Revisiting the Concentration Problem of Vector Fields within a Spherical Cap: A Commuting Differential Operator Solution
Kornél Jahn,Nándor Bokor +1 more
TLDR
In this paper, the authors propose a mixed vector spherical harmonics that are closely related to the functions of Sheppard and Torok and help to reduce the concentration problem of tangential vector fields within a spherical cap to an equivalent scalar problem.Abstract:
We propose a novel basis of vector functions, the mixed vector spherical harmonics that are closely related to the functions of Sheppard and Torok and help us reduce the concentration problem of tangential vector fields within a spherical cap to an equivalent scalar problem. Exploiting an analogy with previous results published by Grunbaum, Longhi and Perlstadt, we construct a differential operator that commutes with the concentration operator of this scalar problem and propose a stable and convenient method to obtain its eigenfunctions. Having obtained the scalar eigenfunctions, the calculation of tangential vector Slepian functions is straightforward.read more
Citations
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Journal ArticleDOI
The Darboux process and time-and-band limiting for matrix orthogonal polynomials☆
Mirta M. Castro,F. A. Grünbaum +1 more
TL;DR: In this article, the authors extend the Darboux process to matrix valued orthogonal polynomials and show that in this case an algebraic miracle that plays a very important role in the classical case survives an application of the so-called Darbouque process in the matrix valued context.
Journal ArticleDOI
Internal and external potential-field estimation from regional vector data at varying satellite altitude
TL;DR: In this paper, the authors develop the theory behind a procedure to invert for planetary potential fields from vector observations collected within a spatially bounded region at varying satellite altitude, which relies on the construction of spatiospectrally localized bases of functions that mitigate the noise amplification caused by downward continuation (from the satellite altitude to the planetary surface).
Posted Content
Scalar and vector Slepian functions, spherical signal estimation and spectral analysis
TL;DR: A theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers in the 1960s, and how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and, particularly for applications in the geosciences.
Journal ArticleDOI
Time and Band Limiting for Matrix Valued Functions, an Example ?
TL;DR: In this paper, it was shown that a bis-pectral property implies that the corresponding global operator admits a commuting local operator, which is a non-commutative analog of the famous prolate spheroidal wave operator.
Book ChapterDOI
Potential-field estimation using scalar and vector slepian functions at satellite altitude
TL;DR: Two approaches to estimate potential fields on a spherical Earth, from gradient data collected at satellite altitude, are presented, one designed for radialcomponent data only and one that incorporates a new type of vectorial spherical Slepian functions that are introduced in this chapter.
References
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Book
Matrix Analysis
Roger A. Horn,Charles R. Johnson +1 more
TL;DR: In this article, the authors present results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrate their importance in a variety of applications, such as linear algebra and matrix theory.
Related Papers (5)
Prolate spheroidal wave functions, Fourier analysis and uncertainty — IV: Extensions to many dimensions; generalized prolate spheroidal functions
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