Pollak, prolate spheroidal wave functions, fourier analysis. Ball prolate spheroidal wave functions in arbitrary dimensions. Three examples of slepians, named after david slepian. Prolate spheroidal wave functions in q fourier analysis. Prolate spheroidal wave functions, fourier analysis, and uncertainty v. In this paper we introduce a new version of the prolate spheroidal wave function using standard methods of qcalculus and we formulate some of its properties. This paper introduces the oblate spheroidal quaternionic wave functions osqwfs, which extend the oblate spheroidal wave functions introduced in the. Prolate spheroidal wave functions pswfs are important functions in. This article requires a subscription to view the full text. Some comments on fourier analysis, uncertainty and. Generalized prolate spheroidal wave functions for offset.
Their mathematical properties are investigated in great detail, and many applications to signal analysis are pointed out. These functions, also known as slepian prolate functions, were bandlimited and exhibited interesting orthogonality relations. Contents 1 introduction 102 2 prolate spheroidal wave functions 105. Slepianprolate spheroidal wave functions, fourier analysis and uncertaintyiv. In two earlier papers in this series, the extent to which a square. Prolate spheroidal wave functions pswfs are the eigenfunctions of the bandlimited operator in one dimension. Later it is discovered that they are essential for the description of electromagnetic wave propagation, for the signal processing and. If you have a subscription you may use the login form below to view the article. 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.
As application we give a qsampling theorem which extrapolates functions defined on q n and 0 prolate spheroidal wave functions, fourier analysis and uncertainty iii. Robust signal reconstruction using the prolate spherical wave. Introduction clifford analysis offers both a generalization of complex analysis in the plane and a re. Landau h j and pollak h o 1962 prolate spheroidal wave functions, fourier analysis and uncertainty iii. Roughly speaking, the result is true for the best choice of the. Prolate spheroidal wave functions of order zero springerlink. Prolate spheroidal wave functions, fourier analysis and uncertainty i slepian 1961 bell system technical journal wiley online library skip to article content. Jan 01, 2004 read wavelets based on prolate spheroidal wave functions, journal of fourier analysis and applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. The former were introduced in a classic paper 7 by david slepian and his collaborators in bell labs as solutions of an energy concentration problem. Prolate spheroidal wave functions and their generalisations, timefrequency concentration problem, bandlimited functions. Pollak manuscript received july 6, 1960 the theory developed in the preceding paperl is applied to a number of. In mathematics, the prolate spheroidal wave functions pswf are a set of orthogonal bandlimited functions.
Clifford analysis, fourier transform, linear canonical transform, offset linear canonical transform, prolate spheroidal wave functions. Reflective prolatespheroidal operators and the kpkdv. Citeseerx document details isaac councill, lee giles, pradeep teregowda. 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, fourier analysis, and uncertainty. Interpolation orthogonal expansion prolate spheroidal wave function. Prolate spheroidal wave functions, fourier analysis and uncertaintyiv.
As such, they play an important role in signal processing, fourier analysis, and. Spatial smoothing in fmri using prolate spheroidal wave functions. This paper investigates the extent to which a time series can be concentrated on a finite index set and also have its spectrum. As application we give a qsampling theorem which extrapolates functions defined on q n and 0 pp 129536. Bell system technical journal, volume 57 1978, 71430. Rokhlin, highfrequency asymptotic expansions for certain prolate spheroidal wave functions, the journal of fourier analysis and applications, 9 2003. The purpose of this paper is to examine the mathematical truth in the engineering intuition that there are approximately 2wt independent signals. These are of great utility in disciplines such as geodesy 28 or cosmology. The dimension of the space of essentially time and bandlimited signals. Apn modified method to return each sk as complex values, the eigenvalues and the weights. Despite their useful properties in many applications, gpsfs are often replaced by crude approximations. Shen, sampling with prolate spheroidal wave functions, sampling theory in signal and image proccesing, vol. Nov 01, 2003 read highfrequency asymptotic expansions for certain prolate spheroidal wave functions, journal of fourier analysis and applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Citeseerx scientific documents that cite the following paper.
Pdf reducing sampling error by prolate spheroidal wave. The theory developed in the preceding paper is applied to a number of questions about timelimited and bandlimited signals. Prolate spheroidal wave functions, fourier analysis and uncertainty i. The magic of the prolate spheroidal functions in various. The prolate spheroidal wave functions, which are a special case of the spheroidal wave. Read wavelets based on prolate spheroidal wave functions, journal of fourier analysis and applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. Spatial smoothing in fmri using prolate spheroidal wave. The magic of the prolate spheroidal functions in various setups.
On the use of discrete prolate spheroidal windows for. Prolate spheroidal wave functions in qfourier analysis. Hypercomplex signal energy concentration in the spatial and. Prolate spheroidal wave functions in q fourier analysis authors. A complete set of bandlimited functions is described which possesses the curious property of being orthogonal over a given finite interval as well as over.
Slepian, d prolate spheroidal wave functions, fourier analysis and uncertaintyv. Key to the analysis are certain sequences, called discrete prolate spheroidal sequences, and certain functions of frequency called discrete prolate spheroidal functions. 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. Bandlimited functions, prolate spheroidal wave functions, and sampling theorems. Bandlimited signal extrapolation using prolate spheroidal. Read highfrequency asymptotic expansions for certain prolate spheroidal wave functions, journal of fourier analysis and applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. 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. As such, they play an important role in signal processing, fourier analysis, and approximation theory. Highfrequency asymptotic expansions for certain prolate. Sep 17, 2003 landau h j and pollak h o 1962 prolate spheroidal wave functions, fourier analysis and uncertainty iii. Abstract the prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property 6.
Prolate spheroidal wave functions, fourier analysis and uncertainty, iii, bell system tech. On the evaluation of prolate spheroidal wave functions and. Computational harmonic analysis references page winter, 2002 course. A bandlimited function y with its fourier transform support on. 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. Introduction in this work we shall be concerned with the construction of prolate spheroidal wave functions pswfs and their associated prolate spheroidal wavelets ps wavelets. 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. Pdf discretetodiscrete prolate spheroidal wave functions and.
The orthogonal basis is spanned by the socalled discrete prolate spheroidal dps sequences. Slepian showed in 15 that timelimited parts of bandlimited sequences span a lowdimensional subspace. Optimal data acquisition in fmri using prolate spheroidal wave functions. Applied and computational harmonic analysis, 22 2007.
They also satisfy a discrete orthogonality relation. Wavelets based on prolate spheroidal wave functions. Prolate spheroidal wave functions, fourier analysis and uncertainty i abstract. They are normalized versions of the solutions to helmholtz wave equation 1 in spheroidal coordinates. 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 cimpa school on real and complex analysis with applications, buea cameroun, 114 may 2011. Journal of classical analysis volume 11, number 1 2017, 121 doi. Lazhar dhaouadi submitted on 18 jul 2007, last revised 9 apr 2008 this version, v2 abstract. 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. International journal of imaging systems and technology, 1262. Abstract generalized prolate spheroidal functions gpsf are the eigen functionsofthetruncatedfouriertransform, restrictedtoddimensional balls in the spatial domain and frequency domain. Computational geometric and boundary value properties of. Zernike polynomials are a basis of orthogonal polynomials on the unit disk that are a natural basis for representing smooth functions. Prolate spheroidal wave function as exact solution of the.
Discretetodiscrete prolate spheroidal wave functions and finite duration discrete fractional fourier transform. In this paper, we introduce the prolate spheroidal wave functions pswfs of real order. Prolate spheroidal wave functions, an introduction to the. Sampling theory approach to prolate spheroidal wavefunctions. Generalized prolate spheroidal functions, gpsf, pswf, truncated fourier transform, implementation. Prolate spheroidal wave functions, fourier analysis and uncertainty iii. Computational harmonic analysis references page winter, 2002. Prolate spheroidal wave functions, fourier analysis and uncertainty. Prolate spheroidal wave functions in qfourier analysis 707. They are eigenfunctions of a timelimiting operation followed by a lowpassing operation. Introduction the prolate spheroidal wave functions pswfs originally appeared during the resolution of the helmholtz equation in the spheroidal coordinate system by variables separation 16. Investigation of the problem of simultaneously concentrating a function and its fourier transform differed from the other problems i have worked on in two.
Some popular basis sets for performing this analysis is the legendre, chebyschev, and fourier series. Lazhar dhaouadi submitted on 18 jul 2007 v1, last revised 9 apr 2008 this version, v2. Prolate spheroidal wave functions, fourier analysis and uncertainty ii. A discrete time series has associated with it an amplitude spectrum which is a periodic function of frequency. Prolate spheroidal wave functions, fourier analysis and. They arise in a number of applications including optics and atmospheric sciences. Wavelets based on prolate spheroidal wave functions, journal. In this paper, we provide a selfcontained reference on zernike polynomials, algorithms for evaluating them, and what appear to be new numerical schemes for quadrature and. The naming suggestion was provided by bob parker of scripps institute of oceanography, who suggested that discrete prolate spheroidal sequences was a mouthful. Properties of the functions are derived and several applications to the representation of signals are made.
Prolate spheroidal wave functions, fourier analysis, and. Pollak on discrete prolate spheroidal wave functions and sequences dpswf, dpss eventually led to the naming of the sequences as slepians. A new friendly method of computing prolate spheroidal wave. Prolate spheroidal wave functions in q fourier analysis, year. Osipov, certain upper bounds on the eigenvalues associated with prolate.
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. In section 5, we illustrate the analysis via several numerical examples. Let denote the time truncation operator, such that if and only if is timelimited within. Some comments on fourier analysis, uncertainty and modeling. Recently, pswfs have been becoming increasingly popular in various areas in which such functions occur this includes physics e.
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