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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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##### References:
 [1] Albertella, A; Sansò, F; Sneeuw, N, Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere, J. Geodesy, 73, 436-447, (1999) · Zbl 1004.86004 [2] Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Croz, J.D., Greenbaum, A., Hammarling, S., McKenney, A.: LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA (1999) · Zbl 0934.65030 [3] Arfken, G.B., Weber, H.J., Harris, F.E.: Mathematical Methods for Physicists: A Comprehensive Guide, 7th edn. Academic Press/Elsevier, Waltham, MA (2012) · Zbl 1239.00005 [4] Bell, B; Percival, DB; Walden, AT, Calculating thomson’s spectral multitapers by inverse iteration, J. Comput. Graph. Stat., 2, 119-130, (1993) [5] Dahlen, FA; Simons, FJ, Spectral estimation on a sphere in geophysics and cosmology, Geophys. J. Int., 174, 774-807, (2008) [6] Das, S; Hajian, A; Spergel, DN, Efficient power spectrum estimation for high resolution CMB maps, Phys. Rev. D, 79, 083008, (2009) [7] Devaney, AJ; Wolf, E, Multipole expansions and plane wave representations of the electromagnetic field, J. Math. Phys., 15, 234-244, (1974) [8] Eshagh, M, Spatially restricted integrals in gradiometric boundary value problems, Artif. Satell., 44, 131-148, (2009) [9] Freeden, W., Schreiner, M.: Spherical Functions of Mathematical Geosciences: A Scalar, Vectorial, and Tensorial Setup. Springer, Berlin (2009) · Zbl 1167.86002 [10] Grünbaum, FA; Longhi, L; Perlstadt, M, Differential operators commuting with finite convolution integral operators: some non-abelian examples, SIAM J. Appl. Math., 42, 941-955, (1982) · Zbl 0497.22012 [11] Han, SC; Ditmar, P, Localized spectral analysis of global satellite gravity fields for recovering time-variable mass redistributions, J. Geod., 82, 423-430, (2008) [12] Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge, UK (1985). Reprinted with corrections 1990 · Zbl 0576.15001 [13] Jahn, K; Bokor, N, Vector Slepian basis functions with optimal energy concentration in high numerical aperture focusing, Opt. Commun., 285, 2028-2038, (2012) [14] Landau, HJ; Pollak, HO, Prolate spheroidal wave functions, Fourier analysis and uncertainty-II, Bell Syst. Tech. J., 40, 65-84, (1961) · Zbl 0184.08602 [15] Landau, HJ; Pollak, HO, Prolate spheroidal wave functions, Fourier analysis and uncertainty-III: the dimension of the space of essentially time- and band-limited signals, Bell Syst. Tech. J., 41, 1295-1336, (1962) · Zbl 0184.08603 [16] Lessig, C; Fiume, E, On the effective dimension of light transport, Comput. Graph. Forum, 29, 1399-1403, (2010) [17] Liu, QH; Xun, DM; Shan, L, Raising and lowering operators for orbital angular momentum quantum numbers, Int. J. Theor. Phys., 49, 2164-2171, (2010) · Zbl 1200.81166 [18] Maniar, H., Mitra, P.P.: The concentration problem for vector fields. Int. J. Bioelectromagn. 7(1), 142-145 (2005). URL http://www.ijbem.net/volume7/number1/pdf/037.pdf · Zbl 1200.81166 [19] Marinucci, D; Peccati, G, Representations of SO(3) and angular polyspectra, J. Multivar. Anal., 101, 77-100, (2010) · Zbl 1216.60027 [20] Moore, NJ; Alonso, MA, Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields, J. Opt. Soc. Am. A, 26, 2211-2218, (2009) [21] Morse, P.M., Feshbach, H.: Methods of Theoretical Physics. Part I. International Series in Pure and Applied Physics. McGraw-Hill, New York (1953) · Zbl 0051.40603 [22] Percival, D.B., Walden, A.T.: Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques. Cambridge University Press, Cambridge, UK (1993). Reprinted with corrections 1998 · Zbl 0796.62077 [23] Plattner, A., Simons, F.J.: Potential-field estimation using scalar and vector slepian functions at satellite altitude. In: Freeden, W., Nashed, M.Z., Sonar T. (eds.) Handbook of Geomathematics, 2nd edn. Springer, Berlin (2014) [24] Plattner, A; Simons, FJ, Spatiospectral concentration of vector fields on a sphere, Appl. Comput. Harmon. Anal., 36, 1-22, (2014) · Zbl 1336.94024 [25] Sheppard, CJR; Török, P, Efficient calculation of electromagnetic diffraction in optical systems using a multipole expansion, J. Mod. Opt., 44, 803-818, (1997) [26] Simons, FJ; Freeden, W (ed.); Nashed, MZ (ed.); Sonar, T (ed.), Slepian functions and their use in signal estimation and spectral analysis, 891-924, (2010), Berlin · Zbl 1197.86039 [27] Simons, FJ; Dahlen, FA, Spherical Slepian functions and the polar gap in geodesy, Geophys. J. Int., 166, 1039-1061, (2006) [28] Simons, FJ; Dahlen, FA; Wieczorek, MA, Spatiospectral concentration on a sphere, SIAM Rev., 48, 504-536, (2006) · Zbl 1117.42003 [29] Simons, FJ; Wang, DV, Spatiospectral concentration in the Cartesian plane, Int. J. Geomath., 2, 1-36, (2011) · Zbl 1226.42017 [30] Slepian, D, Prolate spheroidal wave functions, Fourier analysis and uncertainty-IV: extensions to many dimensions; generalized prolate spheroidal functions, Bell Syst. Tech. J., 43, 3009-3057, (1964) · Zbl 0184.08604 [31] Slepian, D, Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev., 25, 379-393, (1983) · Zbl 0571.94004 [32] Slepian, D; Pollak, HO, Prolate spheroidal wave functions, Fourier analysis and uncertainty-I, Bell Syst. Tech. J., 40, 43-63, (1961) · Zbl 0184.08601 [33] Swarztrauber, P.N.: The vector harmonic transform method for solving partial differential equations in spherical geometry. Mon. Weather Rev. 121(12), 3415-3437 (1993). doi:10.1175/1520-0493(1993)121<3415:TVHTMF>2.0.CO;2 [34] Szegő, G.: Orthogonal Polynomials, AMS Colloquium Publications, vol. 23, 4th edn. American Mathematical Society, Providence, RI (1975) · Zbl 0305.42011 [35] Tygert, M, Recurrence relations and fast algorithms, Appl. Comput. Harmon. Anal., 28, 121-128, (2010) · Zbl 1182.65195 [36] Winch, DE; Roberts, PH, Derivatives of addition theorems for Legendre functions, J. Aust. Math. Soc. B, 37, 212-234, (1995) · Zbl 0859.33011
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