We show that a fractional version of the finite fourier transform may be defined by using prolate spheroidal wave functions of order zero. Fourier expansion of generalized prolate spheroidal wave. Prolate spheroidal wave functions pswfs are known to be useful for analyzing the properties of the finiteextension fourier transform fift. Prolate spheroidal wave functions associated with the quaternionic fourier transform article pdf available in mathematical methods in the applied sciences may 2017 with 105 reads. Other results include a general perturbation scheme for differential equations and the reduction to two dimensions of the case of functions of d 2. Walter and xiaoping shen communicated by yang wang abstract. Prolate spheroidal wave functions, an introduction to the. Slepian, d prolate spheroidal wave functions, fourier analysis and uncertaintyv. Prolate spheroidal wave functions, fourier analysis and uncertainty i. Prolate spheroidal wave functions, fourier analysis and uncertainty. Pdf reducing sampling error by prolate spheroidal wave. Generalized prolate spheroidal wave functions for optical. This suggests that the fourier transform of the autocorrelation sequence of a university of victoria. The present paper finds a new kind of hypercomplex signals whose energy concentration is maximal in both time and frequency under quaternionic fourier transform.
Prolate spheroidal wave function as exact solution of the. Osa generalized prolate spheroidal wave functions for. Spectral analysis of the finite hankel transform and circular. For decades mathematicians, physicists, and engineers have relied on various orthogonal expansions such as fourier, legendre, and chebyschev to solve a variety of problems. Prolate spheroidal wave functions and their generalisations, timefrequency concentration problem, bandlimited functions. Some of these examples are discussed below very briefly.
Optimal data acquisition in fmri using prolate spheroidal wave functions. International journal of imaging systems and technology, 1262. Bandlimited signal extrapolation using prolate spheroidal. Based on the pswf, the finite fractional fourier transform was developed. In this paper, for digital signal processing application, we derive discretetodiscrete prolate spheroidal wave functions. Slepian june 30, 1923 november 29, 2007 was an american mathematician. Prolate spheroidal wave functions and discrete prolate spheroidal sequences the continuous time problem of maximally concentrating a timelimited function to a limited bandwidth was considered by slepian, pollak, and landau and they found that the prolate spheroidal wave functions pswf are maximally concentrated in this regard 811. Moreover, examples of energy concentrated ratios between the truncated. The new wavelets preserve the high energy concentration in both the. Recently, pswfs have been becoming increasingly popular in various areas in which such functions occur this includes physics e. We first discuss a few features of the fourier transform ft, and then we solve the initialvalue problem for the wave equation using the fourier transform. Spatial smoothing in fmri using prolate spheroidal wave functions martin a. Two dimensional prolate spheroidal wave functions for mri.
Pollak, prolate spheroidal wave functions, fourier analysis. Here f x represents the value of the smoothed data at the coordinate point, x, of the image. Prolate spheroidal wave functions, fourier analysis, and. It is therefore reasonable to be interested in expressing the solution of the schrodinger equation in terms of prolate spheroidal wave functions.
The new wavelets preserve the high energy concentration in both the time and frequency domain. This chapter is based in part on abramowitz and stegun 1964, chapter 21 by a. Volkmer department of mathematical sciences, university of wisconsinmilwaukee, milwaukee, wisconsin. Some popular basis sets for performing this analysis is the legendre, chebyschev, and fourier series. In this paper we exploit the orthogonal properties of prolate spheroidal wave functions pswf in the form of a new orthogonal expansion which we have named the slepian series. Pollak manuscript received july 6, 1960 the theory developed in the preceding paperl is applied to a number of questions about timelimited and bandlimited signals. Prolate spheroidal wave functions, fourier analysis and uncertainty ii. More fourier transform theory, especially as applied to solving the wave equation.
Discretetodiscrete prolate spheroidal wave functions and. On the evaluation of prolate spheroidal wave functions and. Discrete prolate spheroidal wave functions and interpolation. Fourier transforms and the wave equation overview and motivation. These functions, also known as slepian prolate functions, were bandlimited and exhibited interesting orthogonality relations. Spatial smoothing using prolate spheroidal wave functions. In this paper, we introduce one family of vectorial prolate spheroidal wave functions of real order. Prolate spheroidal wave functions and applications. We explore a number of issues related to this problem, including the effective computation of the corresponding slepzan functions for. Prolate spheroidal wave functions and their properties.
The singular functions for the problem of recovering a time limited function from its fourier transform in a certain band of frequencies are given, in the simplest case, by the prolate spheroidal wave functions. Prolate spheroidal wave functions whose domain is a portion of the surface of the unit sphere are more generally called slepian functions see also spectral concentration problem. Fast functional magnetic resonance imaging via prolate. Prolate spheroidal wave functions and their properties computation of the pswfs by flammers method uniform estimates of the pswfs and their derivatives applications of the pswfs prolate spheroidal wave functions from pde point of view pswfs as eigenfunctions of a di erential and an integral operator some properties of the pswfs spheroidal. Within the last ten years, computer processing speeds have climbed to the point where generating the function set is not an issue when it comes to their study. Prolate spheroidal wave functions, fourier analysis and. The solution of the schrodinger equation is fundamental for capturing the. Spatial smoothing in fmri using prolate spheroidal wave.
The transform is linear and additive in its index and asymptotically goes over to namiass definition of the fractional fourier transform. These are of great utility in disciplines such as geodesy 28 or cosmology. In other words, the fourier transform of a bandlimited function is compactly supported. Pollak, prolate spheroical wave functions, fourier analysis arid uncertainty, iii. On the use of discrete prolate spheroidal windows for. Fortunately, the inverse fourier transform is very similar to.
A complete set of bandlimited functions is described which possesses the. A significant step in the process is the higher order piecewise polynomial approximation of the overlap integral required for obtaining the expansion coefficients accurately with very high precision. Wavesfourier transforms wikibooks, open books for an. Prolate spheroidal wave functions, fourier analysis and uncertainty, iii, bell system tech. Slepianprolate spheroidal wave functions, fourier analysis and uncertaintyiv. Fourier transform of the kernel function, which in turn can be. Fourier transform of wave function mathematics stack. Pdf prolate spheroidal wave functions associated with the. The article is concerned with a particular multiresolution analysis mra composed of paleywiener spaces. Wedescribe an algorithm for the interpolation ofburst errors in discretetime signals that can be modelled as being bandlimited. Jan 01, 2004 wavelets based on prolate spheroidal wave functions wavelets based on prolate spheroidal wave functions walter, gilbert g shen, xiaoping 20040101 00. Prolate spheroidal wave functions, fourier analysis and uncertainty il by h.
Generalized prolate spheroidal functions, gpsf, pswf, truncated fourier transform, implementation. The sinc function in the sampling theory can be replaced by the prolate spheroidal wave functions pswfs, or slepian functions, since they have much better time localization than the sinc. The new signals are a generalization of the prolate spheroidal wave functions also known as slepian functions to quaternionic space, which are called quaternionic prolate spheroidal. Contents 1 introduction 102 2 prolate spheroidal wave functions 105. Spectral analysis of the finite hankel transform and. Spatial smoothing in fmri using prolate spheroidal wave functions. Prolate spheroidal wave functions pswfs play an important role in various areas, from physics e. This paper investigates the extent to which a time. Wavelets based on prolate spheroidal wave functions wavelets based on prolate spheroidal wave functions walter, gilbert g shen, xiaoping 20040101 00. Generalized prolate spheroidal wave functions for offset. Prolate spheroidal wave functions 811 we next show that. As a special case of this definition, it is shown that the finite fourier transform may be inverted by using.
R okhlin, highfrequency asymptotic expansions for certain prolate spheroidal wave functions, j. Ball prolate spheroidal wave functions in arbitrary dimensions. The new signals are a generalization of the prolate spheroidal wave functions also. Introduction clifford analysis offers both a generalization of complex analysis in the plane and a re. In two earlier papers in this series, the extent to which a square. Derivation of prolate spheroidal wave functions there are several ways to generate the function set, each having its advantages and disadvantages in complexity and precision 3,7,18,24. Soleski,a new friendly method of computing prolate spheroidal wave func tions and wavelets, appl.
Spheroidal wave functions become elementary if gammanpi2. Explicit upper bounds on the eigenvalues associated with. Pdf vectorial ball prolate spheroidal wave functions. Prolate spheroidal wave functions, fourier analysis and uncertainty iii. The generalized prolate spheroidal wave functions has been recently defined by gupta 4 as the solution of the differential equation 2.
Volkmer department of mathematical sciences, university of wisconsinmilwaukee, milwaukee, wisconsin acknowledgements. Prolate spheroidal wave functions, fourier analysis and uncertainty iv. Sep 04, 2016 the present paper finds a new kind of hypercomplex signals whose energy concentration is maximal in both time and frequency under quaternionic fourier transform. A discrete time series has associated with it an amplitude spectrum which is a periodic function of frequency. Wavelets based on prolate spheroidal wave functions, journal.
Pdf prolate spheroidal wave functions associated with. One of the principal reasons for the importance of pswfs is that they are a. Their usual wavelet basis consisting of sinc functions is replaced by one based on prolate spheroidal wave functions pswfs which have much better time localization than the sinc function. Spheroidal wave functions flammer pdf spheroidal wave functions of flammer c. The dimension of the space of essentially time and bandlimited signals. He was colleagues with claude shannon and richard hamming at bell labs. We extend the theory of pswfs for the finiteextension fractional fourier transform, the finiteextension linear canonical transform, and the finiteextension offset linear canonical transform. They are normalized versions of the solutions to helmholtz wave equation 1 in spheroidal coordinates. Prolate spheroidal wave functions, an introduction to the slepian series and its properties ian c. Hypercomplex signal energy concentration in the spatial and. He is best known for his work with algebraic coding theory, probability theory, and distributed source coding.
The magic of the prolate spheroidal functions in various setups. On the n dimensional extension of the discrete prolate. The algorithm correctly restores a mutilated signal that is indeed bandlimited. Prolate spheroidal wave functions associated with the. In this paper, we introduce the prolate spheroidal wave functions pswfs of real order. Fast functional magnetic resonance imaging via prolate wavelets. Its inverse fourier transform, gx, has maximal sig. Clifford analysis, fourier transform, linear canonical transform, offset linear canonical transform, prolate spheroidal wave functions. The magic of the prolate spheroidal functions in various. Spectral methods based on prolate spheroidal wave functions. Then, we define the finite duration discrete fractional fourier transform fidfrft based on it. They are vectorial eigenfunctions of an integral operator related to the finite fourier transform, and solve the divergence free constrained maximum. Wavelets based on prolate spheroidal wave functions.
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