×

zbMATH — the first resource for mathematics

Slepian spatial-spectral concentration on the ball. (English) Zbl 1376.94010
Summary: We formulate and solve the analog of Slepian spatial-spectral concentration problem on the three-dimensional ball. Both the standard Fourier-Bessel and also the Fourier-Laguerre spectral domains are considered since the latter exhibits a number of practical advantages such as spectral decoupling and exact computation. The Slepian spatial and spectral concentration problems are formulated as eigenvalue problems, the eigenfunctions of which form an orthogonal family of concentrated functions. Equivalence between the spatial and spectral problems is shown. The spherical Shannon number on the ball is derived, which acts as the analog of the space-bandwidth product in the Euclidean setting, giving an estimate of the number of concentrated eigenfunctions and thus the dimension of the space of functions that can be concentrated in both the spatial and spectral domains simultaneously. Various symmetries of the spatial region are considered that reduce considerably the computational burden of recovering eigenfunctions, either by decoupling the problem into smaller subproblems or by affording analytic calculations. The family of concentrated eigenfunctions forms a Slepian basis that can be used to represent concentrated signals efficiently. We illustrate our results with numerical examples and show that the Slepian basis indeed permits a sparse representation of concentrated signals.

MSC:
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
Software:
3DEX; sapa
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abramo, L. R.; Reimberg, P. H.; Xavier, H. S., CMB in a box: causal structure and the Fourier-Bessel expansion, Phys. Rev. D, 82, 043510, (Aug. 2010)
[2] Ahn, C. P.; Alexandroff, R.; Allende Prieto, C.; Anderson, S. F.; Anderton, T.; Andrews, B. H.; Aubourg, É.; Bailey, S.; Balbinot, E.; Barnes, R., The ninth data release of the sloan digital sky survey: first spectroscopic data from the SDSS-III baryon oscillation spectroscopic survey, Astrophys. J. Suppl., 203, 21, (Dec. 2012)
[3] Albertella, A.; Sansò, F.; Sneeuw, N., Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere, J. Geod., 73, 9, 436-447, (Jun. 1999)
[4] Amore, P., Asymptotic and exact series representations for the incomplete gamma function, Europhys. Lett., 71, 1, 1-7, (2005)
[5] Boyd, J. P., Approximation of an analytic function on a finite real interval by a bandlimited function and conjectures on properties of prolate spheroidal functions, Appl. Comput. Harmon. Anal., 15, 2, 168-176, (2003) · Zbl 1031.65155
[6] Castro, P. G.; Heavens, A. F.; Kitching, T. D., Weak lensing analysis in three dimensions, Phys. Rev. D, 72, 2, 023516, (Jul. 2005)
[7] Chen, Q.; Gottlieb, D.; Hesthaven, J., Spectral methods based on prolate spheroidal wave functions for hyperbolic pdes, SIAM J. Numer. Anal., 43, 5, 1912-1933, (2005) · Zbl 1101.65100
[8] Cohen, L., Time-frequency distributions—a review, Proc. IEEE, 77, 7, 941-981, (Jul. 1989)
[9] Colton, D.; Kress, R., Inverse acoustic and electromagnetic scattering theory, (1998), Springer-Verlag Berlin · Zbl 0893.35138
[10] Dahlen, F. A.; Simons, F. J., Spectral estimation on a sphere in geophysics and cosmology, Geophys. J. Int., 174, 774-807, (2008)
[11] Daubechies, I., Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inform. Theory, 34, 4, 605-612, (1988) · Zbl 0672.42007
[12] Daubechies, I., The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 36, 5, 961-1005, (1990) · Zbl 0738.94004
[13] de Villiers, G. D.; Marchaud, F. B.T.; Pike, E. R., Generalized Gaussian quadrature applied to an inverse problem in antenna theory, Inverse Probl., 17, 4, 1163-1179, (2001) · Zbl 0982.78016
[14] dunkl, C. F., A Laguerre polynomial orthogonality and the hydrogen atom, Anal. Appl., 01, 02, 177-188, (2003) · Zbl 1047.33003
[15] Jackson, J. I.; Meyer, C. H.; Nishimura, D. G.; Macovski, A., Selection of a convolution function for Fourier inversion using gridding [computerised tomography application], IEEE Trans. Med. Imag., 10, 3, 473-478, (1991)
[16] Kennedy, R. A.; Sadeghi, P., Hilbert space methods in signal processing, (Mar. 2013), Cambridge University Press Cambridge, UK
[17] Landau, H. J., On szegö’s eingenvalue distribution theorem and non-Hermitian kernels, J. Anal. Math., 28, 1, 335-357, (1975) · Zbl 0321.45005
[18] Landau, H. J.; Pollak H, O., Prolate spheroidal wave functions, Fourier analysis and uncertainity. II, Bell Syst. Tech. J., 40, 65-84, (Jan. 1961)
[19] Landau, H. J.; Pollak H, O., Prolate spheroidal wave functions, Fourier analysis and uncertainity. III: the dimension of the space of essentially time- and band-limited signals, Bell Syst. Tech. J., 41, 1295-1336, (1962) · Zbl 0184.08603
[20] Landau, H. J.; Widom, H., Eigenvalue distribution of time and frequency limiting, J. Math. Anal. Appl., 77, 2, 469-481, (1980) · Zbl 0471.47029
[21] Lanusse, F.; Rassat, A.; Starck, J.-L., Spherical 3D isotropic wavelets, Astron. Astrophys., 540, A92, (Apr. 2012)
[22] Leistedt, B.; McEwen, J. D., Exact wavelets on the ball, IEEE Trans. Signal Process., 60, 12, 6257-6269, (2012) · Zbl 1393.94137
[23] Leistedt, B.; Rassat, A.; Rfrgier, A.; Starck, J.-L., 3DEX: a code for fast spherical Fourier-Bessel decomposition of 3D surveys, Astron. Astrophys., 540, A60, (Apr. 2012)
[24] Lemoine, D., The discrete Bessel transform algorithm, J Chem. Phys, 101, 5, 3936-3944, (1994)
[25] Mathews, J.; Breakall, J.; Karawas, G., The discrete prolate spheroidal filter as a digital signal processing tool, IEEE Trans. Acoust. Speech Signal Process., 33, 6, 1471-1478, (1985)
[26] McEwen, J. D.; Wiaux, Y., A novel sampling theorem on the sphere, IEEE Trans. Signal Process., 59, 12, 5876-5887, (Dec. 2011)
[27] Meaney, C., Localization of spherical harmonic expansions, Monatsh. Math., 98, 1, 65-74, (1984) · Zbl 0543.43006
[28] Moore, I. C.; Cada, M., Prolate spheroidal wave functions, an introduction to the Slepian series and its properties, Appl. Comput. Harmon. Anal., 16, 3, 208-230, (2004) · Zbl 1048.65027
[29] Mortlock, D. J.; Challinor, A. D.; Hobson, M. P., Analysis of cosmic microwave background data on an incomplete sky, Mon. Not. R. Astron. Soc., 330, 405-420, (Feb. 2002)
[30] Fernndez, N., Polynomial bases on the sphere, (Advanced Problems in Constructive Approximation, ISNM Inter. Ser. Numer. Math., vol. 142, (2003), Birkhäuser Basel), 39-52
[31] Percival, D.; Walden, A., Spectral analysis for physical applications: multitaper and conventional univariate techniques, (1993), Cambridge University Press Cambridge, New York · Zbl 0796.62077
[32] Pollard, H., Representation of an analytic function by a Laguerre series, Ann. of Math., 48, 2, 358-365, (Apr. 1947)
[33] Sakurai, J. J., Modern quantum mechanics, (1994), Addison Wesley Publishing Company, Inc. Reading, MA
[34] Shkolnisky, Y., Prolate spheroidal wave functions on a disc—integration and approximation of two-dimensional bandlimited functions, Appl. Comput. Harmon. Anal., 22, 2, 235-256, (2007) · Zbl 1117.65041
[35] Shkolnisky, Y.; Tygert, M.; Rokhlin, V., Approximation of bandlimited functions, Appl. Comput. Harmon. Anal., 21, 3, 413-420, (2006) · Zbl 1110.65023
[36] Simons, F. J.; Dahlen, F. A.; Wieczorek, M. A., Spatiospectral concentration on a sphere, SIAM Rev., 48, 3, 504-536, (2006) · Zbl 1117.42003
[37] Simons, F. J.; Loris, I.; Brevdo, E.; Daubechies, I. C., Wavelets and wavelet-like transforms on the sphere and their application to geophysical data inversion, (Wavelets and Sparsity XIV, vol. 81380, (2011), SPIE), 81380X
[38] Simons, F. J.; Loris, I.; Nolet, G.; Daubechies, I. C.; Voronin, S.; Vetter, P. A.; Charlty, J.; Vonesch, C., Solving or resolving global tomographic models with spherical wavelets, and the scale and sparsity of seismic heterogeneity, Geophys. J. Int., 187, 969-988, (2011)
[39] Simons, F. J.; Wang, D. V., Spatiospectral concentration in the Cartesian plane, Int. J. Geomath., 2, 1, 1-36, (2011) · Zbl 1226.42017
[40] Slepian, D., Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV: extensions to many dimensions; generalised prolate spheroidal functions, Bell Syst. Tech. J., 43, 3009-3057, (1964) · Zbl 0184.08604
[41] Slepian, D., Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev., 25, 3, 379-393, (1983) · Zbl 0571.94004
[42] Slepian, D.; Pollak, H. O., Prolate spheroidal wave functions, Fourier analysis and uncertainity. I, Bell Syst. Tech. J., 40, 43-63, (Jan. 1961)
[43] Slepian, D.; Sonnenblick, E., Eigenvalues associated with prolate spheroidal wave functions of zero order, Bell Syst. Tech. J., 44, 8, 1745-1759, (1965) · Zbl 0135.37701
[44] Spergel, D. N.; Bean, R.; Doré, O.; Nolta, M. R.; Bennett, C. L.; Dunkley, J.; Hinshaw, G.; Jarosik, N.; Komatsu, E.; Page, L.; Peiris, H. V.; Verde, L.; Halpern, M.; Hill, R. S.; Kogut, A.; Limon, M.; Meyer, S. S.; Odegard, N.; Tucker, G. S.; Weiland, J. L.; Wollack, E.; Wright, E. L., Three-year wilkinson microwave anisotropy probe (WMAP) observations: implications for cosmology, Astrophys. J., Suppl. Ser., 170, 2, 377-408, (2007)
[45] Teyssier, R.; Pires, S.; Prunet, S.; Aubert, D.; Pichon, C.; Amara, A.; Benabed, K.; Colombi, S.; Refregier, A.; Starck, J.-L., Full-sky weak-lensing simulation with 70 billion particles, Astron. Astrophys., 497, 2, 335-341, (Apr. 2009)
[46] Thomson, D. J., Spectrum estimation and harmonic analysis, Proc. IEEE, 70, 9, 1055-1096, (1982)
[47] Thomson, D. J., Quadratic-inverse spectrum estimates: applications to palaeoclimatology, Philos. Trans. R. Soc. Lond. Ser. A, 332, 1627, 539-597, (1990) · Zbl 0714.62110
[48] Thomson, D. J.; Robbins, M. F.; Maclennan, C. G.; Lanzerotti, L. J., Spectral and windowing techniques in power spectral analyses of geomagnetic data, Phys. Earth Planet. Inter., 12, 23, 217-231, (1976)
[49] Watson, G., A treatise on the theory of Bessel functions, (1995), Cambridge University Press · Zbl 0849.33001
[50] Weniger, E. J., On the analyticity of Laguerre series, J. Phys. A, 41, 42, 425207, (2008) · Zbl 1154.30003
[51] Wieczorek, M. A.; Simons, F. J., Localized spectral analysis on the sphere, Geophys. J. Int., 162, 3, 655-675, (May 2005)
[52] Wieczorek, M. A.; Simons, F. J., Minimum variance multitaper spectral estimation on the sphere, J. Fourier Anal. Appl., 13, 6, 665-692, (2007) · Zbl 1257.33018
[53] Xu, W.; Chamzas, C., On the extrapolation of band-limited functions with energy constraints, IEEE Trans. Acoust. Speech Signal Process., 31, 5, 1222-1234, (1983) · Zbl 0566.94005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.