The Hankel transform H_n[f(x)](q) = int_0^infinity xf(x)J_n(qx)dx is studied for integer n>=-1 and positive parameter q. It is proved that the Hankel transform is given by uniformly and absolutely convergent series in reciprocal powers of q, provided special conditions on the function f(x) and its derivatives are imposed. It is necessary to underline that similar formulas obtained previously are in fact asymptotic expansions only valid when q tends to infinity. If one of the conditions is violated, our series become asymptotic series. The validity of the formulas is illustrated by a number of examples.
The complete Lipschitz-Hankel integrals (LHIs) include the Laplace transforms of the Bessel functions, multiplied by powers. Such Laplace transforms can be evaluated using associated Legendre functions. It is noted that there are errors in published versions of these evaluations, and a merged and emended list of seven transforms is given. Errata for standard reference works, such as the table of Gradshteyn and Ryzhik, are also given. Most of the errors are attributable to inconsistent normalization of the Legendre functions. These transforms can be viewed as limits of incomplete LHIs, which fi…
The aim of this paper is to prove some new uncertainty principles for the windowed Hankel transform. They include uncertainty principle for orthonormal sequence, local uncertainty principle, logarithmic uncertainty principle and Heisenberg-type uncertainty principle. As a side result, we obtain the Shapiro's dispersion theorem for the windowed Hankel transform.
Norm equivalences between a function and its Hankel transform are studied both in the context of weighted Lebesgue spaces with power weights, and in Lorentz spaces. Boas'-type results involving real-valued general monotone functions are obtained. Corresponding results for the Fourier transform are also given.
We exhibit three classes of compactly supported functions which provide reproducing kernels for the Sobolev spaces $H^δ(\R^d)$ of arbitrary order $\,δ>d/2.\,$ Our method of construction is based on a new class of oscillatory integral transforms that incorporate radial Fourier transforms and Hankel transforms.
We investigate necessary and/or sufficient conditions for the pointwise and uniform convergence of the weighted Hankel transforms $$\mathcal{L}^α_{ν,μ}f(r) = r^μ\int_0^\infty (rt)^νf(t) j_α(rt)\, dt, \quad α\geq -1/2, \quad r\geq 0, $$ where $ν,μ\in \mathbb{R}$ are such that $0\leq μ+ν\leq α+3/2$. We subdivide these transforms into two classes in such a way that the uniform convergence criteria is remarkably different on each class. In more detail, we have the transforms satisfying $μ+ν=0$ (such as the classical Hankel transform), that generalize the cosine transform, and those satisfying $0<μ…
In this article we shall study the analytic theory and the representation theoretic interpretations of Hankel transforms and fundamental Bessel kernels of an arbitrary rank over an archimedean field.
Based on known definite integrals of Bessel functions of the first kind, we obtain exact solutions to unknown definite integrals using the method of integral transforms from Hankel's transform.
Using Ramanujan's Master Theorem, two formulas are derived which define the Hankel transforms of order zero with even functions by inverse Mellin transforms, provided these functions and their derivatives obey special conditions. Their validity is illustrated by a number of examples. With a help of one of these formulas, one as yet unknown parametric improper integral of the Bessel function J_0(x) is calculated.
In this paper, we reconsider the large-argument asymptotic expansions of the Hankel, Bessel and modified Bessel functions and their derivatives. New integral representations for the remainder terms of these asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re-expansions for these remainder terms and provide their error estimates. A detailed discussion on the sharpness of our error bounds and their relation to other results in the literature is given. The techniques used in this paper should also generalize to asymptotic expansions which arise fro…
In this paper we investigate algebraic, differential and asymptotic properties of polynomials $p_n(x)$ that are orthogonal with respect to the complex oscillatory weight $w(x)=e^{iωx}$ on the interval $[-1,1]$, where $ω>0$. We also investigate related quantities such as Hankel determinants and recurrence coefficients. We prove existence of the polynomials $p_{2n}(x)$ for all values of $ω>0$, as well as degeneracy of $p_{2n+1}(x)$ at certain values of $ω$ (called kissing points). We obtain detailed asymptotic information as $ω\to\infty$, using recent theory of multivariate highly oscillatory in…
For fixed real numbers $c>0,$ $α>-\frac{1}{2},$ the finite Hankel transform operator, denoted by $\mathcal{H}_c^α$ is given by the integral operator defined on $L^2(0,1)$ with kernel $K_α(x,y)= \sqrt{c xy} J_α(cxy).$ To the operator $\mathcal{H}_c^α,$ we associate a positive, self-adjoint compact integral operator $\mathcal Q_c^α=c\, \mathcal{H}_c^α\, \mathcal{H}_c^α.$ Note that the integral operators $\mathcal{H}_c^α$ and $\mathcal Q_c^α$ commute with a Sturm-Liouville differential operator $\mathcal D_c^α.$ In this paper, we first give some useful estimates and bounds of the eigenfunctions $…
Considering the relationship between two bases in representation space of the three-dimensional proper Lorentz group, we derive some formulas with integrals involving Coulomb wave functions, which can be considered as Fourier, Mellin, $K$-Bessel, Hankel and Mehler-Fock transforms of these functions.
Complex generalized Gauss-Radau quadrature rules for Hankel transforms of integer order
School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院）；Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology（华中科技大学工程建模与科学计算湖北省重点实验室）
This paper explores the regularity properties of an inverse spectral transform for Hilbert--Schmidt Hankel operators on the unit disc. This spectral transform plays the role of action-angles variables for an integrable infinite dimensional Hamiltonian system -- the cubic Szegö equation. We investigate the regularity of functions on the tori supporting the dynamics of this system, in connection with some wave turbulence phenomenon, discovered in a previous work and due to relative small gaps between the actions. We revisit this phenomenon by proving that generic smooth functions and a G $δ$ den…
We study orthogonal polynomials and Hankel determinants generated by a symmetric semi-classical Jacobi weight. By using the ladder operator technique, we derive the second-order nonlinear difference equations satisfied by the recurrence coefficient $β_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials. This enables us to obtain the large $n$ asymptotics of $β_n(t)$ and $\mathrm{p}(n,t)$ based on the result of Kuijlaars et al. [Adv. Math. \textbf{188} (2004) 337-398]. In addition, we show the second-order differential equation satisfied by the orthogonal…
E. Heine in the 19th century studied a system of orthogonal polynomials associated with the weight $\left[x(x-α)(x-β)\right]^{-\frac{1}{2}}$, $x\in[0,α]$, $0<α<β$. A related system was studied by C. J. Rees in 1945, associated with the weight $\left[(1-x^2)(1-k^2x^2)\right]^{-\frac{1}{2}}$, $x\in[-1,1]$, $k^2\in(0,1)$. These are also known as elliptic orthogonal polynomials, since the moments of the weights maybe expressed in terms of elliptic integrals. Such orthogonal polynomials are of great interest because the corresponding Hankel determinant, depending on a parameter $k^2$, where $0<k^2<…
