proof of the Dubrovin conjecture and analysis of the tritronqu\'ee solutions of
We show that the tritronqu\'ee solution of the Painlev\'e equation , which is analytic for large with is pole-free in a region containing the full sector and the disk . This proves in
particular the Dubrovin conjecture, an open problem in the theory of Painlev\'e
transcendents. The method, building on a technique developed in Costin, Huang,
Schlag (2012), is general and constructive. As a byproduct, we obtain the value
of the tritronqu\'ee and its derivative at zero within less than 1/100 rigorous
error bounds
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