488 research outputs found
Voronoi means, moving averages, and power series
We introduce a {\it non-regular} generalisation of the N\"{o}rlund mean, and
show its equivalence with a certain moving average. The Abelian and Tauberian
theorems establish relations with convergent sequences and certain power
series. A strong law of large numbers is also proved
A new note on factored infinite series and trigonometric Fourier series
In this paper, we have proved two main theorems under more weaker conditions dealing with absolute weighted arithmetic mean summability factors of infinite series and trigonometric Fourier series. We have also obtained certain new results on the different absolute summability methods
A new note on factored infinite series and trigonometric Fourier series
In this paper, we have proved two main theorems under more weaker conditions dealing with absolute weighted arithmetic mean summability factors of infinite series and trigonometric Fourier series. We have also obtained certain new results on the different absolute summability methods
Asymptotic spectral theory for nonlinear time series
We consider asymptotic problems in spectral analysis of stationary causal
processes. Limiting distributions of periodograms and smoothed periodogram
spectral density estimates are obtained and applications to the spectral domain
bootstrap are given. Instead of the commonly used strong mixing conditions, in
our asymptotic spectral theory we impose conditions only involving
(conditional) moments, which are easily verifiable for a variety of nonlinear
time series.Comment: Published in at http://dx.doi.org/10.1214/009053606000001479 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Regularity of roots of polynomials
We show that smooth curves of monic complex polynomials , with a compact interval, have absolutely continuous roots in a uniform
way. More precisely, there exists a positive integer and a rational number
, both depending only on the degree , such that if
then any continuous choice of roots of is absolutely continuous with
derivatives in for all , in a uniform way with respect to
. The uniformity allows us to deduce also a multiparameter
version of this result. The proof is based on formulas for the roots of the
universal polynomial in terms of its coefficients which we derive
using resolution of singularities. For cubic polynomials we compute the
formulas as well as bounds for and explicitly.Comment: 32 pages, 2 figures; minor changes; accepted for publication in Ann.
Sc. Norm. Super. Pisa Cl. Sci. (5); some typos correcte
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
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