Prolate Spheroidal Wave Functions of Order Zero

Osipov, Alexander V.; Rokhlin, Vladimir; Xiao, Han

doi:10.1007/978-1-4614-8259-8

Cited by 98 publications

(128 citation statements)

References 0 publications

Supporting

Mentioning

127

Contrasting

Unclassified

Order By: Relevance

“…While the one dimensional Prolate Spheroidal Wave functions received considerable interest in signal processing and mathematics [26], the generalization to multiple dimensions is not straightforward.…”

Section: Multi-dimensional Extensionsmentioning

confidence: 99%

See 1 more Smart Citation

Function Approximation on Arbitrary Domains Using Fourier Extension Frames

Matthysen¹,

Huybrechs²

2018

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Fourier extension is an approximation scheme in which a function on an arbitary bounded domain is approximated using a classical Fourier series on a bounding box. On the smaller domain the Fourier series exhibits redundancy, and it has the mathematical structure of a frame rather than a basis. It is not trivial to construct approximations in this frame using function evaluations in points that belong to the domain only, but one way to do so is through a discrete least squares approximation. The corresponding system is extremely ill-conditioned, due to the redundancy in the frame, yet its solution via a regularized SVD is known to be accurate to very high (and nearly spectral) precision. Still, this computation requires O(N 3 ) operations. In this paper we describe an algorithm to compute such Fourier extension frame approximations in only O(N 2 log 2 N ) operations for general 2D domains. The cost improves to O(N log 2 N ) operations for simpler tensor-product domains. The algorithm exploits a phenomenon called the plunge region in the analysis of time-frequency localization operators, which manifests itself here as a sudden drop in the singular values of the least squares matrix. It is known that the size of the plunge region scales like O(log N ) in one dimensional problems. In this paper we show that for most 2D domains in the fully discrete case the plunge region scales like O(N log N ), proving a discrete equivalent of a result that was conjectured by Widom for a related continuous problem. The complexity estimate depends on the Minkowski or box-counting dimension of the domain boundary, and as such it is larger than O(N log N ) for domains with fractal shape.

show abstract

Section: Multi-dimensional Extensionsmentioning

confidence: 99%

“…Let T Ω and B Λ be as in eqs. (26) and (27). We are interested in the behavior for large n Λ , with constant oversampling γ = n Λ /PR.…”

Section: Bounding η( N R )mentioning

confidence: 99%

Function Approximation on Arbitrary Domains Using Fourier Extension Frames

Matthysen¹,

Huybrechs²

2018

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

show abstract

“…From the proof of Lemma , it follows that f may still be represented by with the coefficient given by , but this representation is valid now only for x ∈ T . If indeed

f \notin ℬ_{boldW}

, the series will certainly not converge in mean square over the whole three‐dimensional Euclidean space.Remark The well‐developed theory of 1D PSWFs was applied to construct quadratures, interpolation, and differentiation formulae for bandlimited signals . The resulting numerical algorithms are satisfactory .…”

Section: Definitions and Properties Of The Prolate Spheroidal Quaternmentioning

confidence: 99%

“…The well-developed theory of 1D PSWFs was applied to construct quadratures, interpolation, and differentiation formulae for bandlimited signals [13,37]. The resulting numerical algorithms are satisfactory [37]. The representation of bandlimited quaternionic functions on regions in higher dimensions is of both theoretical and engineering interest.…”

Section: Remark 35mentioning

confidence: 99%

See 1 more Smart Citation

Constructing prolate spheroidal quaternion wave functions on the sphere

Morais

Kou

2016

Math Methods in App Sciences

View full text Add to dashboard Cite

Over the last years, considerable attention has been paid to the role of the prolate spheroidal wave functions (PSWFs) introduced in the early sixties by D. Slepian and H.O. Pollak to many practical signal and image processing problems. The PSWFs and their applications to wave phenomena modeling, fluid dynamics, and filter design played a key role in this development. In this paper, we introduce the prolate spheroidal quaternion wave functions (PSQWFs), which refine and extend the PSWFs. The PSQWFs are ideally suited to study certain questions regarding the relationship between quaternionic functions and their Fourier transforms. We show that the PSQWFs are orthogonal and complete over two different intervals: the space of square integrable functions over a finite interval and the three‐dimensional Paley–Wiener space of bandlimited functions. No other system of classical generalized orthogonal functions is known to possess this unique property. We illustrate how to apply the PSQWFs for the quaternionic Fourier transform to analyze Slepian's energy concentration problem. We address all of the aforementioned and explore some basic facts of the arising quaternionic function theory. We conclude the paper by computing the PSQWFs restricted in frequency to the unit sphere. The representation of these functions in terms of generalized spherical harmonics is explicitly given, from which several fundamental properties can be derived. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

Reflective prolate‐spheroidal operators and the adelic Grassmannian

Casper

Grünbaum

Yakimov

et al. 2023

Comm Pure Appl Math

View full text Add to dashboard Cite

Beginning with the work of Landau, Pollak and Slepian in the 1960s on time‐band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory, and integrable systems. Previously, such pairs were constructed by ad hoc methods, which essentially worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point W of Wilson's infinite dimensional adelic Grassmannian gives rise to an integral operator , acting on for a contour , which reflects a differential operator with rational coefficients in the sense that on a dense subset of . By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function . The exact size of this algebra with respect to a bifiltration is in turn determined using algebro‐geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property above in place of plain commutativity. Furthermore, we prove that the time‐band limited operators of the generalized Laplace transforms with kernels given by the rank one bispectral functions always reflect a differential operator. A 90° rotation argument is used to prove that the time‐band limited operators of the generalized Fourier transforms with kernels admit a commuting differential operator. These methods produce vast collections of integral operators with prolate‐spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.

show abstract

Prolate Spheroidal Wave Functions of Order Zero

Cited by 98 publications

References 0 publications

Function Approximation on Arbitrary Domains Using Fourier Extension Frames

Function Approximation on Arbitrary Domains Using Fourier Extension Frames

Constructing prolate spheroidal quaternion wave functions on the sphere

Reflective prolate‐spheroidal operators and the adelic Grassmannian

Contact Info

Product

Resources

About