The modified kontorovich-lebedev transform and its application to solving canonical problems of diffraction

The aim of this work is to fill the gap between three-dimensional embedding formulas for problems of diffraction by cones and the modified Smyshlyaev formulas. The three-dimensional embedding formulas express the diffraction effect in the form of iterated integrals over spatial variables. The modified Smyshlyaev formulas express it in the form of a single contour integral with respect to the parameter ν of separation of variables. This situation resembles the theorem of convolution for the Fourier transform: repeated convolutions are expressed by a single integral with respect to frequency. In [1], where the modified Smyshlyaev formulas were introduced for the first time, these formulas were hypothesized and then proved. No regular method for deriving them has been proposed. The extension of the analogy with the Fourier transform to the case of conical problems of diffraction enables one to construct a technique for transformation of contour integrals that can be used for deriving the modified Smyshlyaev formulas directly from the three-dimensional embedding formulas. This extension is performed by introducing an integral transform similar to the Kontorovich-Lebedev transform, for which analogs of the theorem of convolution and the Plancherel formulas can be successfully proved.     

On integral and finite Fourier transforms of continuous <Emphasis Type="Italic">q</Emphasis>-Hermite polynomials

We give an overview of the remarkably simple transformation properties of the continuous q-Hermite polynomials H _n (x|q) of Rogers with respect to the classical Fourier integral transform. The behavior of the q-Hermite polynomials under the finite Fourier transform and an explicit form of the q-extended eigenfunctions of the finite Fourier transform, defined in terms of these polynomials, are also discussed. The text was submitted by the authors in English.     

Convolution of Lorentz Invariant Ultradistributions and Field Theory

A general definition of convolution between two arbitrary four-dimensional Lorentz invariant (fdLi) tempered ultradistributions is given, in both Minkowski and Euclidean space (spherically symmetric tempered Ultradistributions). The product of two arbitrary fdLi distributions of exponential type is defined via the convolution of its corresponding Fourier transforms. Several examples of convolution of two fdLi tempered ultadisrtibutions are given. In particular, we calculate exactly the convolution of two Feynman's massless prapagators. An expression for the Fourier transform of a Lorentz invariant tempered ultradistribution in terms of modified Bessel distributions is obtained in this work (generalization of Bochner's formula to Minkowski space). From the deduction of the convoltion formula, we obtain the generalization to the Minkowski space, of the dimensional regularization of the perturbation theory of Green functions in the Euclidean configuration space given in Erdelyi (Higher Transcendental Functions, 1953). As an example we evaluate the convolution of two n-dimensional complex-mass Wheeler propagators.     

显微数字全息中物光波前重建方法研究和比较

根据全息理论和线性系统理论,采用离轴无透镜傅里叶变换全息记录光路,对利用菲涅耳近似法、基于瑞利—索末菲衍射积分的卷积法以及角谱理论方法数值重建全息图进行了比较研究,并做了计算机模拟验证.结果表明：菲涅耳近似法和角谱方法重建像质比较好,且菲涅耳方法重建速度快;在记录距离极小的情况下,尽管记录距离不满足通常的菲涅耳近似条件,菲涅耳近似公式仍然成立;自由空间脉冲响应的快速傅里叶变换的性质与距离有关,由卷积方法得到的再现像只在某一特定距离下比较理想;对于极小物场、大孔径显微数字全息来说,菲涅耳近似重建方法是较为有效的方法.     

Generalized analytic signal associated with linear canonical transform

Analytic signal is tightly associated with Hilbert transform and Fourier transform. The linear canonical transform is the generalization of many famous linear integral transforms, such as Fourier transform, fractional Fourier transform and Fresnel transform. Based on the parameter (a, b)-Hilbert transform and the linear canonical transform, in this paper, we develop some issues on generalized analytic signal. The generalized analytic signal can suppress the negative frequency components in the linear canonical transform domain. Furthermore, we prove that the kernel function of the inverse linear canonical transform satisfies the generalized analytic condition and get the generalized analytic pairs. We show the generalized Bedrosian theorem is valid in the linear canonical transform domain.     

Light diffraction and the principle of causality

Diffraction formulas for non-monochromatic light are written in the formalism of linear response theory to stress the role of the causality principle. The analysis yields a formula containing a space integral of a time convolution. An example shows when the order of space and time integrations can be exchanged: this is possible only at the end of the optical transient. The system output can then be written as a time function of the space Fresnel or Fourier transform of the input. The white light interferometry experiments (channeled spectra) support the model: they result from it when transients are shorter than the average pulse duration.     

New method for solving fractional partial integro-differential equations by combination of Laplace transform and resolvent kernel method

In this article, we obtained the approximate solution for a new class of Time-Fractional Partial Integro-Differential Equation (TFPIDE) of the Caputo-Volterra type in which the integral is not limited to the convolution type. This new class of TFPIDE is distinct from the common problem with the convolution integral kernel. The general expression of the analytical solution for this special type of TFPIDE was derived using a combination of Laplace transform and the resolvent kernel method. In the process, Laplace transform will transform the equation into a second kind Volterra integral equation in terms of the transformed function. Two main problems in deriving the approximate analytical solutions were identified as Case I and Case II problems. To obtain the approximate solutions for Case I and Case II problems, numerical methods were designed based on approximation of the resolvent kernel with truncated Neumann series as well as approximation of the Laplace transform based on truncated Taylor series. Several numerical examples are presented to indicate the plausibility, mechanism and performance of the proposed methods.     

Discrete Quadratic-Phase Fourier Transform: Theory and Convolution Structures

The discrete Fourier transform is considered as one of the most powerful tools in digital signal processing, which enable us to find the spectrum of finite-duration signals. In this article, we introduce the notion of discrete quadratic-phase Fourier transform, which encompasses a wider class of discrete Fourier transforms, including classical discrete Fourier transform, discrete fractional Fourier transform, discrete linear canonical transform, discrete Fresnal transform, and so on. To begin with, we examine the fundamental aspects of the discrete quadratic-phase Fourier transform, including the formulation of Parseval’s and reconstruction formulae. To extend the scope of the present study, we establish weighted and non-weighted convolution and correlation structures associated with the discrete quadratic-phase Fourier transform.     

Fast algorithm of discrete gyrator transform based on convolution operation

The expression of gyrator transform (GT) is rewritten by using convolution operation, from which GT can be composed of phase-only filtering, Fourier transform and inverse Fourier transform. Therefore, fast Fourier transform (FFT) algorithm can be introduced into the calculation of convolution format of GT in the discrete case. Some simulations are presented in order to demonstrate the validity of the algorithm.     

Quantum-state tomogram from <Emphasis Type="Italic">s</Emphasis>-parameterized quasidistributions

We find an integral transform realizing the connection between the s-parameterized quasidistributions of a quantum state and its corresponding tomogram, which looks like a Radon transform. We show that the kernel of the new Radon transform is a Gaussian function with s-parameterized dispersion. It can be considered as a broadened delta-function appeared in the standard Radon transform.     

Propagation of super-Lorentzian beams through a paraxial ABCD optical system

Based on the Collins integral, analytical propagation formulae of super-Lorentzian (SL) SL₀₁, SL₁₀, and SL₁₁ beams passing through a paraxial ABCD optical system are derived by means of the convolution theorem of the Fourier transform. The propagation properties of the SL₀₁ and SL₁₁ beams in free space are graphically illustrated with numerical examples. The power in the bucket of the SL₀₁ and SL₁₁ beams has been also examined in the far-field plane. This research is useful to the applications of super-Lorentzian beams.     

Multichannel sampling expansion in the linear canonical transform domain and its application to superresolution

The linear canonical transform (LCT) describes the effect of first-order quadratic phase optical system on a wave field. In this paper, we address the problem of signal reconstruction from multichannel samples in the LCT domain based on a new convolution theorem. Firstly, a new convolution structure is proposed for the LCT, which states that a modified ordinary convolution in the time domain is equivalent to a simple multiplication operation for LCT and Fourier transform (FT). Moreover, it is expressible by a one dimensional integral and easy to implement in the designing of filters. The convolution theorem in FT domain is shown to be a special case of our achieved results. Then, a practical multichannel sampling expansion for band limited signal with the LCT is introduced. This sampling expansion which is constructed by the new convolution structure can reduce the effect of spectral leakage and is easy to implement. Last, the potential application of the multichannel sampling is presented to show the advantage of the theory. Especially, the application of multichannel sampling in the context of the image superresolution is also discussed. The simulation results of superresolution are also presented.     

一种改良的啁啾变换算法及其应用

快速啁啾算法引入两次快速傅里叶变换（FFT）及一个解析高斯核,计算复杂度低于卷积算法.通过对啁啾算法实现过程进行的改进,避免了该算法在实现过程中存在的一些问题,比如输出窗口小、信号丢失、计算复杂度稍大等缺点. 把算法用于简单的可求得解析解的系统并与之做比较. 对高斯函数,最大误差通常在10^-15数量级,而对矩形函数,由于受FFT算法计算精度的影响,误差在10^-3数量级,但这并不影响算法的性能. 最后把算法用于一种典型的标量衍射系统及分数傅里叶变换的计算,获得了很好的结果. 关键词： 快速啁啾算法 啁啾Z变换 菲涅耳变换 分数傅里叶变换     

柯林斯公式的D-FFT计算

将柯林斯公式及其逆运算表示为卷积形式,导出对应的传递函数,讨论使用快速傅里叶变换(FFT)计算柯林斯公式时满足取样定理的条件,基于研究结果,给出光波通过一光学系统的衍射场计算及根据衍射场重建入射平面光波场的实例。     

Papoulis-like generalized sampling expansions in fractional Fourier domains and their application to superresolution

We present a generalized convolution theorem in the fractional Fourier domains that preserves the convolution theorem of the conventional Fourier transform. The Papoulis-like generalized sampling expansions in the fractional Fourier domains using this generalized convolution theorem are also derived and it is shown that the classical generalized Papoulis sampling expansion is a special case of it. Its application in the context of the image superresolution is also discussed.     

Propagation of a Lorentz-Gauss beam through a misaligned optical system

Based on the generalized integral formula and the convolution theorem of the Fourier transform, an analytical propagation formula of a Lorentz-Gauss beam passing through a misaligned paraxial optical system is derived. As numerical examples, the propagation properties of a Lorentz-Gauss beam through a misaligned thin lens with the lateral displacement and the angle displacement are graphically illustrated, respectively. The influences of the lateral displacement and the angle displacement of the misaligned thin lens on the normalized light intensity and the phase distributions are also examined, respectively.     

Convolution of n-Dimensional Tempered Ultradistributions and Field Theory

In this work, a general definition of convolution between two arbitrary tempered ultradistributions is given. When one of the tempered ultradistributions is rapidly decreasing this definition coincides with the definition of J. Sebastiao e Silva. In the four-dimensional case, when the tempered ultradistributions are even in the variables k ⁰ and , we obtain an expression for the convolution, which is more suitable for practical applications. The product of two arbitrary even (in the variables x ⁰ and r) four-dimensional distributions of exponential type is defined via the convolution of its corresponding Fourier transforms. With this definition of convolution, we treat the problem of singular products of Green Functions in Quantum Field Theory (for Renormalizable as well as for nonrenormalizable theories). Several examples of convolution of two tempered ultradistributions are given. In particular, we calculate the convolution of two massless Wheeler's propagators and the convolution of two complex mass Wheeler's propagators.     

Fourier transform moiré tomography for high-sensitivity mapping asymmetric 3-D temperature field

Fourier transform evaluation for moiré deflectogram is proposed to automatically map the temperature field. The moiré deflectogram is generated by conventional deflectometer and is analyzed by means of Fourier transform algorithm. The convolution backprojection algorithm is used for the optical tomography. Asymmetric 3-D gas temperature distribution for a given layer is reconstructed.     

A fast wave superposition spectral method with complex radius vector combined with two-dimensional fast Fourier transform algorithm for acoustic radiation of axisymmetric bodies

Based on the two-dimensional fast Fourier transform (2D FFT) algorithm, a wave superposition spectral method with complex radius vector has been proposed to efficiently analyze the acoustic radiation from an axisymmetric body. First, the complex Fourier series are used along both circumferential and meridian directions, to expand the integral kernel function and unknown source strength density distributed function. Then, by means of the rectangular integral formula, the radiation sound pressure is described in the form of two-dimensional discrete Fourier transform and generalized through 2D FFT algorithm. Finally, several numerical examples are performed to verify the accuracy and efficiency of the present method. Comparing with the other existing analysis ways, the present method has three different characteristics: (1) there is no singular integral in the numerical computation; (2) the unique solution can be given for all eigen wavenumbers owing to the application of the virtual boundary technology with complex radius vector; and (3) the computational efficiency is improved remarkably because all Fourier terms are calculated simultaneously through 2D FFT algorithm.