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Papers by Tatiana Levitina
In recent times, considerable attention was focused on the computation of the eigenvalues and eig... more In recent times, considerable attention was focused on the computation of the eigenvalues and eigenfunctions of the Finite (truncated) Fourier Transform (FFT). An incentive to that a series of papers by Slepian and his collaborators at Bell labs was. The latest results published by Walter and Shen on sampling with the prolate spheroidal functions — the 1D FFT eigenfunctions — will necessary produce a new wave of interest. Thus the possibility to extend their approach to the 2D case looks very promising and important in image processing. The associated eigenvalues are also important in many practical applications, in particular, in order to estimate the accuracy loss caused by truncation of the Fourier transform of a two–dimensional signal. Of special importance is the property of the 1D– and 2D FFT eigenvalues that starting from a certain index, they all are zero up to a very high accuracy. The number of ’non-zero’ eigenvalues is often called the degree of freedom of the related Pal...
Preliminary test calculations are presented to illustrate the facilities of a new modification of... more Preliminary test calculations are presented to illustrate the facilities of a new modification of the Filter Diagonalization method that serves to analyze a signal spectrum within a selected energy range. The modification employs as filtering the eigenfunctions of the Finite Fourier Transform, or prolates, which due to their special properties are superior to other filters.
To analyze a harmonic signal spectrum within a selected energy range, a mod- i¯cation of the Filt... more To analyze a harmonic signal spectrum within a selected energy range, a mod- i¯cation of the Filter Diagonalization method is applied that employs for flltering the eigenfunctions of the Finite Fourier Transform, or prolates. In addition to the advantages brought with prolates to the Filter Diagonalization, that have already been discussed in our recent publications, the presented modi¯cation allows spec- tral analysis of discrete signal data even if the latter is sampled on a nonuniform time grid.
The Filter diagonalization technique using exact eigenfunctions of the finite Fourier transform i... more The Filter diagonalization technique using exact eigenfunctions of the finite Fourier transform is discussed and improved. A previously developed computational method based on the Walter-Shen sampling formula is advanced and extended.
Publication in the conference proceedings of SampTA, Bremen, Germany, 2013
Computational Biology and Chemistry, 2003
To analyze a harmonic signal spectrum within a selected energy range, a modification of the Filte... more To analyze a harmonic signal spectrum within a selected energy range, a modification of the Filter Diagonalization method is applied that employs for filtering the eigenfunctions of the Finite Fourier Transform, orprolates. In addition to the advantages brought with prolates to the Filter Diagonalization, that have already been discussed in our recent publications, the presented modification allows spectral analysis of discrete signal data even if the latter is sampled on a nonuniform time grid.
The Filter diagonalization technique using exact iegenfunctions of the finite Fourier Transform i... more The Filter diagonalization technique using exact iegenfunctions of the finite Fourier Transform is discussed and improved. A previously developed computational method based on the Walter-Shen sampling formula isadvanced and extended.
During the last years the interest in Finite Fourier Transform (FFT) eigenfunctions, often referr... more During the last years the interest in Finite Fourier Transform (FFT) eigenfunctions, often referred to as ’prolates’, has increased significantly among scientists both in the field of quantum chemistry as well as in the signal processing community. These prolates are band-limited and highly concentrated at a finite time-interval. Both features are acquired by the convolution of a band-limited function with a prolate. This will permit the interpolation of such a convolution using the Walter and Shen sampling formula 1 essentially simplifying the computations 2 . The Fourier transform of the convolution may not necessarily be continuous and the concentration interval is twice as large as that of the prolate 3 . Rigorous error estimates are given as dependent on the truncation limit and the accuracy achieved is tested by numerical examples.
Sampling Theory in Signal and Image Processing, 2015
Sampling Theory in Signal and Image Processing, 2012
In this paper, we study the properties of the eigenfunctions of the finite Hankel transform. We d... more In this paper, we study the properties of the eigenfunctions of the finite Hankel transform. We deduce a sampling series in terms of these functions for Hankel-band-limited signals and derive bounds for the truncation error of the sampling series.
Applied Mathematics and Computation, 2020
In this paper, we present the Abramov approach for the numerical simulation of the whispering gal... more In this paper, we present the Abramov approach for the numerical simulation of the whispering gallery modes in prolate spheroids. The main idea of this approach is the Newton-Raphson technique combined with the quasi-time marching. In the first step, a solution of a simpler problem, as an initial guess for the Newton-Raphson iterations, is provided. Then, step-by-step, this simpler problem is converted into the original problem, while the quasi-time parameter τ runs from τ = 0 to τ = 1. While following the involved imaginary path two numerical approaches are realized, the first is based on the Prüfer angle technique, the second on high order finite difference schemes.
Journal of Computational Methods in Sciences and Engineering, 2004
A new powerful approach to compute the eigenfunctions of the finite two-dimensional Fourier trans... more A new powerful approach to compute the eigenfunctions of the finite two-dimensional Fourier transform is developed and analysed. The numerical technique is a generalization of an earlier method developed for ordinary prolate spheroidal wave functions. Special considerations are given to the problem of singularities and the procedure to locate the eigenvalues. It is demonstrated that the computations are fundamentally improved by the introduction of appropriate auxiliary differential equations.
Computational Methods in Sciences and Engineering 2003, 2003
ABSTRACT
In recent times, considerable attention was focused on the computation of the eigenvalues and eig... more In recent times, considerable attention was focused on the computation of the eigenvalues and eigenfunctions of the Finite (truncated) Fourier Transform (FFT). An incentive to that a series of papers by Slepian and his collaborators at Bell labs was. The latest results published by Walter and Shen on sampling with the prolate spheroidal functions — the 1D FFT eigenfunctions — will necessary produce a new wave of interest. Thus the possibility to extend their approach to the 2D case looks very promising and important in image processing. The associated eigenvalues are also important in many practical applications, in particular, in order to estimate the accuracy loss caused by truncation of the Fourier transform of a two–dimensional signal. Of special importance is the property of the 1D– and 2D FFT eigenvalues that starting from a certain index, they all are zero up to a very high accuracy. The number of ’non-zero’ eigenvalues is often called the degree of freedom of the related Pal...
Preliminary test calculations are presented to illustrate the facilities of a new modification of... more Preliminary test calculations are presented to illustrate the facilities of a new modification of the Filter Diagonalization method that serves to analyze a signal spectrum within a selected energy range. The modification employs as filtering the eigenfunctions of the Finite Fourier Transform, or prolates, which due to their special properties are superior to other filters.
To analyze a harmonic signal spectrum within a selected energy range, a mod- i¯cation of the Filt... more To analyze a harmonic signal spectrum within a selected energy range, a mod- i¯cation of the Filter Diagonalization method is applied that employs for flltering the eigenfunctions of the Finite Fourier Transform, or prolates. In addition to the advantages brought with prolates to the Filter Diagonalization, that have already been discussed in our recent publications, the presented modi¯cation allows spec- tral analysis of discrete signal data even if the latter is sampled on a nonuniform time grid.
The Filter diagonalization technique using exact eigenfunctions of the finite Fourier transform i... more The Filter diagonalization technique using exact eigenfunctions of the finite Fourier transform is discussed and improved. A previously developed computational method based on the Walter-Shen sampling formula is advanced and extended.
Publication in the conference proceedings of SampTA, Bremen, Germany, 2013
Computational Biology and Chemistry, 2003
To analyze a harmonic signal spectrum within a selected energy range, a modification of the Filte... more To analyze a harmonic signal spectrum within a selected energy range, a modification of the Filter Diagonalization method is applied that employs for filtering the eigenfunctions of the Finite Fourier Transform, orprolates. In addition to the advantages brought with prolates to the Filter Diagonalization, that have already been discussed in our recent publications, the presented modification allows spectral analysis of discrete signal data even if the latter is sampled on a nonuniform time grid.
The Filter diagonalization technique using exact iegenfunctions of the finite Fourier Transform i... more The Filter diagonalization technique using exact iegenfunctions of the finite Fourier Transform is discussed and improved. A previously developed computational method based on the Walter-Shen sampling formula isadvanced and extended.
During the last years the interest in Finite Fourier Transform (FFT) eigenfunctions, often referr... more During the last years the interest in Finite Fourier Transform (FFT) eigenfunctions, often referred to as ’prolates’, has increased significantly among scientists both in the field of quantum chemistry as well as in the signal processing community. These prolates are band-limited and highly concentrated at a finite time-interval. Both features are acquired by the convolution of a band-limited function with a prolate. This will permit the interpolation of such a convolution using the Walter and Shen sampling formula 1 essentially simplifying the computations 2 . The Fourier transform of the convolution may not necessarily be continuous and the concentration interval is twice as large as that of the prolate 3 . Rigorous error estimates are given as dependent on the truncation limit and the accuracy achieved is tested by numerical examples.
Sampling Theory in Signal and Image Processing, 2015
Sampling Theory in Signal and Image Processing, 2012
In this paper, we study the properties of the eigenfunctions of the finite Hankel transform. We d... more In this paper, we study the properties of the eigenfunctions of the finite Hankel transform. We deduce a sampling series in terms of these functions for Hankel-band-limited signals and derive bounds for the truncation error of the sampling series.
Applied Mathematics and Computation, 2020
In this paper, we present the Abramov approach for the numerical simulation of the whispering gal... more In this paper, we present the Abramov approach for the numerical simulation of the whispering gallery modes in prolate spheroids. The main idea of this approach is the Newton-Raphson technique combined with the quasi-time marching. In the first step, a solution of a simpler problem, as an initial guess for the Newton-Raphson iterations, is provided. Then, step-by-step, this simpler problem is converted into the original problem, while the quasi-time parameter τ runs from τ = 0 to τ = 1. While following the involved imaginary path two numerical approaches are realized, the first is based on the Prüfer angle technique, the second on high order finite difference schemes.
Journal of Computational Methods in Sciences and Engineering, 2004
A new powerful approach to compute the eigenfunctions of the finite two-dimensional Fourier trans... more A new powerful approach to compute the eigenfunctions of the finite two-dimensional Fourier transform is developed and analysed. The numerical technique is a generalization of an earlier method developed for ordinary prolate spheroidal wave functions. Special considerations are given to the problem of singularities and the procedure to locate the eigenvalues. It is demonstrated that the computations are fundamentally improved by the introduction of appropriate auxiliary differential equations.
Computational Methods in Sciences and Engineering 2003, 2003
ABSTRACT