We propose and implement a quasi-discrete Hankel transform algorithm based on Dini series expansion （DQDHT） in this paper. By making use of the property that the zero-order Bessel function derivative J0^1（0）=0, th...We propose and implement a quasi-discrete Hankel transform algorithm based on Dini series expansion （DQDHT） in this paper. By making use of the property that the zero-order Bessel function derivative J0^1（0）=0, the DQDHT can be used to calculate the values on the symmetry axis directly. In addition, except for the truncated treatment of the input function, no other approximation is made, thus the DQDHT satisfies the discrete Parsevat theorem for energy conservation, implying that it has a high numerical accuracy. Further, we have performed several numerical tests. The test results show that the DQDHT has a very high numerical accuracy and keeps energy conservation even after thousands of times of repeating the transform either in a spatial domain or in a frequency domain. Finally, as an example, we have applied the DQDHT to the nonlinear propagation of a Gaussian beam through a Kerr medium system with cylindrical symmetry. The calculated results are found to be in excellent agreement with those based on the conventional 2D-FFT algorithm, while the simulation based on the proposed DQDHT takes much less computing time.展开更多
In Phys. Lett. A 313 （2003） 343 we have found that the self-recipràcal Hankel transformation （HT） is embodied in quantum mechanics by a transform between two entangled state representations of continuum varia...In Phys. Lett. A 313 （2003） 343 we have found that the self-recipràcal Hankel transformation （HT） is embodied in quantum mechanics by a transform between two entangled state representations of continuum variables. In this work we study Hankel transforms and properties of Bessel function via entangled state representations＇ transformation in quantum mechanics.展开更多
Based on our preceding works of how to relate the mathematical Hankel transform to quantum mechanical representation transform and how to express the Bessel equation by an operator identity in some appropriate represe...Based on our preceding works of how to relate the mathematical Hankel transform to quantum mechanical representation transform and how to express the Bessel equation by an operator identity in some appropriate representations we propose the concept of quantum mechanical Hankel transform with regard to quantum state vectors. Then we discuss its new applications.展开更多
The Hankel transform is widely used to solve various engineering and physics problems,such as the representation of electromagnetic field components in the medium,the representation of dynamic stress intensity factors...The Hankel transform is widely used to solve various engineering and physics problems,such as the representation of electromagnetic field components in the medium,the representation of dynamic stress intensity factors,vibration of axisymmetric infinite membrane and displacement intensity factors which all involve this type of integration.However,traditional numerical integration algorithms cannot be used due to the high oscillation characteristics of the Bessel function,so it is particularly important to propose a high precision and efficient numerical algorithm for calculating the integral of high oscillation.In this paper,the improved Gaver-Stehfest(G-S)inverse Laplace transform method for arbitrary real-order Bessel function integration is presented by using the asymptotic characteristics of the Bessel function and the accumulation of integration,and the optimized G-S coefficients are given.The effectiveness of the algorithm is verified by numerical examples.Compared with the linear transformation accelerated convergence algorithm,it shows that the G-S inverse Laplace transform method is suitable for arbitrary real order Hankel transform,and the time consumption is relatively stable and short,which provides a reliable calculation method for the study of electromagnetic mechanics,wave propagation,and fracture dynamics.展开更多
Using the parametrized entangled state representations we have found a generalized Hankel transformationwith the integral kernel being a combination of Bessel functions.This generalized Hankel transformation correspon...Using the parametrized entangled state representations we have found a generalized Hankel transformationwith the integral kernel being a combination of Bessel functions.This generalized Hankel transformation corresponds tothe appropriate quantum mechanical representation transformation.展开更多
Based on the fact that the quantum mechanical version of Hankel transform kernel（the Bessel function） is just the transform between ｜q, r〉 and（s, r′｜, two induced entangled state representations are given, and ...Based on the fact that the quantum mechanical version of Hankel transform kernel（the Bessel function） is just the transform between ｜q, r〉 and（s, r′｜, two induced entangled state representations are given, and working with them we derive fractional squeezing-Hankel transform（FrSHT） caused by the operator e（-iα）（a1-a-2-＋a-1a-2）e-（-iπa2-a2）, which is an entangled fractional squeezing transform operator. The additive rule of the FrSHT can be explicitly proved.展开更多
This paper presents an accurate and efficient method for the computation of resonant frequencies and radiation patterns of a dual frequency stacked circular microstrip antenna.The problem is first formulated using th...This paper presents an accurate and efficient method for the computation of resonant frequencies and radiation patterns of a dual frequency stacked circular microstrip antenna.The problem is first formulated using the Hankel transform domain approach and expressions are obtained for the Green's function in the Hankel transform domain,which relates the electric surface currents on the circular disks and tangential electric field components on the surfaces of the substrates. Then Galerkin's method together with Parsebal's relation for Hankel transformation is used to solve for the unknown currents, In the derivation process,the resonant frequencies are numerically determined as a function of the radii of two circular disks and thicknesses and relative permittivies of two substrates.Finally,the far zone radiation patterns are directly obtained from the Green's function and the currents. The numerical results for the resonant frequencies and radiation patterns are in excellent agreement with the available experimental data corroborating the accuracy of the present method.展开更多
A local Hankel transformation of order 1/2 is defined for every finite place of the field of rational numbers.Its inversion formula and the Plancherel type theorem are obtained.A Connes type trace formula is given for...A local Hankel transformation of order 1/2 is defined for every finite place of the field of rational numbers.Its inversion formula and the Plancherel type theorem are obtained.A Connes type trace formula is given for each local Hankel transformation of order 1/2.An S-local Connes type trace formula is derived for the S-local Hankel transformation of order 1/2.These formulas are generalizations of Connes' corresponding trace formulas in 1999.展开更多
A new stable numerical method,based on Chebyshev wavelets for numerical evaluation of Hankel transform,is proposed in this paper.The Chebyshev wavelets are used as a basis to expand a part of the integrand,r f(r),appe...A new stable numerical method,based on Chebyshev wavelets for numerical evaluation of Hankel transform,is proposed in this paper.The Chebyshev wavelets are used as a basis to expand a part of the integrand,r f(r),appearing in the Hankel transform integral.This transforms the Hankel transform integral into a Fourier-Bessel series.By truncating the series,an efficient and stable algorithm is obtained for the numerical evaluations of the Hankel transforms of order ν>−1.The method is quite accurate and stable,as illustrated by given numerical examples with varying degree of random noise terms εθ_(i) added to the data function f(r),where θ_(i) is a uniform random variable with values in[−1,1].Finally,an application of the proposed method is given for solving the heat equation in an infinite cylinder with a radiation condition.展开更多
The general expressions of finite Hankel transform are naturally deduced with the help of the property of Bessel functions.The equations in this paper can degenerate into three kinds of boundaries since all the coeffi...The general expressions of finite Hankel transform are naturally deduced with the help of the property of Bessel functions.The equations in this paper can degenerate into three kinds of boundaries since all the coefficients in the boundary conditions are taken into consideration.The results can be adopted in solving physics problems involving the finite Hankel transform.展开更多
This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic ...This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed.展开更多
The Hankel transform is an important transform. In this paper, westudy the wavelets associated with the Hankel transform, thendefine the Weyl transform of the wavelets. We give criteria of itsboundedness and compactne...The Hankel transform is an important transform. In this paper, westudy the wavelets associated with the Hankel transform, thendefine the Weyl transform of the wavelets. We give criteria of itsboundedness and compactness on the Lp-spaces.展开更多
基金supported by the National Natural Science Foundation of China (Grant Nos 10674045 and 60538010)the National Natural Science Foundation of Hunan Province,China (Grant No 08jj3001)
文摘We propose and implement a quasi-discrete Hankel transform algorithm based on Dini series expansion （DQDHT） in this paper. By making use of the property that the zero-order Bessel function derivative J0^1（0）=0, the DQDHT can be used to calculate the values on the symmetry axis directly. In addition, except for the truncated treatment of the input function, no other approximation is made, thus the DQDHT satisfies the discrete Parsevat theorem for energy conservation, implying that it has a high numerical accuracy. Further, we have performed several numerical tests. The test results show that the DQDHT has a very high numerical accuracy and keeps energy conservation even after thousands of times of repeating the transform either in a spatial domain or in a frequency domain. Finally, as an example, we have applied the DQDHT to the nonlinear propagation of a Gaussian beam through a Kerr medium system with cylindrical symmetry. The calculated results are found to be in excellent agreement with those based on the conventional 2D-FFT algorithm, while the simulation based on the proposed DQDHT takes much less computing time.
基金The project supported by National Natural Science Foundation of China under Grant No. 10475056 and the President Foundation of the Chinese Academy of Sciences
文摘In Phys. Lett. A 313 （2003） 343 we have found that the self-recipràcal Hankel transformation （HT） is embodied in quantum mechanics by a transform between two entangled state representations of continuum variables. In this work we study Hankel transforms and properties of Bessel function via entangled state representations＇ transformation in quantum mechanics.
基金The project supported by the President Foundation of the Chinese Academy of Sciences and the National Natural Science Foundation of China under Grant No. 10574060
文摘Based on our preceding works of how to relate the mathematical Hankel transform to quantum mechanical representation transform and how to express the Bessel equation by an operator identity in some appropriate representations we propose the concept of quantum mechanical Hankel transform with regard to quantum state vectors. Then we discuss its new applications.
基金Supported by the National Natural Science Foundation of China(42064004,12062022,11762017,11762016)
文摘The Hankel transform is widely used to solve various engineering and physics problems,such as the representation of electromagnetic field components in the medium,the representation of dynamic stress intensity factors,vibration of axisymmetric infinite membrane and displacement intensity factors which all involve this type of integration.However,traditional numerical integration algorithms cannot be used due to the high oscillation characteristics of the Bessel function,so it is particularly important to propose a high precision and efficient numerical algorithm for calculating the integral of high oscillation.In this paper,the improved Gaver-Stehfest(G-S)inverse Laplace transform method for arbitrary real-order Bessel function integration is presented by using the asymptotic characteristics of the Bessel function and the accumulation of integration,and the optimized G-S coefficients are given.The effectiveness of the algorithm is verified by numerical examples.Compared with the linear transformation accelerated convergence algorithm,it shows that the G-S inverse Laplace transform method is suitable for arbitrary real order Hankel transform,and the time consumption is relatively stable and short,which provides a reliable calculation method for the study of electromagnetic mechanics,wave propagation,and fracture dynamics.
基金National Natural Science Foundation of China under Grant Nos.10475056 and 10775097
文摘Using the parametrized entangled state representations we have found a generalized Hankel transformationwith the integral kernel being a combination of Bessel functions.This generalized Hankel transformation corresponds tothe appropriate quantum mechanical representation transformation.
基金Project supported by the National Natural Science Foundation of China(Grant No.11304126)the Natural Science Foundation of Jiangsu Province,China(Grant No.BK20130532)
文摘Based on the fact that the quantum mechanical version of Hankel transform kernel（the Bessel function） is just the transform between ｜q, r〉 and（s, r′｜, two induced entangled state representations are given, and working with them we derive fractional squeezing-Hankel transform（FrSHT） caused by the operator e（-iα）（a1-a-2-＋a-1a-2）e-（-iπa2-a2）, which is an entangled fractional squeezing transform operator. The additive rule of the FrSHT can be explicitly proved.
文摘This paper presents an accurate and efficient method for the computation of resonant frequencies and radiation patterns of a dual frequency stacked circular microstrip antenna.The problem is first formulated using the Hankel transform domain approach and expressions are obtained for the Green's function in the Hankel transform domain,which relates the electric surface currents on the circular disks and tangential electric field components on the surfaces of the substrates. Then Galerkin's method together with Parsebal's relation for Hankel transformation is used to solve for the unknown currents, In the derivation process,the resonant frequencies are numerically determined as a function of the radii of two circular disks and thicknesses and relative permittivies of two substrates.Finally,the far zone radiation patterns are directly obtained from the Green's function and the currents. The numerical results for the resonant frequencies and radiation patterns are in excellent agreement with the available experimental data corroborating the accuracy of the present method.
文摘A local Hankel transformation of order 1/2 is defined for every finite place of the field of rational numbers.Its inversion formula and the Plancherel type theorem are obtained.A Connes type trace formula is given for each local Hankel transformation of order 1/2.An S-local Connes type trace formula is derived for the S-local Hankel transformation of order 1/2.These formulas are generalizations of Connes' corresponding trace formulas in 1999.
文摘A new stable numerical method,based on Chebyshev wavelets for numerical evaluation of Hankel transform,is proposed in this paper.The Chebyshev wavelets are used as a basis to expand a part of the integrand,r f(r),appearing in the Hankel transform integral.This transforms the Hankel transform integral into a Fourier-Bessel series.By truncating the series,an efficient and stable algorithm is obtained for the numerical evaluations of the Hankel transforms of order ν>−1.The method is quite accurate and stable,as illustrated by given numerical examples with varying degree of random noise terms εθ_(i) added to the data function f(r),where θ_(i) is a uniform random variable with values in[−1,1].Finally,an application of the proposed method is given for solving the heat equation in an infinite cylinder with a radiation condition.
基金supported by the National Natural Scientific Foundation of China (Grant Nos.10972103 and 10902055)the National Science Foundation for Postdoctoral Scientists of China (Grant No.20070411046)
文摘The general expressions of finite Hankel transform are naturally deduced with the help of the property of Bessel functions.The equations in this paper can degenerate into three kinds of boundaries since all the coefficients in the boundary conditions are taken into consideration.The results can be adopted in solving physics problems involving the finite Hankel transform.
基金supported by CSIR,New Delhi(Grant No.25(240)/15/EMR-Ⅱ)
文摘This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed.
基金This work was supported by the National Natural Science Foundation of China(Grant No.90104004)973 Project of China(Grant No.G1999075105).
文摘The Hankel transform is an important transform. In this paper, westudy the wavelets associated with the Hankel transform, thendefine the Weyl transform of the wavelets. We give criteria of itsboundedness and compactness on the Lp-spaces.
