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A discretization of the continuous fourier transform
Rafael G. Campos,L. Z. Juárez +1 more
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In this paper, a finite representation of the continuous Fourier transform is obtained in a linear space of dimensionN by using some properties of the Hermite polynomialsHN(x) and its zeros.Abstract:
A finite representation of the continuous Fourier transform is obtained in a linear space of dimensionN for the Hermite functions by using some properties of the Hermite polynomialsHN(x) and its zeros. TheN-dimensional vectors representing a Hermite function and its Fourier-trnasformed function converge to their exact continuous values whenN→∞. The genericN×N matrix representing the kernel of the continuous transform exp [−ipx], preserves the conspicuous properties of this function, and their entries approach the values that this kernel takes atp=xj andx=xk, wherexj andxk are two zeros ofHN(x) whenN→∞. These properties lead to propose this kind of matrices as finite representations of the kernel of the continuous Fourier transform for functions other than the Hermite ones, yielding certain quadrature formula for this transform. Some numerical examples of this application are shown and they are compared with those obtained through standard procedures.read more
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A fast algorithm for the linear canonical transform
Rafael G. Campos,Jared Figueroa +1 more
TL;DR: An O(NlogN) algorithm to compute the LCT is obtained by using a chirp-FFT-chirp transformation yielded by a convergent quadrature formula for the fractional Fourier transform to give a unitary discrete LCT in closed form.
Journal ArticleDOI
A New Formulation of the Fast Fractional Fourier Transform
TL;DR: This work derives a Gaussian-like quadrature of the continuous fractional Fourier transform from a bilinear form of eigenvectors of the matrix associated to the recurrence equation of the Hermite polynomials, which becomes a more accurate version of the FFT and can be used for nonperiodic functions.
Journal ArticleDOI
Coincident frequencies and relative phases among brain activity and hormonal signals.
TL;DR: Findings suggest that the procedure applied here provides a method to analyze typical frequencies, or periods and phases between signals with the same period, and generates specific patterns for brain signals and hormones and relations among them.
Journal ArticleDOI
A quadrature formula for the Hankel transform
TL;DR: A quadrature formula for a certain Fourier-Bessel transform and for the Hankel transform of order ν>−1 is presented and the error function produced by this algorithm is shown to be of a smaller order than 1/N.
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Quadrature formulas for integrals transforms generated by orthogonal polynomials
TL;DR: In this paper, the authors obtained quadrature formulas for integral transforms generated by a family of orthogonal polynomials, their asymptotic expressions and bilinear generating functions.
References
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Journal ArticleDOI
Properties of the zeros of the classical polynomials and of the Bbessel functions
TL;DR: Many algebraic equations satisfied by the zeros of the classical polynomials and of the Bessel functions are reported in this article, some of which are collected from recent papers; several of them are new; most of them display remarkable diophantine features.
Journal ArticleDOI
Matrices, differential operators, and polynomials
TL;DR: In this article, the Hermite polynomials (Hermite, Laguerre, Lagrange, Gegenbauer, Jacobi, and Jacobi) are represented as matrices whose eigenvalues and eigenvectors are given, fully or in part, by very simple formulas.
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On a method for computing eigenvalues and eigenfunctions of linear differential operators
TL;DR: In this paper, a simple method is presented to compute the eigenvalues and the Eigenfunctions of second-order linear differential operators, with homogeneous boundary conditions, both in a finite interval and on the semi-line.