On the Eigenfunctions of the Finite Hankel Transform (original) (raw)
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Journal of Fourier Analysis and Applications, 2017
For fixed real numbers c > 0, α > − 1 2 , the finite Hankel transform operator, denoted by H α c is given by the integral operator defined on L 2 (0, 1) with kernel K α (x, y) = √ cxyJ α (cxy). To the operator H α c , we associate a positive, self-adjoint compact integral operator Q α c = c H α c H α c. Note that the integral operators H α c and Q α c commute with a Sturm-Liouville differential operator D α c. In this paper, we first give some useful estimates and bounds of the eigenfunctions ϕ (α) n,c of H α c or Q α c. These estimates and bounds are obtained by using some special techniques from the theory of Sturm-Liouville operators, that we apply to the differential operator D α c. If (µ n,α (c)) n and λ n,α (c) = c |µ n,α (c)| 2 denote the infinite and countable sequence of the eigenvalues of the operators H (α) c and Q α c , arranged in the decreasing order of their magnitude, then we show an unexpected result that for a given integer n ≥ 0, λ n,α (c) is decreasing with respect to the parameter α. As a consequence, we show that for α ≥ 1 2 , the λ n,α (c) and the µ n,α (c) have a super-exponential decay rate. Also, we give a lower decay rate of these eigenvalues. As it will be seen, the previous results are essential tools for the analysis of a spectral approximation scheme based on the eigenfunctions of the finite Hankel transform operator. Some numerical examples will be provided to illustrate the results of this work.
Bounding Functions via N-th order Hankel Transform
2004
In this presentation we establish new bounds on functions, which have their n-th order Hankel Transform bandlimited. This class of functions is proved to be also Fourier Transform bandlimited. N-th order Hankel Transforms and the consequent bounds are important for the reconstruction in CAT, MRI (Magnetic Resonance Imaging) etc. Indeed many algorithms of modern tomography use the appropriate forms of Hankel Transform. For example, we may refer the Optimal Hankel Transform Reconstruction [1], the Rhombus Hankel Transform Reconstruction [5] etc. The new bounds, presented in this paper, simplifies the according Hankel Transform Reconstruction techniques. Actually a criterion based on these bounds can reject terms of the reconstruction with weak influence on image quality.
Theory and operational rules for the discrete Hankel transform
Journal of the Optical Society of America A, 2015
Previous definitions of a discrete Hankel transform (DHT) have focused on methods to approximate the continuous Hankel integral transform. In this paper, we propose and evaluate the theory of a DHT that is shown to arise from a discretization scheme based on the theory of Fourier–Bessel expansions. The proposed transform also possesses requisite orthogonality properties which lead to invertibility of the transform. The standard set of shift, modulation, multiplication, and convolution rules are derived. In addition to the theory of the actual manipulated quantities which stand in their own right, this DHT can be used to approximate the continuous forward and inverse Hankel transform in the same manner that the discrete Fourier transform is known to be able to approximate the continuous Fourier transform.
Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions
Journal of Computational and Applied Mathematics, 2009
In this paper, we develop two practical methods for the computation of the eigenvalues as well as the eigenfunctions of the finite Hankel transform operator. These different eigenfunctions are called circular prolate spheroidal wave functions (CPSWFs). This work is motivated by the potential applications of the CPSWFs as well as the development of practical methods for computing their values. Also, in this work, we should prove that the CPSWFs form an orthonormal basis of the space of Hankel band-limited functions, an orthogonal basis of L 2 ([0, 1]) and an orthonormal system of L 2 ([0, +∞[). Our computation of the CPSWFs and their associated eigenvalues is done by the use of two different methods. The first method is based on a suitable matrix representation of the finite Hankel transform operator. The second method is based on the use of an efficient quadrature method based on a special family of orthogonal polynomials. Also, we give two Maple programs that implement the previous two methods. Finally, we present some numerical results that illustrate the results of this work.
On a generalized finite Hankel transform
Applied Mathematics and Computation, 2007
In the present work we introduce a finite integral transform involving combination of Bessel functions as kernel under prescribed conditions. The corresponding inversion formula and some properties of this transform have also been given. Three problems of heat conduction in an infinite, a semi-infinite and a finite circular cylinder bounded by given surfaces with radiation-type boundary value conditions have been solved by applying this transform.
An uncertainty principle for Hankel transforms
Proceedings of the American Mathematical Society, 1999
There exists a generalized Hankel transform of order α ≥ −1/2 on R, which is based on the eigenfunctions of the Dunkl operator Tαf (x) = f (x) + α + 1 2 f(x) − f(−x) x , f ∈ C 1 (R). For α = −1/2 this transform coincides with the usual Fourier transform on R. In this paper the operator Tα replaces the usual first derivative in order to obtain a sharp uncertainty principle for generalized Hankel transforms on R. It generalizes the classical Weyl-Heisenberg uncertainty principle for the position and momentum operators on L 2 (R); moreover, it implies a Weyl-Heisenberg inequality for the classical Hankel transform of arbitrary order α ≥ −1/2 on [0, ∞[.
Application of the generalized shift operator to the Hankel transform
SpringerPlus, 2014
It is well known that the Hankel transform possesses neither a shift-modulation nor a convolution-multiplication rule, both of which have found many uses when used with other integral transforms. In this paper, the generalized shift operator, as defined by Levitan, is applied to the Hankel transform. It is shown that under this generalized definition of shift, both convolution and shift theorems now apply to the Hankel transform. The operation of a generalized shift is compared to that of a simple shift via example.
On the zeros of basic finite Hankel transforms
Journal of Mathematical Analysis and Applications, 2006
In this article we prove that the basic finite Hankel transform whose kernel is the third-type Jackson q-Bessel function has only infinitely many real and simple zeros, provided that q satisfies a condition additional to the standard one. We also study the asymptotic behavior of the zeros. The obtained results are applied to investigate the zeros of q-Bessel functions as well as the zeros of q-trigonometric functions. A basic analog of a theorem of G. Pólya (1918) on the zeros of sine and cosine transformations is also given.