502 research outputs found
Explicit Hermite-type eigenvectors of the discrete Fourier transform
The search for a canonical set of eigenvectors of the discrete Fourier
transform has been ongoing for more than three decades. The goal is to find an
orthogonal basis of eigenvectors which would approximate Hermite functions --
the eigenfunctions of the continuous Fourier transform. This eigenbasis should
also have some degree of analytical tractability and should allow for efficient
numerical computations. In this paper we provide a partial solution to these
problems. First, we construct an explicit basis of (non-orthogonal)
eigenvectors of the discrete Fourier transform, thus extending the results of
[7]. Applying the Gramm-Schmidt orthogonalization procedure we obtain an
orthogonal eigenbasis of the discrete Fourier transform. We prove that the
first eight eigenvectors converge to the corresponding Hermite functions, and
we conjecture that this convergence result remains true for all eigenvectors.Comment: 21 pages, 4 figures, 1 tabl
Wiener-Hopf factorization and distribution of extrema for a family of L\'{e}vy processes
In this paper we introduce a ten-parameter family of L\'{e}vy processes for
which we obtain Wiener-Hopf factors and distribution of the supremum process in
semi-explicit form. This family allows an arbitrary behavior of small jumps and
includes processes similar to the generalized tempered stable, KoBoL and CGMY
processes. Analytically it is characterized by the property that the
characteristic exponent is a meromorphic function, expressed in terms of beta
and digamma functions. We prove that the Wiener-Hopf factors can be expressed
as infinite products over roots of a certain transcendental equation, and the
density of the supremum process can be computed as an exponentially converging
infinite series. In several special cases when the roots can be found
analytically, we are able to identify the Wiener-Hopf factors and distribution
of the supremum in closed form. In the general case we prove that all the roots
are real and simple, and we provide localization results and asymptotic
formulas which allow an efficient numerical evaluation. We also derive a
convergence acceleration algorithm for infinite products and a simple and
efficient procedure to compute the Wiener-Hopf factors for complex values of
parameters. As a numerical example we discuss computation of the density of the
supremum process.Comment: Published in at http://dx.doi.org/10.1214/09-AAP673 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On extrema of stable processes
We study the Wiener--Hopf factorization and the distribution of extrema for
general stable processes. By connecting the Wiener--Hopf factors with a certain
elliptic-like function we are able to obtain many explicit and general results,
such as infinite series representations and asymptotic expansions for the
density of supremum, explicit expressions for the Wiener--Hopf factors and the
Mellin transform of the supremum, quasi-periodicity and functional identities
for these functions, finite product representations in some special cases and
identities in distribution satisfied by the supremum functional.Comment: Published in at http://dx.doi.org/10.1214/10-AOP577 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Asymptotic approximations to the Hardy-Littlewood function
The function was introduced by Hardy
and Littlewood [10] in their study of Lambert summability, and since then it
has attracted attention of many researchers. In particular, this function has
made a surprising appearance in the recent disproof by Alzer, Berg and
Koumandos [1] of a conjecture by Clark and Ismail [2]. More precisely, Alzer
et. al. [1] have shown that the Clark and Ismail conjecture is true if and only
if for all . It is known that is unbounded in the
domain from above and below, which disproves the Clark and
Ismail conjecture, and at the same time raises a natural question of whether we
can exhibit at least one point for which . This turns out to
be a surprisingly hard problem, which leads to an interesting and non-trivial
question of how to approximate for very large values of . In this
paper we continue the work started by Gautschi in [7] and develop several
approximations to for large values of . We use these approximations
to find an explicit value of for which .Comment: 16 pages, 3 figures, 2 table
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