A Parseval equation and a generalized finite Hankel transformation

summary:In this paper, we study the finite Hankel transformation on spaces of ge\-ne\-ra\-lized functions by developing a new procedure. We consider two Hankel type integral transformations $h_\mu$ and $h_\mu ^{\ast }$ connected by the Parseval equation $$\sum_{n=0}^{\infty }(h_\mu f)(n)(h_\mu ^{\ast } \varphi )(n)= \int_{0}^{1}f(x)\varphi (x)\, dx.$$ A space $S_\mu$ of functions and a space $L_\mu$ of complex sequences are introduced. $h_\mu ^{\ast }$ is an isomorphism from $S_\mu$ onto $L_\mu$ when $\mu \geq -\frac{1}{2}$. We propose to define the generalized finite Hankel transform $h'_\mu f$ of $f\in S'_\mu$ by $$ \langle (h'_\mu f), ((h_\mu ^{\ast } \varphi )(n))_{n=0}^{\infty }\rangle =\langle f,\varphi \rangle, \quad \text{for } \varphi \in S_\mu . $

Fast multi-dimensional scattered data approximation with Neumann boundary conditions

An important problem in applications is the approximation of a function $f$ from a finite set of randomly scattered data $f(x_j)$. A common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials $\{e^{2\pi i kx}\}$. This leads to fast numerical algorithms, but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data, an assumption that is rarely satisfied in practice. To overcome this drawback we impose Neumann boundary conditions on the data. This implies the use of cosine polynomials $\cos (\pi kx)$ as basis functions. We show that scattered data approximation using cosine polynomials leads to a least squares problem involving certain Toeplitz+Hankel matrices. We derive estimates on the condition number of these matrices. Unlike other Toeplitz+Hankel matrices, the Toeplitz+Hankel matrices arising in our context cannot be diagonalized by the discrete cosine transform, but they still allow a fast matrix-vector multiplication via DCT which gives rise to fast conjugate gradient type algorithms. We show how the results can be generalized to higher dimensions. Finally we demonstrate the performance of the proposed method by applying it to a two-dimensional geophysical scattered data problem

Effects of MHD on the Unsteady Rotating Flow of a Generalized Maxwell Fluid with Oscillating Gradient Between Coaxial Cylinders

The aim of  this paper is studied the effect of magnetic field on the unsteady rotating flow of a generalized Maxwell fluid with fractional derivative between two infinite straight circular cylinder .The velocity field and the shear stress are obtained by means of discrete Laplace transform and finite Hankel transform. The exact solution for the velocity field and the shear stress that have been obtained by integral and series form in terms of the generalized G functions and Mitting –leffer function .the graphs are plotted to show the effects of the fractional parameter on the fluid dynamic characteristics with MHD on the velocity and shear stress

Pitt's inequalities and uncertainty principle for generalized Fourier transform

We study the two-parameter family of unitary operators $$\mathcal{F}_{k,a}=\exp\Bigl(\frac{i\pi}{2a}\,(2\langle k\rangle+{d}+a-2 )\Bigr) \exp\Bigl(\frac{i\pi}{2a}\,\Delta_{k,a}\Bigr),$$ which are called $(k,a)$-generalized Fourier transforms and defined by the $a$-deformed Dunkl harmonic oscillator $\Delta_{k,a}=|x|^{2-a}\Delta_{k}-|x|^{a}$, $a>0$, where $\Delta_{k}$ is the Dunkl Laplacian. Particular cases of such operators are the Fourier and Dunkl transforms. The restriction of $\mathcal{F}_{k,a}$ to radial functions is given by the $a$-deformed Hankel transform $H_{\lambda,a}$. We obtain necessary and sufficient conditions for the weighted $(L^{p},L^{q})$ Pitt inequalities to hold for the $a$-deformed Hankel transform. Moreover, we prove two-sided Boas--Sagher type estimates for the general monotone functions. We also prove sharp Pitt's inequality for $\mathcal{F}_{k,a}$ transform in $L^{2}(\mathbb{R}^{d})$ with the corresponding weights. Finally, we establish the logarithmic uncertainty principle for $\mathcal{F}_{k,a}$.Comment: 16 page

The asymptotics a Bessel-kernel determinant which arises in Random Matrix Theory

In Random Matrix Theory the local correlations of the Laguerre and Jacobi Unitary Ensemble in the hard edge scaling limit can be described in terms of the Bessel kernel (containing a parameter $\alpha$). In particular, the so-called hard edge gap probabilities can be expressed as the Fredholm determinants of the corresponding integral operator restricted to the finite interval [0, R]. Using operator theoretic methods we are going to compute their asymptotics as R goes to infinity under certain assumption on the parameter $\alpha$.Comment: 50 page