Bessel Expansions Research Articles

How to compute multivariate Bessel expansions

We develop a constructive method for computing explicitly multivariate Bessel expansions of the type ∑m≥1αm∏i=1kJμi(ζmxi)(ζmxi)μi,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{m\\ge 1} \\alpha _m \\prod _{i=1}^k \\frac{J_{\\mu _i}(\\zeta _m x_i)}{(\\zeta _m x_i)^{\\mu _i}}, \\end{aligned}$$\\end{document}assuming that for a particular value η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document} a closed expression for the single-variable Bessel expansion ∑m≥1αmJη(ζmx)(ζmx)η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\sum _{m\\ge 1}\\alpha _m \\frac{J_{\\eta }(\\zeta _m x)}{(\\zeta _m x)^\\eta } \\end{aligned}$$\\end{document}as a power series of x2j\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x^{2j}$$\\end{document}, j∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$j\\in \\mathbb {N}$$\\end{document}, is known. Using the method we compute in a closed form a bunch of examples of multivariate Bessel expansions.

On Derivatives, Riesz Transforms and Sobolev Spaces for Fourier–Bessel expansions

We study the problem of an appropriate choice of derivatives associated with discrete Fourier-Bessel expansions. We introduce a new so-called essential measure Fourier-Bessel setting, where the relevant derivative is simply the ordinary derivative. Then we investigate Riesz transforms and Sobolev spaces in this context. Our main results are $L^p$-boundedness of the Riesz transforms (even in a multi-dimensional situation) and an isomorphism between the Sobolev and Fourier-Bessel potential spaces. Moreover, throughout the paper we collect various comments concerning two other closely related Fourier-Bessel situations that were considered earlier in the literature. We believe that our observations shed some new light on analysis of Fourier-Bessel expansions.

$$L^p$$-Boundedness of Stein’s Square Functions Associated with Fourier–Bessel Expansions

In this paper we prove \(L^p\) estimates for Stein’s square functions associated with Fourier–Bessel expansions. Furthermore, we prove transference results for square functions from Fourier–Bessel series to Hankel transforms. Actually, these are transference results for vector-valued multipliers from discrete to continuous in the Bessel setting. As a consequence, we deduce the sharpness of the range of p for the \(L^p\)-boundedness of Fourier–Bessel Stein’s square functions from the corresponding property for Hankel–Stein square functions. Finally, we deduce \(L^p\) estimates for Fourier–Bessel multipliers from that ones we have got for our Stein square functions.

Inequalities for second-order Riesz transforms associated with Bessel expansions

Inequalities for second-order Riesz transforms associated with Bessel expansions

A few properties of the third Jackson q-Bessel function

The Hahn–Exton (third Jackson) q-Bessel function attracts the attention of many researchers in mathematics. We add a few more properties of this important function, such as some identities and upper bounds involving this function, in relation with the coefficients of the corresponding basic Fourier–Bessel expansions.

Hardy spaces for Fourier-Bessel expansions

We study Hardy spaces for Fourier--Bessel expansions associated with Bessel operators on $((0,1), x^{2\nu+1}\, dx)$ and $((0,1), dx)$. We define Hardy spaces $H^1$ as the sets of $L^1$-functions for which their maximal functions for the corresponding Poisson semigroups belong to $L^1$. Atomic characterizations are obtained.

A Fast Analysis-Based Discrete Hankel Transform Using Asymptotic Expansions

A fast and numerically stable algorithm is described for computing the discrete Hankel transform of order $0$ as well as evaluating Schlömilch and Fourier--Bessel expansions in $\mathcal{O}(N(\log N)^2/\log\!\log N)$ operations. The algorithm is based on an asymptotic expansion for Bessel functions of large arguments, the fast Fourier transform, and the Neumann addition formula. All the algorithmic parameters are selected from error bounds to achieve a near-optimal computational cost for any accuracy goal. Numerical results demonstrate the efficiency of the resulting algorithm.

Potential operators associated with Jacobi and Fourier–Bessel expansions

We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier–Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those 1≤p,q≤∞, for which the potential operators are of strong type (p,q), of weak type (p,q) and of restricted weak type (p,q). These results may be thought of as analogues of the celebrated Hardy–Littlewood–Sobolev fractional integration theorem in the Jacobi and Fourier–Bessel settings. As an ingredient of our line of reasoning, we also obtain sharp estimates of the Poisson kernel related to Fourier–Bessel expansions.

Differential theory for anisotropic cylindrical objects with an arbitrary cross section

We extend the differential theory to anisotropic cylindrical structures with an arbitrary cross section. Two cases have to be distinguished. When the anisotropic cylinders do not contain the origin, the scattering matrix of the device is calculated from the extended differential theory with the help of the scattering matrix propagation algorithm. The fields outside the cylinders are described by Fourier-Bessel expansions. When the origin is located in one cylinder, the fields inside the cylinder are expressed from a semi-analytical theory related to a homogeneous anisotropic medium. In this second case, the formalism of the scattering matrix propagation algorithm is not exactly the same and requires suitable change. The numerical results are in good agreement with the ones obtained for the diffraction by one circular cylinder. The theory is then applied on the diffraction by an elliptical cylinder.

The Bochner–Riesz means for Fourier–Bessel expansions: Norm inequalities for the maximal operator and almost everywhere convergence

In this paper, we develop a thorough analysis of the boundedness properties of the maximal operator for the Bochner–Riesz means related to the Fourier–Bessel expansions. For this operator, we study weighted and unweighted inequalities in the spaces Lp((0,1),x2ν+1dx). Moreover, weak and restricted weak type inequalities are obtained for the critical values of p. As a consequence, we deduce the almost everywhere pointwise convergence of these means.

Representation of Photonic Crystals and Their Localized Modes Through the Use of Fourier–Bessel Expansions

Photonic crystal (PhC) geometry is typically characterized by its translational symmetry. However, it can be treated based on the rotationally symmetry through the use of Fourier-Bessel expansions about the center of rotation. Fourier-Bessel expansions of the inverse dielectric of the structures and the transverse electric (TE) localized modes extract the rotational symmetry that is present. The results show that in PhCs and photonic quasi-crystals, the localized <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</sub> field contains only the rotational orders that correspond to the rotational order of the dielectric plus and minus the rotational order of the corresponding perfect state. This relationship indicates the potential for simplification within the master equation in cylindrical coordinates that will require further examination.

An axisymmetric elastic analysis for circular sandwich panels with functionally graded cores

This paper presents an analytical solution in the framework of the elasticity theory, which is useful in describing the elastic bending response of axisymmetric circular sandwich panels with functionally graded material cores and homogeneous face-sheets. The Young’s modulus of the core is assumed to be exponentially dependant on the transverse direction and the Poisson’s ratio as well as uniform and equal to the face-sheets ratio. The elastic solution is obtained using a Plevako representation, which reduces the problem to the search of potential functions satisfying linear fourth-order partial differential equations. We explicitly obtain the analytical solution by writing the potential functions as Fourier Bessel expansions with respect to the radial coordinate. A comparative study of functionally graded versus a homogeneous sandwich core is presented by considering the first term of the expansion as the loading condition. In this way, the solution is written in a closed form and furnishes a benchmark to accurately investigate the agreement with the structural theory results.

LEGENDRE, BESSEL AND BOUBAKER POLYNOMIALS THEORETICAL EXPRESSIONS OF LOW TEMPERATURE PROFILE IN A PYROLYSIS SPRAY MODEL: CASE OF GAUSSIAN DEPOSITED LAYER

In this paper, a solution to heat equation in a spherical model is proposed. The model takes into account conjointly the boundary conditions and the transfer equations in a spray device. The deposited layer is supposed to be Gaussian and not uniform like most recent studies. The solution, using Legendre, Bessel and Boubaker polynomials expansions, presents the advantage of being continuous and indefinitely derivable. These features make the resolution connectable to any heat transfer similar model.

Littlewood–Paley–Stein [formula omitted]-functions for Fourier–Bessel expansions

g k -Functions related to the Poisson semigroup of Fourier–Bessel expansions are defined for each k ⩾ 1 . It is proved that these g k -functions are Calderón–Zygmund operators in the sense of the associated space of homogeneous type, hence their mapping properties follow from the general theory.

Transferring Boundedness from Conjugate Operators Associated with Jacobi, Laguerre, and Fourier–Bessel Expansions to Conjugate Operators in the Hankel Setting

In this paper we establish transference results showing that the boundedness of the conjugate operator associated with Hankel transforms on Lorentz spaces can be deduced from the corresponding boundedness of the conjugate operators defined on Laguerre, Jacobi, and Fourier–Bessel settings. Our result also allows us to characterize the power weights in order that conjugation associated with Laguerre, Jacobi, and Fourier–Bessel expansions define bounded operators between the corresponding weighted Lp spaces.

Dynamic Response of Rapidly Heated Cylindrical Rods: Longitudinal and Flexural Behavior

A very fast temperature increase, produced by a nonuniform heat generation, induces in a simply supported, isotropic, cylindrical rod both longitudinal and flexural vibrations. This paper presents an analytical method to study these vibrations and determine the stresses they provoke. The proposed procedure relies on three main steps: an exact solution for the temperature field is first obtained, by means of Fourier–Bessel expansions; quasistatic thermal stresses are then computed as a function of the calculated temperature distribution, making use of the thermoelastic displacement potential and of the solution to the equivalent isothermal two-dimensional stress problem; finally, longitudinal and flexural vibrations excited by an equivalent thermal force and thermal bending moment are determined using the mode-summation method. The influence of thermal shock duration on the maximum value of the longitudinal dynamic stress and of the ratio between the characteristic thermal time and structural response time on the dynamic bending deflection is analyzed and discussed. Finally, a comparison between the analytical model and experimental measurements is presented. The analytical model described in this paper allows the complete evaluation, within the linear elastic domain, of quasistatic and dynamic thermal stresses induced in an isotropic cylindrical rod by rapid internal heating.

The Impact of Non-Isotropic Scattering and Directional Antennas on MIMO Multicarrier Mobile Communication Channels

In outdoor environments, waves propagate non- isotropically because of non-uniform distribution of scatterers. In addition directional antennas, which are efficient for communication applications, have considerable impact on the response of the communication channel. The impact of non-isotropic propagation and directional antennas is more significant when the system employs multiple antennas. In this paper, we propose a space-time-frequency cross-correlation function (CCF) for multiple-input multiple-output multicarrier channels in a two-dimensional (2D) random scattering medium. The expression of the CCF turns out to be a multiplication of three CCFs. Two of these terms characterize the impact of mobile station and base station, respectively and are linear Bessel expansions, where the coefficients are given by the linear convolution of the Fourier series coefficients (FSC)s of the employed antenna patterns and the FSCs of the probability density function describing 2D non-isotropic environment. The third expression characterizes the impact of the wireless channel. In the stationary case, we derive the expression of the fading channel power spectrum (CPS) in terms of the non-isotropic pdf and antenna patterns. Using the expression of the CCF, we also calculate coherence-bandwidth (CB)/coherence-time (CT) as a function of carrier-frequency/time separations. Our numerical results show a good fit to the available experimental results obtained for the CB/CT of realistic outdoor channels.

Weighted inequalities for the Bochner–Riesz means related to the Fourier–Bessel expansions

We prove weighted inequalities for the Bochner–Riesz means for Fourier–Bessel series with more general weights w ( x ) than previously considered power weights. These estimates are given by using the local A p theory and Hardy's inequalities with weights. Moreover, we also obtain weighted weak type ( 1 , 1 ) inequalities. The case when w ( x ) = x a is sketched and follows as a corollary of the main result.

Conjugacy for Fourier–Bessel expansions

We define and investigate the conjugate operator for Fourier&#8211;Bessel expansions. Weighted norm and weak type $(1,1)$ inequalities are proved for this operator by using a local version of the Calderón&#8211;Zygmund theory, with weights in most cases m

The Bochner–Riesz means for Fourier–Bessel expansions

The Bochner–Riesz means for Fourier–Bessel expansions are analyzed. We prove a uniform two-weight inequality, with potential weights, for these means. The result provides necessary and sufficient conditions for boundedness. Moreover, we obtain some corollaries regarding the convergence of these means and the boundedness of other operators related to the Fourier–Bessel series.