Abhayapala, T. D., Pollock, T. S., and Kennedy, R. A. (2003) “Characterization of 3D spatial wireless channels,” In Proc. IEEE Vehicular Technology Conference (Fall), VTC2003-Fall, vol. 1, pp. 123-127. Orlando, FL.Google Scholar

Aczel, A. D. (2001) The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity.Washington Square Press, New York, NY.

Aizerman, A., Braverman, E. M., and Rozoner, L. I. (1964) “Theoretical foundations of the potential function method in pattern recognition learning,”Automation and Remote Control, vol. 25, pp. 821-837.Google Scholar

Antoine, J.-P. and Vandergheynst, P. (1999) “Wavelets on the 2-sphere: A group-theoretical approach,”Appl. Comput. Harmon. Anal., vol. 7, no. 3, pp. 262-291.CrossRefGoogle Scholar

Armitage, C. and Wandelt, B. D. (2004) “Deconvolution map-making for cosmic microwave background observations,”Phys. Rev. D, vol. 70, no. 12, pp. 123007:1-123007:7.Google Scholar

Aronszajn, N. (1950) “Theory of reproducing kernels,”Trans. Amer. Math. Soc., vol. 68, no. 3, pp. 337-404.CrossRefGoogle Scholar

Audet, P. (2011) “Directional wavelet analysis on the sphere: Application to gravity and topography of the terrestrial planets,”J. Geophys. Res., vol. 116.Google Scholar

Bell, E. (1986) Men of Mathematics.Simon & Schuster, New York, NY.

Bernkopf, M. (2008) “Schmidt, Erhard,” Complete Dictionary of Scientific Biography http://www.encyclopedia.com/doc/1G2-2830903888.html.

Bouniakowsky, V. (1859) “Sur quelques inegalites concernant les integrales ordinaires et les integrales aux differences finies,”Memoires de l'Acad. de St.-Petersbourg (VIP Serie), vol. 1, no. 9.Google Scholar

Brechbuhler, C. H., Gerig, G., and Kubler, O. (1995) “Parametrization of closed surfaces for 3-D shape description,”Comput. Vision Image Understand., vol. 61, no. 2, pp. 154-170.CrossRefGoogle Scholar

Bulow, T. (2004) “Spherical diffusion for 3D surface smoothing,”IEEE Trans. Pattern Anal. Mach. Intell., vol. 26, no. 12, pp. 1650-1654.CrossRefGoogle Scholar

Cajori, F. (1985) A History of Mathematics.Chelsea Publishing Company, London, 4th edn. (the first edition was published by The Macmillian Company, New York in 1893).

Cohen, L. (1989) “Time-frequency distributions-a review,”Proc. IEEE, vol. 77, no. 7, pp. 941-981.CrossRefGoogle Scholar

Cohen, L. (1995) Time-Frequency Analysis: Theory and Applications.Prentice-Hall, New York, NY.

Colton, D. and Kress, R. (1998) Inverse Acoustic and Electromagnetic Scattering Theory.Springer-Verlag, Berlin, 2nd edn.

Courant, R. and Hilbert, D. (1966) Methods of Mathematical Physics. Volume 1.Interscience Publishers, New York, NY (the first German edition was published by Julius Springer, Berlin in 1924, and the first English edition by Interscience Publishers, New York in 1953).

Cucker, F. and Smale, S. (2002) “On the mathematical foundations of learning,”Bull. Am. Math. Soc., New Ser., vol. 39, no. 1, pp. 1-49.CrossRefGoogle Scholar

Cui, J. and Freeden, W. (1997) “Equidistribution on the sphere,”SIAM J. Sci. Comput., vol. 18, no. 2, pp. 595-609.CrossRefGoogle Scholar

Dahlen, F. A. and Simons, F. J. (2008) “Spectral estimation on a sphere in geophysics and cosmology,”Geophys. J. Int., vol. 174, no. 3, pp. 774-807.CrossRefGoogle Scholar

Dauben, J. W. (1990) Georg Cantor: His Mathematics and Philosophy of the Infinite.Princeton University Press, Princeton, NJ.

Debnath, L. and Mikusinski, P. (1999) Introduction to Hilbert Spaces with Applications.Academic Press, San Diego, CA, 2nd edn.

Driscoll, J. R. and Healy, D. M. Jr. (1994) “Computing Fourier transforms and convolutions on the 2-sphere,”Adv. Appl. Math., vol. 15, no. 2, pp. 202-250.CrossRefGoogle Scholar

Durak, L. and Arikan, O. (2003) “Short-time Fourier transform: two fundamental properties and an optimal implementation,”IEEE Trans. Signal Process., vol. 51, no. 5, pp. 1231-1242.CrossRefGoogle Scholar

Edmonds, A. R. (1957) Angular Momentum in Quantum Mechanics.Princeton University Press, Princeton, NJ.

Farmelo, G. (2009) The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom.Basic Books, New York, NY.

Fischer, E. (1907) “Sur la convergence en moyenne,”Comptes rendus de I'Academie des sciences, Paris, vol. 144, pp. 1022-1024.Google Scholar

Franks, L. E. (1969) Signal Theory.Prentice-Hall, Englewood Cliffs, NJ.

Freeden, W. and Michel, V. (1999) “Constructive approximation and numerical methods in geodetic research today. An attempt at a categorization based on an uncertainty principle,”J. Geodesy, vol. 73, no. 9, pp. 452-465.CrossRefGoogle Scholar

Gorski, K. M., Hivon, E., Banday, A. J., Wandelt, B. D., Hansen, F. K., Reinecke, M., and Bartelmann, M. (2005) “HEALPix: A Framework for high-resolution discretization and fast analysis of data distributed on the sphere,”Astrophys. J., vol. 622, no. 2, pp. 759-771.CrossRefGoogle Scholar

Han, C., Sun, B., Ramamoorthi, R., and Grinspun, E. (2007) “Frequency domain normal map filtering,”ACM Trans. Graph., vol. 26, no. 3, pp. 28:1-28:12.Google Scholar

Harris, F. J. (1978) “On the use of windows for harmonic analysis with the discrete Fourier transform,”Proc. IEEE, vol. 66, no. 1, pp. 51-83.CrossRefGoogle Scholar

Helmberg, G. (1969) Introduction to Spectral Theory in Hilbert Space.John Wiley & Sons, New York, NY (reprinted in 2008 by Dover Publications, Mineola, NY).

Huang, W., Khalid, Z., and Kennedy, R. A. (2011) “Efficient computation of spherical harmonic transform using parallel architecture of CUDA,” In Proc. Int. Conf. Signal Process. Commun. Syst., ICSPCS'2011, p. 6. Honolulu, HI.

Kennedy, R. A., Lamahewa, T. A., and Wei, L. (2011) “On azimuthally symmetric 2-sphere convolution,”Digit. Signal Process., vol. 5, no. 11, pp. 660-666.CrossRefGoogle Scholar

Kennedy, R. A., Sadeghi, P., Abhayapala, T. D., and Jones, H. M. (2007) “Intrinsic limits of dimensionality and richness in random multipath fields,”IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2542-2556.CrossRefGoogle Scholar

Khalid, Z., Durrani, S., Kennedy, R. A., and Sadeghi, P. (2012a) “Spatio-spectral analysis on the sphere using spatially localized spherical harmonics transform,”IEEE Trans. Signal Process., vol. 60, no. 3, pp. 1487-1492.Google Scholar

Khalid, Z., Kennedy, R. A., Durrani, S., Sadeghi, P., Wiaux, Y., and McEwen, J. D. (2013a) “Fast directional spatially localized spherical harmonic transform,” IEEE Trans. Signal Process. (in print).

Khalid, Z., Kennedy, R. A., and Sadeghi, P. (2012b) “Efficient computation of commutative anisotropic convolution on the 2-sphere,” In Proc. Int. Conf. Signal Process. Commun. Syst., ICSPCS'2012, p. 6. Gold Coast, Australia.

Khalid, Z., Sadeghi, P., Kennedy, R. A., and Durrani, S. (2013b) “Spatially varying spectral filtering of signals on the unit sphere,”IEEE Trans. Signal Process., vol. 61, no. 3, pp. 530-544.Google Scholar

Koks, D. (2006) “Using rotations to build aerospace coordinate systems,” Tech. Rep. DSTO-TN-0640, Defence Science and Technology Organisation, Australia, Electronic Warfare and Radar Division, Systems Sciences Laboratory.

Kreyszig, E. (1978) Introductory Functional Analysis with Applications.John Wiley & Sons, New York, NY.

Landau, H. J. and Pollak, H. O. (1962) “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., vol. 41, no. 4, pp. 1295-1336.CrossRefGoogle Scholar

Landau, H. J., Slepian, D., and Pollak, H. O. (1961) “Prolate spheroidal wave functions, Fourier analysis and uncertainty-II,”Bell Syst. Tech. J., vol. 40, no. 1, pp. 65-84.CrossRefGoogle Scholar

Lebedev, N. N. (1972) Special Functions and Their Applications.Dover Publications, New York, NY (revised edition, translated from the Russian and edited by Richard A. Silverman, unabridged and corrected republication).

Mate, L. (1989) Hilbert Space Methods in Science and Engineering.Adam Hilger, IOP Publishing, Bristol and New York.

McEwen, J. D., Hobson, M. P., Mortlock, D. J., and Lasenby, A. N. (2007) “Fast directional continuous spherical wavelet transform algorithms,”IEEE Trans. Signal Process., vol. 55, no. 2, pp. 520-529.CrossRefGoogle Scholar

McEwen, J. D. and Wiaux, Y. (2011) “A novel sampling theorem on the sphere,”IEEE Trans. Signal Process., vol. 59, no. 12, pp. 5876-5887.CrossRefGoogle Scholar

Mercer, J. (1909) “Functions of positive and negative type, and their connection with the theory of integral equations,”Phil. Trans. R. Soc. Lond., vol. 209, no. 456, pp. 415-446.Google Scholar

Messiah, A. (1961) Quantum Mechanics.North-Holland, Amsterdam (translated from French by J. Potter).

Narcowich, F. J. and Ward, J. D. (1996) “Non-stationary wavelets on the m-sphere for scattered data,”Appl. Comput. Harmonic Anal., vol. 3, no. 4, pp. 324-336.CrossRefGoogle Scholar

Ng, K.-W. (2005) “Full-sky correlation functions for CMB experiments with asymmetric window functions,”Phys. Rev. D, vol. 71, no. 8, pp. 083009:1-083009:9.Google Scholar

Oppenheim, A. V., Willsky, A. S., and Nawab, S. H. (1996) Signals and Systems.Prentice-Hall, Upper Saddle River, NJ.

Papoulis, A. (1977) Signal Analysis.McGraw-Hill, New York, NY.

Pietsch, A. (2010) “Erhard Schmidt and his contributions to functional analysis,”Math. Nachr., vol. 283, no. 1, pp. 6-20.CrossRefGoogle Scholar

Pollock, T. S., Abhayapala, T. D., and Kennedy, R. A. (2003) “Introducing space into MIMO capacity calculations,”J. Telecommun. Syst., vol. 24, no. 2, pp. 415-436.CrossRefGoogle Scholar

Rafaely, B. (2004) “Plane-wave decomposition of the sound field on a sphere by spherical convolution,”J. Acoust. Soc. Am., vol. 116, no. 4, pp. 2149-2157.CrossRefGoogle Scholar

Reid, C. (1986) Hilbert-Courant.Springer-Verlag, New York, NY (previously published as two separate volumes: Hilbert by Constance Reid, Springer-Verlag, 1970; Courant in Gottingen and New York: The Story of an Improbable Mathematician by Constance Reid, Springer-Verlag, 1976).

Riesz, F. (1907) “Sur les systemes orthogonaux de fonctions,”Comptes rendus de I'Academie des sciences, Paris, vol. 144, pp. 615-619.Google Scholar

Riesz, F. and Sz.-Nagy, B. (1990) Functional Analysis.Dover Publications, Mineola, NY, 2nd edn. (unabridged Dover republication of the edition published by Frederick Ungar Publishing Co., New York, 1955; translated from the original second French edition 1953).

Riley, K. F., Hobson, M. P., and Bence, S. J. (2006) Mathematical Methods for Physics and Engineering.Cambridge University Press, Cambridge, UK, 3rd edn.

Risbo, T. (1996) “Fourier transform summation of Legendre series and D-functions,”J. Geodesy, vol. 70, no. 7, pp. 383-396.CrossRefGoogle Scholar

Sadeghi, P., Kennedy, R. A., and Khalid, Z. (2012) “Commutative anisotropic convolution on the 2-sphere,”IEEE Trans. Signal Process., vol. 60, no. 12, pp. 6697-6703.CrossRefGoogle Scholar

Sakurai, J. J. (1994) Modern Quantum Mechanics.Addison-Wesley Publishing Company, Inc., Reading, MA.

Schroder, P. and Sweldens, W. (2000) “Spherical wavelets: Efficiently representing functions on a sphere,” In Sweldens, R. Haagmans, eds., Wavelets in the Geosciences, vol. 90 of Lecture Notes in Earth Sciences, pp. 158-188. Springer, Berlin (reprinted from Computer Graphics Proceedings, 1995, 161-172, ACM Siggraph).

Seon, K.-I. (2006) “Smoothing of all-sky survey map with Fisher-von Mises function,”J. Korean Phys. Soc., vol. 48, no. 3, pp. 331-334.Google Scholar

Simons, F. J., Dahlen, F. A., and Wieczorek, M. A. (2006) “Spatiospec-tral concentration on a sphere,”SIAM Rev., vol. 48, no. 3, pp. 504-536.CrossRefGoogle Scholar

Simons, M., Solomon, S. C., and Hager, B. H. (1997) “Localization of gravity and topography: constraints on the tectonics and mantle dynamics of Venus,”Geophys. J. Int., vol. 131, no. 1, pp. 24-44.CrossRefGoogle Scholar

Slepian, D. (1978) “Prolate spheroidal wave functions, Fourier analysis and uncertainty-V: Discrete time case,”Bell Syst. Tech. J., vol. 57, no. 5, pp. 3009-3057.Google Scholar

Slepian, D. and Pollak, H. O. (1961) “Prolate spheroidal wave functions, Fourier analysis and uncertainty-I,”Bell Syst. Tech. J., vol. 40, no. 1, pp. 43-63.CrossRefGoogle Scholar

Sloane, N. J. A. (2001) “The On-Line Encyclopedia of Integer Sequences,” http://oeis.org/A059343 (table of Hermite polynomial coefficients).

Sloane, N. J. A. (2002) “The On-Line Encyclopedia of Integer Sequences,” http://oeis.org/A008316 (table of Legendre polynomial coefficients).

Spergel, D. N., Bean, R., Dore, 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., and Wright, E. L. (2007) “Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Implications for cosmology,”The Astrophysical Journal Supplement Series, vol. 170, no. 2, pp. 377-408.CrossRefGoogle Scholar

Starck, J.-L., Moudden, Y., Abrial, P., and Nguyen, M. (2006) “Wavelets, ridgelets and curvelets on the sphere,”Astronom. and Astrophys., vol. 446, no. 3, pp. 1191-1204.CrossRefGoogle Scholar

Stewart, I. (1973) Galois Theory.Chapman and Hall, London.

Strang, G. (2007) Computational Science and Engineering.Wellesley-Cambridge Press, Wellesley, MA.

Tegmark, M., Hartmann, D. H., Briggs, M. S., and Meegan, C. A. (1996) “The angular power spectrum of BATSE 3B gamma-ray bursts,”Astrophys. J., vol. 468, pp. 214-224.CrossRefGoogle Scholar

Trapani, S. and Navaza, J. (2006) “Calculation of spherical harmonics and Wigner d functions by FFT. Applications to fast rotational matching in molecular replacement and implementation into AMoRe,”Acta Cryst. A, vol. 62, no. 4, pp. 262-269.CrossRefGoogle Scholar

Wandelt, B. D. and Gorski, K. M. (2001) “Fast convolution on the sphere,”Phys. Rev. D, vol. 63, no. 12, p. 123002.Google Scholar

Watson, G. N. (1995) A Treatise on the Theory of Bessel Functions.Cambridge University Press, Cambridge, UK.

Wei, L., Kennedy, R. A., and Lamahewa, T. A. (2011) “Quadratic vari-ational framework for signal design on the 2-sphere,”IEEE Trans. Signal Process., vol. 59, no. 11, pp. 5243-5252.CrossRefGoogle Scholar

Wiaux, Y., Jacques, L., and Vandergheynst, P. (2005) “Correspondence principle between spherical and Euclidean wavelets,”Astrophys. J., vol. 632, no. 1, pp. 15-28.CrossRefGoogle Scholar

Wiaux, Y., Jacques, L., Vielva, P., and Vandergheynst, P. (2006) “Fast directional correlation on the sphere with steerable filters,”Astrophys. J., vol. 652, no. 1, pp. 820-832.CrossRefGoogle Scholar

Wiaux, Y., McEwen, J. D., Vandergheynst, P., and Blanc, O. (2008) “Exact reconstruction with directional wavelets on the sphere,”Mon. Not. R. Astron. Soc., vol. 388, no. 2, pp. 770-788.CrossRefGoogle Scholar

Wieczorek, M. (2007) “Gravity and topography of the terrestrial planets,”Treatise on Geophysics, vol. 10, pp. 165-206.Google Scholar

Wieczorek, M. A. and Simons, F. J. (2005) “Localized spectral analysis on the sphere,”Geophys. J. Int., vol. 131, no. 1, pp. 655-675.CrossRefGoogle Scholar

Wigner, E. P. (1959) Group Theory and its Application to the Quantum Mechanics of Atomic Spectra.Academic Press, New York, NY (translated from the original German version first published in 1931; edited by J. J. Griffin, expanded and improved edition).

Wigner, E. P. (1965) “On the matrices which reduce the Kronecker products of representations of S. R. groups,” In Sweldens, Biedenharn, L. C. and Sweldens, H. van Dam, eds., Quantum Theory of Angular Momentum, pp. 87-133. Academic Press, New York, NY (reprinted from a 1940 unpublished manuscript).

Yeo, B. T. T., Ou, W., and Golland, P. (2008) “On the construction of invertible filter banks on the 2-sphere,”IEEE Trans. Image Process., vol. 17, no. 3, pp. 283-300.CrossRefGoogle Scholar

Yu, P., Grant, P. E., Yuan, Q., Xiao, H., Segonne, F., Pienaar, R., Busa, E., Pacheco, J., Makris, N., Buckner, R. L., Golland, P., and Fischl, B. (2007) “Cortical surface shape analysis based on spherical wavelets,”IEEE Trans. Med. Imag., vol. 26, no. 4, pp. 582-597.CrossRefGoogle Scholar

Zuber, M. T., Smith, D. E., Solomon, S. C., Muhleman, D. O., Head, J. W., Garvin, J. B., Abshire, J. B., and Bufton, J. L. (1992) “The Mars observer laser altimeter investigation,”J. Geophys. Res., vol. 97, no. E5, pp. 7781-7797.CrossRefGoogle Scholar