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