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Revisiting the concentration problem of vector fields within a spherical cap: a commuting differential operator solution. (English) Zbl 1317.33010
The paper is devoted to the concentration problem of tangential vector fields within a spherical cap. First, the authors introduce new vector fields, connected to conventional vector spherical harmonics, that they call mixed vector spherical harmonics and they discuss their orthogonality properties. Since the mixed vector spherical harmonics form a complete basis of the Hilbert space of square integrable tangential vector fields, they use this property to expand an arbitrary tangential vector field in terms of this basis and define bandlimitedness when the expansion is a finite sum. In this work it is shown that the vectorial concentration problem within a spherical cap can be reduced to equivalent one-dimensional scalar concentration problems of various orders. The eigenvalue spectrum of the concentration operator is analyzed and an illustration on the scalar eigenfunctions is given. The authors propose a fast and numerically stable way to calculate the eigenfunctions by using a differential operator that commutes with the scalar concentration operator obtained previously. Thus they obtain in a direct way the tangential vector Slepian functions.

MSC:
33C47 Other special orthogonal polynomials and functions
33C55 Spherical harmonics
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
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LAPACK; sapa
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