4,384 research outputs found
Sharp approximations to the Bernoulli periodic functions by trigonometric polynomials
We obtain optimal trigonometric polynomials of a given degree that
majorize, minorize and approximate in the
Bernoulli periodic functions. These are the periodic analogues of two works of
F. Littmann that generalize a paper of J. Vaaler. As applications we provide
the corresponding Erd\"{o}s-Tur\'{a}n-type inequalities, approximations to
other periodic functions and bounds for certain Hermitian forms.Comment: 14 pages. Accepted for publication in the J. Approx. Theory. V2 has
additional references and some typos correcte
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
Bandlimited approximations to the truncated Gaussian and applications
In this paper we extend the theory of optimal approximations of functions in the -metric by entire functions of prescribed
exponential type (bandlimited functions). We solve this problem for the
truncated and the odd Gaussians using explicit integral representations and
fine properties of truncated theta functions obtained via the maximum principle
for the heat operator. As applications, we recover most of the previously known
examples in the literature and further extend the class of truncated and odd
functions for which this extremal problem can be solved, by integration on the
free parameter and the use of tempered distribution arguments. This is the
counterpart of the work \cite{CLV}, where the case of even functions is
treated.Comment: to appear in Const. Appro
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
Hermite polynomials in asymptotic representations of generalized Bernoulli,Euler, Bessel and Buchholz polynomials
This is a second paper on finite exact representations of certain polynomials in terms of Hermite polynomials. The representations have asymptotic properties and include new limits of the polynomials, again in terms of Hermite polynomials. This time we consider the generalized Bernoulli, Euler, Bessel and Buchholz polynomials. The asymptotic approximations of these polynomials are valid for large values of a certain parameter. The representations and limits include information on the zero distribution of the polynomials. Graphs are given that indicate the accuracy of the first term approximations
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