Spatiospectral concentration in the Cartesian plane

  • Frederik J. Simons
  • Dong V. Wang
Original Paper


We pose and solve the analogue of Slepian’s time–frequency concentration problem in the two-dimensional plane, for applications in the natural sciences. We determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the plane, or, alternatively, of strictly spacelimited functions that are optimally concentrated in the Fourier domain. The Cartesian Slepian functions can be found by solving a Fredholm integral equation whose associated eigenvalues are a measure of the spatiospectral concentration. Both the spatial and spectral regions of concentration can, in principle, have arbitrary geometry. However, for practical applications of signal representation or spectral analysis such as exist in geophysics or astronomy, in physical space irregular shapes, and in spectral space symmetric domains will usually be preferred. When the concentration domains are circularly symmetric in both spaces, the Slepian functions are also eigenfunctions of a Sturm–Liouville operator, leading to special algorithms for this case, as is well known. Much like their one-dimensional and spherical counterparts with which we discuss them in a common framework, a basis of functions that are simultaneously spatially and spectrally localized on arbitrary Cartesian domains will be of great utility in many scientific disciplines, but especially in the geosciences.


Bandlimited function Commuting differential operator Concentration problem Eigenvalue problem Spectral analysis Reproducing kernel Spherical harmonics Sturm–Liouville problem 

Mathematics Subject Classification (2000)

42B99 41A30 86-08 47B15 46E22 34B24 45B05 33C55 


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  1. Abramowitz M., Stegun I.A.: Handbook of Mathematical Functions. Dover, New York (1965)Google Scholar
  2. Albertella A., Sacerdote F.: Using Slepian functions for local geodetic computations. Boll. Geod. Sci. Aff. 60(1), 1–14 (2001)Google Scholar
  3. Albertella A., Sansò F., Sneeuw N.: Band-limited functions on a bounded spherical domain: the Slepian problem on the sphere. J. Geod. 73, 436–447 (1999)zbMATHCrossRefGoogle Scholar
  4. Amirbekyan A., Michel V., Simons F.J.: Parameterizing surface-wave tomographic models with harmonic spherical splines. Geophys. J. Int. 174(2), 617 (2008). doi: 10.1111/j.1365-246X.2008.03809.x CrossRefGoogle Scholar
  5. Aronszajn N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68(3), 337–404 (1950)MathSciNetzbMATHCrossRefGoogle Scholar
  6. Audet P., Mareschal J.-C.: Wavelet analysis of the coherence between Bouguer gravity and topography: application to the elastic thickness anisotropy in the Canadian Shield. Geophys. J. Int. 168, 287–298 (2007). doi: 10.1111/j.1365-246X.2006.03231.x CrossRefGoogle Scholar
  7. Bell B., Percival D.B., Walden A.T.: Calculating Thomson’s spectral multitapers by inverse iteration. J. Comput. Graph. Stat. 2(1), 119–130 (1993)MathSciNetGoogle Scholar
  8. Bertero M., De Mol C., Pike E.R.: Linear inverse problems with discrete data. I. General formulation and singular system analysis. Inverse Probl. 1, 301–330 (1985a). doi: 10.1088/0266-5611/1/4/004 MathSciNetzbMATHCrossRefGoogle Scholar
  9. Bertero M., De Mol C., Pike E.R.: Linear inverse problems with discrete data. II. Stability and regularisation. Inverse Probl. 1, 301–330 (1985b). doi: 10.1088/0266-5611/1/4/004 MathSciNetzbMATHCrossRefGoogle Scholar
  10. Beylkin G., Monzón L.: On generalized Gaussian quadratures for exponentials and their applications. Appl. Comput. Harmon. Anal. 12, 332–372 (2002). doi: 10.1006/acha.2002.0380 MathSciNetzbMATHCrossRefGoogle Scholar
  11. Beylkin G., Sandberg K.: Wave propagation using bases for bandlimited functions. Wave Motion 41(3), 263–291 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  12. Borcea L., Papanicolaou G., Vasquez F.G.: Edge illumination and imaging of extended reflectors. SIAM J. Imaging Sci. 1(1), 75–114 (2008). doi: 10.1137/07069290X MathSciNetzbMATHCrossRefGoogle Scholar
  13. Bouwkamp C.J.: On spheroidal wave functions of order zero. J. Math. Phys. 26, 79–92 (1947)MathSciNetzbMATHGoogle Scholar
  14. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  15. Boyd J.P.: Prolate spheroidal wavefunctions as an alternative to Chebyshev and Legendre polynomials for spectral element and pseudospectral algorithms. J. Comput. Phys. 199(2), 688–716 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  16. Brander O., DeFacio B.: A generalisation of Slepian’s solution for the singular value decomposition of filtered Fourier transforms. Inverse Probl. 2, L9–L14 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Bronez T.P.: Spectral estimation of irregularly sampled multidimensional processes by generalized prolate spheroidal sequences. IEEE Trans. Acoust. Speech Signal Process. 36(12), 1862–1873 (1988)zbMATHCrossRefGoogle Scholar
  18. Chambodut A., Panet I., Mandea M., Diament M., Holschneider M., Jamet O.: Wavelet frames: an alternative to spherical harmonic representation of potential fields. Geophys. J. Int. 163(3), 875–899 (2005)CrossRefGoogle Scholar
  19. Chen Q.Y., Gottlieb D., Hesthaven J.S.: Spectral methods based on prolate spheroidal wave functions for hyperbolic PDEs. Wave Motion 43(5), 1912–1933 (2005)MathSciNetzbMATHGoogle Scholar
  20. Coifman R.R., Lafon S.: Geometric harmonics: a novel tool for multiscale out-of-sample extension of empirical functions. Appl. Comput. Harmon. Anal. 21, 31–52 (2006). doi: 10.1016/j.acha.2005.07.005 MathSciNetzbMATHCrossRefGoogle Scholar
  21. Dahlen F.A., Simons F.J.: Spectral estimation on a sphere in geophysics and cosmology. Geophys. J. Int. 174, 774–807 (2008). doi: 10.1111/j.1365-246X.2008.03854.x CrossRefGoogle Scholar
  22. Dahlen F.A., Tromp J.: Theoretical Global Seismology. Princeton University Press, Princeton, NJ (1998)Google Scholar
  23. Daubechies I.: Time–frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory 34, 605–612 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  24. Daubechies, I.: Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial & Applied Mathematics, Philadelphia, PA (1992)Google Scholar
  25. Daubechies I., Paul T.: Time–frequency localisation operators—a geometric phase space approach. II. The use of dilations. Inverse Probl. 4(3), 661–680 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 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, 1163–1179 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  27. de Villiers G.D., Marchaud F.B.T., Pike E.R.: Generalized Gaussian quadrature applied to an inverse problem in antenna theory: II. The two-dimensional case with circular symmetry. Inverse Probl. 19, 755–778 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  28. Delsarte P., Janssen A.J.E.M., Vries L.B.: Discrete prolate spheroidal wave functions and interpolation. SIAM J. Appl. Math. 45(4), 641–650 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  29. Donoho D.L., Stark P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49(3), 906–931 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  30. Edmonds A.R.: Angular Momentum in Quantum Mechanics. Princeton University Press, Princeton, NJ (1996)zbMATHGoogle Scholar
  31. Evans A.J., Andrews-Hanna J.C., Zuber M.T.: Geophysical limitations on the erosion history within Arabia Terra. J. Geophys. Res. 115, E05007 (2010). doi: 10.1029/2009JE003469 CrossRefGoogle Scholar
  32. Faÿ G., Guilloux F., Betoule M., Cardoso J.-F., Delabrouille J., Jeune M.L.: CMB power spectrum estimation using wavelets. Phys. Rev. D 78, 083013 (2008). doi: 10.1103/PhysRevD.78.083013 CrossRefGoogle Scholar
  33. Fengler M.J., Freeden W., Kohlhaas A., Michel V., Peters T.: Wavelet modeling of regional and temporal variations of the earth’s gravitational potential observed by GRACE. J. Geod. 81(1), 5–15 (2007). doi: 10.1007/s00190-006-0040-1 zbMATHGoogle Scholar
  34. Flandrin P.: Temps-Fréquence, 2nd edn. Hermès, Paris (1998)zbMATHGoogle Scholar
  35. Freeden W., Michel V.: Constructive approximation and numerical methods in geodetic research today—an attempt at a categorization based on an uncertainty principle. J. Geod. 73(9), 452–465 (1999)zbMATHCrossRefGoogle Scholar
  36. Freeden W., Windheuser U.: Combined spherical harmonic and wavelet expansion—a future concept in Earth’s gravitational determination. Appl. Comput. Harmon. Anal. 4, 1–37 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  37. Freeden W., Gervens T., Schreiner M.: Constructive Approximation on the Sphere. Clarendon Press, Oxford (1998)zbMATHGoogle Scholar
  38. Golub G.H., van Loan C.F.: Matrix Computations, 2nd edn. Johns Hopkins University Press, Baltimore, MD (1989)zbMATHGoogle Scholar
  39. Gradshteyn I.S., Ryzhik I.M.: Tables of Integrals, Series, and Products, 6th edn. Academic Press, San Diego, CA (2000)Google Scholar
  40. Grünbaum F.A.: Eigenvectors of a Toeplitz matrix: discrete version of the prolate spheroidal wave functions. SIAM J. Algebraic Discrete Methods 2(2), 136–141 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  41. Grünbaum F.A., Longhi L., Perlstadt M.: Differential operators commuting with finite convolution integral operators: some non-Abelian examples. SIAM J. Appl. Math. 42(5), 941–955 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  42. Hall B.C., Mitchell J.J.: Coherent states on spheres. J. Math. Phys. 43(3), 1211–1236 (2002)MathSciNetCrossRefGoogle Scholar
  43. Han S.-C.: Improved regional gravity fields on the Moon from Lunar Prospector tracking data by means of localized spherical harmonic functions. J. Geophys. Res. 113, E11012 (2008). doi: 10.1029/2008JE003166 CrossRefGoogle Scholar
  44. Han S.-C., Ditmar P.: Localized spectral analysis of global satellite gravity fields for recovering time-variable mass redistributions. J. Geod. 82(7), 423–430 (2007). doi: 10.1007/s00190-007-0194-5 Google Scholar
  45. Han S.-C., Simons F.J.: Spatiospectral localization of global geopotential fields from the Gravity Recovery and Climate Experiment GRACE reveals the coseismic gravity change owing to the 2004 Sumatra-Andaman earthquake. J. Geophys. Res. 113, B01405 (2008). doi: 10.1029/2007JB004927 CrossRefGoogle Scholar
  46. Han S.-C., Rowlands D.D., Luthcke S.B., Lemoine F.G.: Localized analysis of satellite tracking data for studying time-variable Earth’s gravity fields. J. Geophys. Res. 113, B06401 (2008a). doi: 10.1029/2007JB005218 CrossRefGoogle Scholar
  47. Han S.-C., Sauber J., Luthcke S.B., Ji C., Pollitz F.F.: Implications of postseismic gravity change following the great 2004 Sumatra-Andaman earthquake from the regional harmonic analysis of GRACE inter-satellite tracking data. J. Geophys. Res. 113, B11413 (2008b). doi: 10.1029/2008JB005705 CrossRefGoogle Scholar
  48. Han S.-C., Mazarico E., Lemoine F.G.: Improved nearside gravity field of the Moon by localizing the power law constraint. Geophys. Res. Lett. 36, L11203 (2009). doi: 10.1029/2009GL038556 CrossRefGoogle Scholar
  49. Hanssen A.: Multidimensional multitaper spectral estimation. Signal Process. 58, 327–332 (1997)zbMATHCrossRefGoogle Scholar
  50. Harig C., Zhong S., Simons F.J.: Constraints on upper-mantle viscosity inferred from the flow-induced pressure gradient across a continental keel. Geochem. Geophys. Geosyst. 11(6), Q06004 (2010). doi: 10.1029/2010GC003038 CrossRefGoogle Scholar
  51. Holschneider M., Chambodut A., Mandea M.: From global to regional analysis of the magnetic field on the sphere using wavelet frames. Phys. Earth Planet. Interiors 135, 107–124 (2003)CrossRefGoogle Scholar
  52. Jackson J.I., Meyer C.H., Nishimura D.G., Macovski A.: Selection of a convolution function for Fourier inversion using gridding. IEEE Trans. Med. Imaging 10(3), 473–478 (1991)CrossRefGoogle Scholar
  53. Jeffreys H., Jeffreys B.S.: Methods of Mathematical Physics, 3rd edn. Cambridge University Press, Cambridge (1988)Google Scholar
  54. Karoui A., Moumni T.: New efficient methods of computing the prolate spheroidal wave functions and their corresponding eigenvalues. Appl. Comput. Harmon. Anal. 24(3), 269–289 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  55. Kennedy, R.A., Zhang, W., Abhayapala, T.D.: Spherical harmonic analysis and model-limited extrapolation on the sphere: integral equation formulation. In: Proceedings of the IEEE International Conference on Signal Processing and Communication Systems, pp. 1–6. IEEE (2008). doi: 10.1109/ICSPCS.2008.4813702
  56. Khare K., George N.: Sampling theory approach to prolate spheroidal wavefunctions. J. Phys. A Math. Gen. 36, 10011–10021 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  57. Kido M., Yuen D.A., Vincent A.P.: Continuous wavelet-like filter for a spherical surface and its application to localized admittance function on Mars. Phys. Earth Planet. Interiors 135, 1–14 (2003)CrossRefGoogle Scholar
  58. Kirby J.F., Swain C.J.: Mapping the mechanical anisotropy of the lithosphere using a 2D wavelet coherence, and its application to Australia. Phys. Earth Planet. Interiors 158(2–4), 122–138 (2006). doi: 10.1016/j.pepi.2006.03.022 CrossRefGoogle Scholar
  59. Kowalski K., Rembieliński J.: Quantum mechanics on a sphere and coherent states. J. Phys. A Math. Gen. 33, 6035–6048 (2000)zbMATHCrossRefGoogle Scholar
  60. Lai M.J., Shum C.K., Baramidze V., Wenston P.: Triangulated spherical splines for geopotential reconstruction. J. Geod. 83, 695–708 (2009). doi: 10.1007/s00190-008-0283-0 CrossRefGoogle Scholar
  61. Landau H.J.: On the eigenvalue behavior of certain convolution equations. Trans. Am. Math. Soc. 115, 242–256 (1965)zbMATHCrossRefGoogle Scholar
  62. Landau H.J., Pollak H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—II. Bell Syst. Tech. J. 40(1), 65–84 (1961)MathSciNetzbMATHGoogle Scholar
  63. Landau H.J., Pollak H.O.: 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(4), 1295–1336 (1962)MathSciNetzbMATHGoogle Scholar
  64. Lilly J.M., Park J.: Multiwavelet spectral and polarization analyses of seismic records. Geophys. J. Int. 122, 1001–1021 (1995)CrossRefGoogle Scholar
  65. Lindquist M.A., Zhang C.H., Glover G., Shepp L., Yang Q.X.: A generalization of the two-dimensional prolate spheroidal wave function method for nonrectilinear MRI data acquisition methods. IEEE Trans. Image Process. 15(9), 2792–2804 (2006). doi: 10.1109/TIP.2006.877314 MathSciNetCrossRefGoogle Scholar
  66. Liu T.-C., van Veen B.D.: Multiple window based minimum variance spectrum estimation for multidimensional random fields. IEEE Trans. Signal Process. 40(3), 578–589 (1992). doi: 10.1109/78.120801 CrossRefGoogle Scholar
  67. Ma J., Rokhlin V., Wandzura S.: Generalized Gaussian quadrature rules for systems of arbitrary functions. SIAM J. Numer. Anal. 33(3), 971–996 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  68. Mallat S.: A Wavelet Tour of Signal Processing. Academic Press, San Diego, CA (1998)zbMATHGoogle Scholar
  69. Maniar H., Mitra P.P.: The concentration problem for vector fields. Int. J. Bioelectromagn. 7(1), 142–145 (2005)Google Scholar
  70. Marinucci D., Pietrobon D., Balbi A., Baldi P., Cabella P., Kerkyacharian G., Natoli P., Picard D., Vittorio N.: Spherical needlets for cosmic microwave background data analysis. Monthly Notices R. Astron. Soc. 383(2), 539–545 (2008). doi: 10.1111/j.1365-2966.2007.12550.x Google Scholar
  71. McEwen J.D., Hobson M.P., Mortlock D.J., Lasenby A.N.: Fast directional continuous spherical wavelet transform algorithms. IEEE Trans. Signal Process. 55(2), 520–529 (2007)MathSciNetCrossRefGoogle Scholar
  72. Michel V., Wolf K.: Numerical aspects of a spline-based multiresolution recovery of the harmonic mass density out of gravity functionals. Geophys. J. Int. 173, 1–16 (2008). doi: 10.1111/j.1365-246X.2007.03700.x CrossRefGoogle Scholar
  73. Miranian L.: Slepian functions on the sphere, generalized Gaussian quadrature rule. Inverse Probl. 20, 877–892 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  74. Mitra P.P., Maniar H.: Concentration maximization and local basis expansions (LBEX) for linear inverse problems. IEEE Trans. Biomed Eng. 53(9), 1775–1782 (2006)CrossRefGoogle Scholar
  75. Moore I.C., Cada M.: Prolate spheroidal wave functions, an introduction to the Slepian series and its properties. Appl. Comput. Harmon. Anal. 16, 208–230 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  76. Narcowich F.J., Ward J.D.: Nonstationary wavelets on the m-sphere for scattered data. Appl. Comput. Harmon. Anal. 3, 324–336 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  77. Nashed M.Z., Walter G.G.: General sampling theorems for functions in Reproducing Kernel Hilbert Spaces. Math. Control Signals Syst. 4, 363–390 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  78. Nyström E.J.: Über die praktische Auflösung von Integralgleichungen mit Anwendungen auf Randwertaufgaben. Acta Math. 54, 185–204 (1930)MathSciNetzbMATHCrossRefGoogle Scholar
  79. Olhede S., Walden A.T.: Generalized Morse wavelets. IEEE Trans. Signal Process. 50(11), 2661–2670 (2002)MathSciNetCrossRefGoogle Scholar
  80. Olhede S.C., Metikas G.: The monogenic wavelet transform. IEEE Trans. Signal Process. 57(9), 3426–3441 (2009). doi: 10.1109/TSP.2009.2023397 MathSciNetCrossRefGoogle Scholar
  81. Panet I., Chambodut A., Diament M., Holschneider M., Jamet O.: New insights on intraplate volcanism in French Polynesia from wavelet analysis of GRACE, CHAMP, and sea surface data. J. Geophys. Res. 111, B09403 (2006). doi: 10.1029/2005JB004141 CrossRefGoogle Scholar
  82. Papoulis A.: A new algorithm in spectral analysis and band-limited extrapolation. IEEE-CS 22(9), 735–742 (1975)MathSciNetGoogle Scholar
  83. Parks, T.W., Shenoy, R.G.: Time–frequency concentrated basis functions. In: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 5, pp. 2459–2462. IEEE (1990)Google Scholar
  84. Parlett B.N., Wu W.-D.: Eigenvector matrices of symmetric tridiagonals. Numer. Math. 44, 103–110 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  85. Percival D.B., Walden A.T.: Spectral Analysis for Physical Applications, Multitaper and Conventional Univariate Techniques. Cambridge University Press, New York (1993)zbMATHCrossRefGoogle Scholar
  86. Percival D.B., Walden A.T.: Wavelet Methods for Time Series Analysis. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  87. Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.: Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd edn. Cambridge University Press, New York (1992)Google Scholar
  88. Ramesh P.S., Lean M.H.: Accurate integration of singular kernels in boundary integral formulations for Helmholtz equation. Int. J. Numer. Methods Eng. 31, 1055–1068 (1991)zbMATHCrossRefGoogle Scholar
  89. Riedel K.S., Sidorenko A.: Minimum bias multiple taper spectral estimation. IEEE Trans. Signal Process. 43(1), 188–195 (1995)CrossRefGoogle Scholar
  90. Saito N.: Data analysis and representation on a general domain using eigenfunctions of Laplacian. Appl. Comput. Harmon. Anal. 25, 68–97 (2007). doi: 10.1016/j.acha.2007.09.005 CrossRefGoogle Scholar
  91. Schmidt M., Han S.-C., Kusche J., Sanchez L., Shum C.K.: Regional high-resolution spatiotemporal gravity modeling from GRACE data using spherical wavelets. Geophys. Res. Lett. 33(8), L0840 (2006). doi: 10.1029/2005GL025509 Google Scholar
  92. Schmidt M., Fengler M., Mayer-Gürr T., Eicker A., Kusche J., Sánchez L., Han S.-C.: Regional gravity modeling in terms of spherical base functions. J. Geod. 81(1), 17–38 (2007). doi: 10.1007/s00190-006-0101-5 zbMATHGoogle Scholar
  93. Schott, J.-J., Thébault, E.: Modelling the Earths magnetic field from global to regional scales. In: Mandea, M., Korte, M. (eds.) Geomagnetic Observations and Models, vol. 5 of IAGA Special Sopron Book Series. Springer, Berlin (2011)Google Scholar
  94. Shepp L., Zhang C.-H.: Fast functional magnetic resonance imaging via prolate wavelets. Appl. Comput. Harmon. Anal. 9(2), 99–119 (2000). doi: 10.1006/acha.2000.0302 MathSciNetzbMATHCrossRefGoogle Scholar
  95. Shkolnisky Y.: Prolate spheroidal wave functions on a disc—integration and approximation of two-dimensional bandlimited functions. Appl. Comput. Harmon. Anal. 22, 235–256 (2007). doi: 10.1016/j.acha.2006.07.002 MathSciNetzbMATHCrossRefGoogle Scholar
  96. Shkolnisky Y., Tygert M., Rokhlin V.: Approximation of bandlimited functions. Appl. Comput. Harmon. Anal. 21, 413–420 (2006). doi: 10.1016/j.acha.2006.05.001 MathSciNetzbMATHCrossRefGoogle Scholar
  97. Simons, F.J.: Slepian functions and their use in signal estimation and spectral analysis. In: Freeden, W., Nashed, M.Z., Sonar, T. (eds.) Handbook of Geomathematics, chap. 30, pp. 891–923. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-01546-5_30
  98. Simons F.J., Dahlen F.A.: Spherical Slepian functions and the polar gap in geodesy. Geophys. J. Int. 166, 1039–1061 (2006). doi: 10.1111/j.1365-246X.2006.03065.x CrossRefGoogle Scholar
  99. Simons, F.J., Dahlen, F.A.: A spatiospectral localization approach to estimating potential fields on the surface of a sphere from noisy, incomplete data taken at satellite altitudes. In: Van de Ville, D., Goyal, V.K., Papadakis, M. (eds.) Wavelets XII, vol. 6701, p. 670117. SPIE (2007). doi: 10.1117/12.732406
  100. Simons, F.J., van der Hilst, R.D., Zuber, M.T.: Spatio-spectral localization of isostatic coherence anisotropy in Australia and its relation to seismic anisotropy: Implications for lithospheric deformation. J. Geophys. Res. 108(B5) 2250. doi: 10.1029/2001JB000704
  101. Simons F.J., Dahlen F.A., Wieczorek M.A.: Spatiospectral concentration on a sphere. SIAM Rev. 48(3), 504–536 (2006). doi: 10.1137/S0036144504445765 MathSciNetzbMATHCrossRefGoogle Scholar
  102. Simons, F.J., Hawthorne, J.C., Beggan, C.D.: Efficient analysis and representation of geophysical processes using localized spherical basis functions. In: Goyal, V.K., Papadakis, M., Van de Ville, D. (eds.) Wavelets XIII, vol. 7446, p. 74460G. SPIE (2009). doi: 10.1117/12.825730
  103. Simons M., Solomon S.C., Hager B.H.: Localization of gravity and topography: constraints on the tectonics and mantle dynamics of Venus. Geophys. J. Int. 131, 24–44 (1997)CrossRefGoogle Scholar
  104. Slepian D.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—IV. Extensions to many dimensions; generalized prolate spheroidal functions. Bell Syst. Tech. J. 43(6), 3009–3057 (1964)MathSciNetzbMATHGoogle Scholar
  105. Slepian D.: On bandwidth. Proc. IEEE 64(3), 292–300 (1976)MathSciNetCrossRefGoogle Scholar
  106. Slepian D.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—V. The discrete case. Bell Syst. Tech. J. 57, 1371–1429 (1978)zbMATHGoogle Scholar
  107. Slepian D.: Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev. 25(3), 379–393 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  108. Slepian D., Pollak H.O.: Prolate spheroidal wave functions, Fourier analysis and uncertainty—I. Bell Syst. Tech. J. 40(1), 43–63 (1961)MathSciNetzbMATHGoogle Scholar
  109. Slepian D., Sonnenblick E.: Eigenvalues associated with prolate spheroidal wave functions of zero order. Bell Syst. Tech. J. 44(8), 1745–1759 (1965)MathSciNetzbMATHGoogle Scholar
  110. Tegmark M.: A method for extracting maximum resolution power spectra from galaxy surveys. Astrophys. J. 455, 429–438 (1995)CrossRefGoogle Scholar
  111. Tegmark M.: A method for extracting maximum resolution power spectra from microwave sky maps. Monthly Notices R. Astron. Soc. 280, 299–308 (1996)Google Scholar
  112. Thomson D.J.: Spectrum estimation and harmonic analysis. Proc. IEEE 70(9), 1055–1096 (1982)CrossRefGoogle Scholar
  113. Thomson D.J.: Quadratic-inverse spectrum estimates: applications to paleoclimatology. Phil. Trans. R. Soc. Lond. Ser. A 332(1627), 539–597 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  114. Tricomi F.G.: Integral Equations, 5th edn. Interscience, New York (1970)Google Scholar
  115. VanDe Ville D., Unser M.: Complex wavelet bases, steerability, and the Marr-like pyramid. IEEE Trans. Image Process. 17(11), 2063–2080 (2008). doi: 10.1109/TIP.2008.2004797 MathSciNetCrossRefGoogle Scholar
  116. Van De Ville D., Philips W., Lemahieu I.: On the N-dimensional extension of the discrete prolate spheroidal window. IEEE Trans. Signal Process. 9(3), 89–91 (2002)Google Scholar
  117. Walden A.T.: Improved low-frequency decay estimation using the multitaper spectral-analysis method. Geophys. Prospect. 38, 61–86 (1990)CrossRefGoogle Scholar
  118. Walter G., Soleski T.: A new friendly method of computing prolate spheroidal wave functions and wavelets. Appl. Comput. Harmon. Anal. 19, 432–443 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  119. Walter G.G., Shen X.: Wavelets based on prolate spheroidal wave functions. J. Fourier Anal. Appl. 10(1), 1–26 (2004). doi: 10.1007/s00041-004-8001-7 MathSciNetzbMATHCrossRefGoogle Scholar
  120. Walter G.G., Shen X.: Wavelet like behavior of Slepian functions and their use in density estimation. Commun. Stat. Theory Methods 34(3), 687–711 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  121. Walter G.G., Soleski T.: Error estimates for the PSWF method in MRI. Contemp. Math. 451, 262 (2008)MathSciNetGoogle Scholar
  122. Wei, L., Kennedy, R.A., Lamahewa, T.A.: Signal concentration on unit sphere: an azimuthally moment weighting approach. In: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, pp. 1–4. IEEE (2010)Google Scholar
  123. Wieczorek M.A., Simons F.J.: Localized spectral analysis on the sphere. Geophys. J. Int. 162(3), 655–675 (2005). doi: 10.1111/j.1365-246X.2005.02687.x CrossRefGoogle Scholar
  124. Wieczorek M.A., Simons F.J.: Minimum-variance spectral analysis on the sphere. J. Fourier Anal. Appl. 13(6), 665–692 (2007). doi: 10.1007/s00041-006-6904-1 MathSciNetzbMATHCrossRefGoogle Scholar
  125. Wingham D.J.: The reconstruction of a band-limited function and its Fourier transform from a finite number of samples at arbitrary locations by Singular Value Decomposition. IEEE Trans. Signal Process. 40(3), 559–570 (1992). doi: 10.1109/78.120799 CrossRefGoogle Scholar
  126. Xiao H., Rokhlin V., Yarvin N.: Prolate spheroidal wavefunctions, quadrature and interpolation. Inverse Probl. 17, 805–838 (2001). doi: 10.1088/0266-5611/17/4/315 MathSciNetzbMATHCrossRefGoogle Scholar
  127. Yang Q.X., Lindquist M.A., Shepp L., Zhang C.-H., Wang J., Smith M.B.: Two dimensional prolate spheroidal wave functions for MRI. J. Magn. Reson. 158, 43–51 (2002)CrossRefGoogle Scholar
  128. Yao K.: Application of reproducing kernel Hilbert spaces—bandlimited signal models. Inf. Control 11(4), 429–444 (1967)zbMATHCrossRefGoogle Scholar
  129. Zhang X.: Wavenumber spectrum of very short wind waves: an application of two-dimensional Slepian windows to spectral estimation. J. Atmos. Ocean. Technol. 11, 489–505 (1994)CrossRefGoogle Scholar
  130. Zhou Y., Rushforth C.K., Frost R.L.: Singular value decomposition, singular vectors, and the discrete prolate spheroidal sequences. Proc. IEEE Int. Conf. Acoust. Speech Signal Process. 9(1), 92–95 (1984)Google Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of GeosciencesPrinceton UniversityPrincetonUSA
  2. 2.Department of Statistics and Operations ResearchThe University of North Carolina at Chapel HillChapel HillUSA

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