Landau's necessary density conditions for the Hankel transform

We will prove an analogue of Landau's necessary conditions [Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117 (1967).] for spaces of functions whose Hankel transform is supported in a measurable subset S of the positive semi-axis. As a special case, necessary density conditions for the existence of Fourier-Bessel frames are obtained.Comment: To appear on J. Funct. Analysi

Amostragem uniforme e não-uniforme

Mestrado em Psicologia ForenseNeste trabalho vamos apresentar alguns conceitos básicos da teoria da amostragem uniforme e da não-uniforme, e sua ligação à teoria de interpolação. Numa primeira parte focaremos essencialmente os conceitos de amostragem clássica de Nyquist e de Bessel, bem como a respectiva ligação à interpolação de Lagrange e de Bessel. Na segunda parte, estudaremos a implementação numérica do método de amostragem por intermédio de funções q-Bessel (introduzido por D. Abreu [2]). O ramo da análise que estuda este tipo de funções é conhecido como q-cálculo. Foram implementadas as termos básicos, como sejam, o q-factorial (equivalente ao factorial clássico), o cálculo das funções q-Bessel, suas derivadas e respectivo método de amostragem. No final deste trabalho será dedicado a exemplos numéricos deste método.In this thesis we present Basic concepts of uniform and non-uniform sampling theory and their connection with interpolation theory. In the first part we will study the connection between the classic Nyquistsampling and Bessel-sampling and Lagrange- and Besselinterpolation. In the second part we study the numerical implementation of the sampling method using q-Bessel function introduced by D. Abreu [2]. To this end we implement basic terms of the q-Calculus, such as the q-factorial (equivalent to the classic factorial), the computation of q-Bessel functions and their derivatives, as well as the sampling method itself. In the end we present numerical examples for this method

Series Representations and Approximation of some Quantile Functions appearing in Finance

It has long been agreed by academics that the inversion method is the method of choice for generating random variates, given the availability of a cheap but accurate approximation of the quantile function. However for several probability distributions arising in practice a satisfactory method of approximating these functions is not available. The main focus of this thesis will be to develop Taylor and asymptotic series representations for quantile functions of the following probability distributions; Variance Gamma, Generalized Inverse Gaussian, Hyperbolic, -Stable and Snedecor’s F distributions. As a secondary matter we briefly investigate the problem of approximating the entire quantile function. Indeed with the availability of these new analytic expressions a whole host of possibilities become available. We outline several algorithms and in particular provide a C++ implementation for the variance gamma case. To our knowledge this is the fastest available algorithm of its sort

Fourier Transforms

The 21st century ushered in a new era of technology that has been reshaping everyday life, simplifying outdated processes, and even giving rise to entirely new business sectors. Today, contemporary users of products and services expect more and more personalized products and services that can meet their unique needs. In that sense, it is necessary to further develop existing methods, adapt them to new applications, or even discover new methods. This book provides a thorough review of some methods that have an increasing impact on humanity today and that can solve different types of problems even in specific industries. Upgrading with Fourier Transformation gives a different meaning to these methods that support the development of new technologies and have a good projected acceleration in the future

Studies in Interpolation and Approximation of Multivariate Bandlimited Functions

The focus of this dissertation is the interpolation and approximation of multivariate bandlimited functions via sampled (function) values. The first set of results investigates polynomial interpolation in connection with multivariate bandlimited functions. To this end, the concept of a uniformly invertible Riesz basis is developed (with examples), and is used to construct Lagrangian polynomial interpolants for particular classes of sampled square-summable data. These interpolants are used to derive two asymptotic recovery and approximation formulas. The first recovery formula is theoretically straightforward, with global convergence in the appropriate metrics; however, it becomes computationally complicated in the limit. This complexity is sidestepped in the second recovery formula, at the cost of requiring a more local form of convergence. The second set of results uses oversampling of data to establish a multivariate recovery formula. Under additional restrictions on the sampling sites and the frequency band, this formula demonstrates a certain stability with respect to sampling errors. Computational simplifications of this formula are also given

The exponentially convergent trapezoidal rule

It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators