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Geometric Space–Frequency Analysis on Manifolds

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Abstract

This paper gives a survey of methods for the construction of space–frequency concentrated frames on Riemannian manifolds with bounded curvature, and the applications of these frames to the analysis of function spaces. In this general context, the notion of frequency is defined using the spectrum of a distinguished differential operator on the manifold, typically the Laplace–Beltrami operator. Our exposition starts with the case of the real line, which serves as motivation and blueprint for the material in the subsequent sections. After the discussion of the real line, our presentation starts out in the most abstract setting proving rather general sampling-type results for appropriately defined Paley–Wiener vectors in Hilbert spaces. These results allow a handy construction of Paley–Wiener frames in \(L_2(\mathbf {M})\), for a Riemann manifold of bounded geometry, essentially by taking a partition of unity in frequency domain. The discretization of the associated integral kernels then gives rise to frames consisting of smooth functions in \(L_2(\mathbf {M})\), with fast decay in space and frequency. These frames are used to introduce new norms in corresponding Besov spaces on \(\mathbf {M}\). For compact Riemannian manifolds the theory extends to \(L_p\) and associated Besov spaces. Moreover, for compact homogeneous manifolds, one obtains the so-called product property for eigenfunctions of certain operators and proves cubature formulae with positive coefficients which allow to construct Parseval frames that characterize Besov spaces in terms of coefficient decay. The general theory is exemplified with the help of various concrete and relevant examples which include the unit sphere and the Poincaré half plane.

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References

  1. Antoine, J.-P., Rosca, D., Vandergheynst, P.: Wavelet transform on manifolds: old and new approaches. Appl. Comput. Harmon. Anal. 28(2), 189–202 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bahouri, Hajer: Gallagher, Isabelle: Paraproduit sur le groupe de Heisenberg et applications. (French) [Paraproduct on the Heisenberg group and applications]. Rev. Mat. Iberoamericana 17(1), 69–105 (2001)

    Article  MathSciNet  Google Scholar 

  3. Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D.: Subsampling needlet coefficients on the sphere. Bernoulli 15, 438–463 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D.: Asymptotics for spherical needlets. Ann. Stat. 37(3), 1150–1171 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Benedetto, J.: Frame Decompositions, Sampling, And Uncertainty Principle Inequalities. Wavelets: Mathematics and applications. Studies in Advanced Mathematics, pp. 247–304. CRC, Boca Raton (1994)

  6. Bergh, J., Löfström, J.: Interpolation Spaces. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  7. Bernstein, S., Ebert, S.: Wavelets on \(S^{3}\) and \(SO(3)\): their construction, relation to each other and Radon transform of wavelets on \(SO(3)\). Math. Methods Appl. Sci. 33, 1895–1909 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Bernstein, S., Ebert, S., Pesenson, I.Z.: Generalized splines for Radon transform on compact Lie groups with applications to crystallography. J. Fourier Anal. Appl. 19, 144–166 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bernstein, S., Pesenson, I.Z.: The Radon Transform on SO(3): Motivations, Generalizations, Discretization, Geometric Analysis and Integral Geometry. Contemporary Mathematics, vol. 598, pp. 77–96. American Mathematical Society, Providence (2013)

  10. Birman, M., Solomyak, M.: Spectral Theory of Selfadjoint Operators in Hilbert Space. D. Reidel Publishing Co., Dordrecht (1987)

    MATH  Google Scholar 

  11. Boas, R.: Entire Functions. Academic Press, New York (1954)

    MATH  Google Scholar 

  12. Bui, H.Q., Duong, X.T., Yan, L.: Calderon reproducing formulas and new Besov spaces associated with operators. Adv. Math. 229(4), 2449–2502 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Butzer, P., Berens, H.: Semi-Groups of Operators and Approximation. Springer, Berlin (1967)

    Book  MATH  Google Scholar 

  14. Butzer, P.L., Scherer, K.: Jackson and Bernstein-type inequalities for families of commutative operators in Banach spaces. J. Approx. Theory 5, 308–342 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  15. Calixto, M., Guerrero, J., Sanchez-Monreal, J.C.: Sampling theorem and discrete Fourier transform on the hyperboloid. J. Fourier Anal. Appl. 17(2), 240–264 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Christensen, J., Olafsson, G.: Examples of coorbit spaces for dual pairs. Acta Appl. Math. 107, 25–48 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Christensen, J., Olafsson, G.: Coorbit spaces for dual pairs. Appl. Comput. Harmon. Anal. 31(2), 303–324 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Christensen, J.: Sampling in reproducing kernel Banach spaces on Lie groups. J. Approx. Theory 164(1), 179–203 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Coifman, R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogenes. Lecture Notes in Mathematics, vol. 242. Springer, Berlin (1971)

  20. Coifman, R.R., Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal. 21, 53–94 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Coulhon, T., Kerkyacharian, G., Petrushev, P.: Heat kernel generated frames in the setting of Dirichlet spaces. J. Fourier Anal. Appl. 18(5), 995–1066 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer, New York (2013)

    Book  MATH  Google Scholar 

  23. Dahlke, S., Dahmen, W., Weinreich, I., et al.: Multiresolution analysis and wavelets on \(S^{2}\) and \(S^{3}\). Numer. Funct. Anal. Optim. 16, 19–41 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dahlke, S., Steidl, G., Teschke, G.: Coorbit spaces and Banach frames on homogeneous spaces with applications to the sphere. Adv. Comput. Math. 21, 147–180 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. DeVore, R., Lorentz, G.: Constructive Approximation. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  26. Duffin, R., Schaeffer, A.: A class of nonharmonic Fourier series. Trans. AMS 72, 341–366 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  27. Durastanti, C., Fantaye, Y., Hansen, F., Marinucci, D., Pesenson, I.Z.: A simple proposal for radial 3D needlets. Phys. Rev. D 90, 103532 (2014)

    Article  Google Scholar 

  28. Ebata, M., Eguchi, M., Koizumi, S., Kumahara, K.: On sampling formulas on symmetric spaces. J. Fourier Anal. Appl. 12(1), 1–15 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ebata, M., Eguchi, M., Koizumi, S., Kumahara, K.: Analogues of sampling theorems for some homogeneous spaces. Hiroshima Math. J. 36(1), 125–140 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Ehler, M., Filbir, F., Mhaskar, H.N.: Locally learning biomedical data using diffusion frames. J. Comput. Biol. 19(11), 1251–1264 (2012)

    Article  MathSciNet  Google Scholar 

  31. Feichtinger, H., Gröchenig, K.: Theory and practice of irregular sampling. In: Benedetto, J.J., Frazier, M.W. (eds.) Wavelets: Mathematics and Applications. Studies in Advanced Mathematics, pp. 305–363. CRC, Boca Raton (1994)

  32. Feichtinger, H., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions. I. J. Funct. Anal. 86(2), 307–340 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  33. Feichtinger, H., Gröchenig, K.: Banach spaces related to integrable group representations and their atomic decompositions. II. Monatsh. Math. 108(2–3), 129–148 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  34. Feichtinger, H., Pesenson, I.: Iterative Recovery of Band Limited Functions on Manifolds. Contemporary Mathematics, vol. 345, pp. 137–153. AMS, Providence (2004)

  35. Feichtinger, H., Pesenson, I.: A reconstruction method for band-limited signals on the hyperbolic plane. Sampl. Theory Signal Image Process. 4(2), 107–119 (2005)

    MathSciNet  MATH  Google Scholar 

  36. Filbir, F., Mhaskar, H.: A quadrature formula for diffusion polynomials corresponding to a generalized heat kernel. J. Fourier Anal. Appl. 16(5), 629–657 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Filbir, F., Mhaskar, H.N.: Marcinkiewicz-Zygmund measures on manifolds. J. Complex. 27(6), 568–596 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  39. Führ, H.: Abstract Harmonic Analysis of Continuous Wavelet Transforms. Lecture Notes in Mathematics, vol. 1863. Springer, Berlin (2005)

  40. Führ, H.: Painless Gabor expansions on homogeneous manifolds. Appl. Comput. Harmon. Anal. 26(2), 200–211 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Führ, H., Gröchenig, K.: Sampling theorems on locally compact groups from oscillation estimates. Math. Z. 255(1), 177–194 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Führ, H., Mayeli, A.: Homogeneous Besov spaces on stratified Lie groups and their wavelet characterization. J. Funct. Spac. Appl., 2012, p.Art. ID 523586, 41

  43. Führ, H., Pesenson, I.: Poincaré and Plancherel-Polya inequalities in harmonic analysis on weighted combinatorial graphs. SIAM J. Discret. Math. 27(4), 2007–2028 (2013)

    Article  MATH  Google Scholar 

  44. Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere, with Applications to Geomathematics. Clarendon Press, Oxford (1998)

    MATH  Google Scholar 

  45. Freeden, W., Volker, M.: Multiscale Potential Theory. Birkhäuser, Boston (2004)

    Book  MATH  Google Scholar 

  46. Furioli, G., Melzi, C., Veneruso, A.: Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth. Math. Nachr. 279, 1028–1040 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  47. Geller, D., Mayeli, A.: Continuous wavelets on compact manifolds. Math. Z. 262, 895–927 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Geller, D., Mayeli, A.: Nearly tight frames and space-frequency analysis on compact manifolds. Math. Z. 263, 235–264 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  49. Geller, D., Mayeli, A.: Besov spaces and frames on compact manifolds. Indiana Univ. Math. J. 58(5), 2003–2042 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  50. Geller, D., Marinucci, D.: Mixed needlets. J. Math. Anal. Appl. 375, 610–630 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. Geller, D., Pesenson, I.: Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21(2), 334–371 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. Geller, D., Pesenson, I.: n-Widths and Approximation Theory on Compact Riemannian Manifolds, Commutative and Noncommutative Harmonic Analysis and Applications. Contemporary Mathematics, vol. 603, pp. 111–122. American Mathematical Society, Providence (2013)

  53. Geller, D., Pesenson, I.: Kolmogorov and linear widths of Balls in Sobolev spaces on compact manifolds. Math. Scand. 115(1), 96–122 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  54. Gröchenig, K.: Describing functions: atomic decompositions versus frames. Monatsh. Math. 112(3), 1–41 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  55. Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)

    Book  MATH  Google Scholar 

  56. Han, Y., Müller, D., Yang, D.: Littlewood–Paley characterizations for Hardy spaces on spaces of homogeneous type. Math. Nachr. 279, 1505–1537 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  57. Han, Y., Müller, D., Yang, D.: A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Caratheodory Spaces. Abstract and Applied Analysis, vol. 2008. Hindawi Publishing Corporation, Cairo (2009)

  58. Hebey, E.: Sobolev Spaces on Riemannian Manifolds. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  59. Helgason, S.: Geometric Analysis on Symmetric Spaces. Mathematical Surveys and Monographs. AMS, Providence (2009)

    MATH  Google Scholar 

  60. Hörmander, L.: The Analysis of Linear Partial Differential Operators. III. Pseudo-differential Operators. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  61. Kempf, A.: Covariant information-density cutoff in curved space-time. Phys. Rev. Lett. 92(22), 221301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  62. Kempf, A., Martin, R.T.W.: Information theory, spectral geometry, and quantum gravity. Phys. Rev. Lett. 100(2), 021304 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  63. Kempf, A., Chatwin-Davies, A., Martin, R.T.W.: A fully covariant information-theoretic ultraviolet cutoff for scalar fields in expanding Friedmann Robertson Walker spacetimes. J. Math. Phys. 54(2), 022301 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  64. Kerkyacharian, G., Petrushev, P.: Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces (2012) arXiv:1210.6237

  65. Klainerman, S., Rodnianski, I.: A geometric approach to the Littlewood-Paley theory. Geom. Funct. Anal. 16(1), 126–163 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  66. Krein, S., Petunin, Y., Semenov, E.: Interpolation of Linear Operators. Translations of Mathematical Monographs, vol. 54. AMS, Providence (1982)

  67. Krein, S., Pesenson, I.: Interpolation Spaces and Approximation on Lie Groups. The Voronezh State University, Voronezh (1990). (Russian)

    MATH  Google Scholar 

  68. Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problem and Applications. Springer, Berlin (1975)

    MATH  Google Scholar 

  69. Maggioni, M., Mhaskar, H.N.: Diffusion polynomial frames on metric measure spaces. Appl. Comput. Harmon. Anal. 24(3), 329–353 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  70. Mallat, S.: Group invariant scattering. Commun. Pure Appl. Math. 65(10), 1331–1398 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  71. Marinucci, D., Peccati, G.: Random Fields on the Sphere. Representation, Limit Theorems and Cosmological Applications. London Mathematical Society Lecture Note Series, vol. 389. Cambridge University Press, Cambridge (2011)

  72. Marinucci, D., et al.: Spherical needlets for CMB data analysis. Mon. Not. R. Astron. Soc. 383, 539–545 (2008)

    Article  Google Scholar 

  73. Müller, D., Yang, D.: A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces. Forum Math. 21, 259–298 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  74. Narcowich, F.J., Petrushev, P., Ward, J.: Localized tight frames on spheres. SIAM J. Math. Anal. 38, 574–594 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  75. Narcowich, F.J., Petrushev, P., Ward, J.: Decomposition of Besov and Triebel-Lizorkin spaces on the sphere. J. Funct. Anal. 238, 530–564 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  76. Nikol’skii, S.: Approximation of Functions of Several Variables and Imbedding Theorems. Springer, Berlin (1975)

    Book  MATH  Google Scholar 

  77. Nursultanov, E., Ruzhansky, M., Tikhonov, S.: Nikolskii inequality and Besov, Triebel-Lizorkin, Wiener and Beurling spaces on compact homogeneous manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. (2014, to appear). arXiv:1403.3430

  78. Nursultanov, E., Ruzhansky, M., Tikhonov, S.: Nikolskii inequality and functional classes on compact Lie groups, Funct. Anal. Appl. 49, 226–229 (2015)

  79. Ortega-Cerda, J., Pridhnani, B.: Beurling-Landau’s density on compact manifolds. J. Funct. Anal. 263(7), 2102–2140 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  80. Pasquale, A.: A Paley-Wiener theorem for the inverse spherical transform. Pac. J. Math. 193, 143–176 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  81. Peetre, J., Sparr, G.: Interpolation on normed Abelian groups. Ann. Mat. Pura Appl. 92, 217–262 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  82. Pesenson, I.: Interpolation spaces on Lie groups, (Russian) Dokl. Akad. Nauk SSSR 246(6), 1298–1303 (1979)

  83. Pesenson, I.: Nikolskii-Besov spaces connected with representations of Lie groups, (Russian) Dokl. Akad. Nauk SSSR 273/1 (1983), 45–49; Engl. Transl. in Soviet Math. Dokl. 28 (1983)

  84. Pesenson, I.: Abstract theory of Nikolskii-Besov spaces, Izv. Vyssh. Uchebn. Zaved. Mat. 59–70 (1988); Engl. Transl. in Soviet Mathematics, 32/6 (1988)

  85. Pesenson, I.: The best approximation in a representation space of a Lie group. Dokl. Acad. Nauk USSR, v. 302(5), 1055–1059 (1988). (Engl. Transl. in Soviet Math. Dokl. 38/2 (1989), 384–388)

  86. Pesenson, I.: Approximations in the representation space of a Lie group. Izv. Vyssh. Uchebn. Zaved. Mat. 7, 43–50 (1990). translation in Soviet Math. (Iz. VUZ) 34/7 (1990), 49–57

  87. Pesenson, I.: The Bernstein inequality in representations of Lie groups. Dokl. Akad. Nauk SSSR 313(4), 803–806 (1990). translation in Soviet Math. Dokl. 42/1 (1991), 87–90

  88. Pesenson, I.: Lagrangian splines, spectral entire functions and Shannon-Whittaker theorem on manifolds. Temple Univ. Res. Rep. 95–87, 1–28 (1995)

    Google Scholar 

  89. Pesenson, I.: Sampling of Paley-Wiener functions on stratified groups. J. Fourier Anal. Appl. 4, 269–280 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  90. Pesenson, I.: Reconstruction of Paley-Wiener Functions on the Heisenberg Group, Voronezh Winter Mathematical Schools. American Mathematical Society Translations: Series 2, vol. 184, pp. 207–216. American Mathematical Society, Providence (1998)

  91. Pesenson, I.: A reconstruction formula for band limited functions in \(L_{2}(R^{d})\). Proc. Am. Math. Soc. 127(12), 3593–3600 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  92. Pesenson, I.: A sampling theorem on homogeneous manifolds. Trans. Am. Math. Soc. 352(9), 4257–4269 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  93. Pesenson, I.: Sampling of band limited vectors. J. Fourier Anal. Appl. 7(1), 93–100 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  94. Pesenson, I., Grinberg, E.: Inversion of the Spherical Radon Transform by a Poisson Type Formula, Radon Transforms and Tomography (South Hadley, MA, 2000). Contemporary Mathematics, vol. 278, pp. 137–146. American Mathematical Society, Providence (2001)

  95. Pesenson, I.: An approach to spectral problems on Riemannian manifolds. Pac. J. Math. 215(1), 183–199 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  96. Pesenson, I.: Poincaré-type inequalities and reconstruction of Paley-Wiener functions on manifolds. J. Geom. Anal. 4(1), 101–121 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  97. Pesenson, I.: Variational splines on Riemannian manifolds with applications to integral geometry. Adv. Appl. Math. 33(3), 548–572 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  98. Pesenson, I.: Band limited functions on quantum graphs. Proc. Am. Math. Soc. 133(12), 3647–3655 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  99. Pesenson, I.: Deconvolution of band limited functions on symmetric spaces. Houst. J. Math. 32(1), 183–204 (2006)

    MathSciNet  MATH  Google Scholar 

  100. Pesenson, I.: Analysis of band-limited functions on quantum graphs. Appl. Comput. Harmon. Anal. 21(2), 230–244 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  101. Pesenson, I.: Frames in Paley-Wiener Spaces on Riemannian Manifolds. Integral Geometry and Tomography. Contemporary Mathematics, vol. 405, pp. 137–153. American Mathematical Society, Providence (2006)

  102. Pesenson, I.: Plancherel-Polya-type inequalities for entire functions of exponential type in Lp(Rd). J. Math. Anal. Appl. 330(2), 1194–1206 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  103. Pesenson, I.: Bernstein-Nikolski inequality and Riesz interpolation Formula on compact homogeneous manifolds. J. Approx. Theory 150(2), 175–198 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  104. Pesenson, I.: Sampling in Paley-Wiener spaces on combinatorial graphs. Trans. Am. Math. Soc. 360(10), 5603–5627 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  105. Pesenson, I.: A Discrete Helgason-Fourier Transform for Sobolev and Besov Functions on Noncompact Symmetric Spaces. Contemporary Mathematics, vol. 464. American Mathematical Society, Providence (2008)

  106. Pesenson, I.Z.: Paley-Wiener approximations and multiscale approximations in Sobolev and Besov spaces on manifolds. J. Geom. Anal. 4(1), 101–121 (2009)

    MathSciNet  MATH  Google Scholar 

  107. Pesenson, I.Z.: Bernstein-Nikolskii and Plancherel-Polya inequalities in Lp-norms on non-compact symmetric spaces. Math. Nachr. 282(2), 253–269 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  108. Pesenson, I.Z., Zayed, A.: Paley-Wiener subspace of vectors in a Hilbert space with applications to integral transforms. J. Math. Anal. Appl. 353(2), 566–582 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  109. Pesenson, I.Z.: Variational splines and Paley-Wiener spaces on combinatorial graphs. Constr. Approx. 29(1), 1–21 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  110. Pesenson, I.Z., Pesenson, M.Z.: Approximation of Besov vectors by Paley-Wiener vectors in Hilbert spaces, Approximation Theory XIII: San Antonio 2010 (Springer Proceedings in Mathematics), by Marian Neamtu and Larry Schumaker, pp. 249–263

  111. Pesenson, I.Z., Pesenson, M.Z.: Sampling, filtering and sparse approximations on combinatorial graphs. J. Fourier Anal. Appl. 16(6), 921–942 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  112. Pesenson, I.Z., Geller, D.: Cubature formulas and discrete Fourier transform on compact manifolds in From Fourier Analysis and Number Theory to Radon Transforms and Geometry. In Memory of Leon Ehrenpreis (Developments in Mathematics 28) by H.M. Farkas, R.C. Gunning, M.I. Knopp and B.A. Taylor, Springer NY (2013)

  113. Pesenson, I.Z.: Paley-Wiener-Schwartz nearly Parseval Frames on Noncompact Symmetric Spaces, Commutative and Noncommutative Harmonic Analysis and Applications. Contemporary Mathematics, vol. 603. American Mathematical Society, Providence (2013)

  114. Pesenson, I.: Multiresolution analysis on compact Riemannian manifolds, In: Multiscale Analysis and Nonlinear Dynamics: From Genes to the Brain ( Reviews of Nonlinear Dynamics and Complexity) M. Z. Pesenson (Ed.), H.G. Schuster (Series Editor) Wiley (2013), pp. 65–83

  115. Pesenson, I.Z.: Approximations in \(L_{p}\)-norms and Besov spaces on compact manifolds. Contemp. Math. 650, 199–210 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  116. Pesenson, I.Z.: Sampling, splines and frames on compact manifolds. Int. J. Geomath. 6(1), 43–81 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  117. Pesenson, I.Z.: Splines and Wavelets on Geophysically Relevant Manifolds. Handbook of Geomathematics. Springer, Berlin (2014)

    Book  Google Scholar 

  118. Pesenson, I.Z.: Boas-Type Formulas and Sampling in Banach Spaces with Applications to Analysis on Manifolds, in New Perspectives on Approximation and Sampling Theory. Springer International Publishing, Switzerland (2014)

    MATH  Google Scholar 

  119. Pesenson, I.Z.: Sampling formulas for groups of operators in Banach spaces. Sampl. Theory Signal Image Process. 14(1), 1–16 (2015)

    MathSciNet  MATH  Google Scholar 

  120. Pesenson, I.Z.: Average sampling and frames on bounded domains. J. Complex. 31(5), 675–688 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  121. Pesenson, I.Z.: Parseval frames for subelliptic spaces on compact homogeneous manifolds. In: International Conference on Harmonic Analysis and Applications, The Graduate Center, CUNY, NY, 2015

  122. Pesenson, I.Z.: Estimates of Kolmogorov, Gelfand and linear n-widths on compact Riemannian manifolds, accepted by Proceedings of AMS

  123. Peyre, G.: Manifold models for signals and images. Comput. Vis. Image Underst. 113, 249–260 (2009)

    Article  Google Scholar 

  124. Plancherel, M., Polya, G.: Fonctions entieres et integrales de Fourier multiples. Comment. Math. Helv. 9, 224–248 (1937)

    Article  MathSciNet  MATH  Google Scholar 

  125. Plancherel, M., Polya, G.: Fonctions entieres et integrales de Fourier multiples. Comment. Math. Helv. 10, 110–163 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  126. Riesz, M.: Les fonctions conjuguees et les series de Fourier. C.R. Acad. Sci. 178, 1464–1467 (1924)

  127. Skrzypczak, L.: Wavelet frames, Sobolev embeddings and negative spectrum of Schrödinger operators on manifolds with bounded geometry. J. Fourier Anal. Appl. 14(3), 415–442 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  128. Stenzel, B.: A reconstruction theorem for Riemannian symmetric spaces of noncompact type. J. Fourier Anal. Appl. 15(6), 839–856 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  129. Sogge, S.: Fourier Integrals in Classical Analysis. Cambridge University Press, Cambridge (1993)

    Book  MATH  Google Scholar 

  130. Seeger, A., Sogge, C.D.: On the boundedness of functions of (pseudo-) differential operators on compact manifolds. Duke Math. J. 59, 709–736 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  131. Taylor, M.: Pseudodifferential Operators. Princeton University Press, Princeton (1981)

    MATH  Google Scholar 

  132. Terras, A.: Harmonic Analysis on Symmetric Spaces and Applications. Springer, Berlin (1985)

    Book  MATH  Google Scholar 

  133. Triebel, H.: Spaces of Hardy-Sobolev-Besov type on complete Riemannian manifolds. Ark. Mat. 24, 299–337 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  134. Triebel, H.: Function spaces on Lie groups. J. Lond. Math. Soc. 35, 327–338 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  135. Triebel, H.: Theory of Function Spaces II. Monographs in Mathematics, vol. 84. Birkhäuser, Basel (1992)

  136. Yang, D., Zhou, Y.: New properties of Besov and Triebel-Lizorkin spaces on RD-spaces. Manuscr. Math. 134, 59–90 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the ESI (Erwin Schrödinger Institute, University Vienna), where the joint work for this paper has begun after the workshop on Time-Frequency Analysis (Spring 2014) and CIRM (Centre international de recontres mathematique, Luminy, Marseille), where the three authors had the chance to continue their work on this manuscript during the period of Hans Feichtinger’s Morlet Chair (winter term 2014/2015). The third author was supported in part by the National Geospatial-Intelligence Agency University Research Initiative (NURI), Grant HM1582-08-1-0019.

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Authors and Affiliations

  1. Faculty of Mathematics, NuHAG, University of Vienna, Wien, Austria

    Hans G. Feichtinger

  2. Lehrstuhl A für Mathematik, RWTH Aachen, Aachen, Germany

    Hartmut Führ

  3. Department of Mathematics, Temple University, Philadelphia, PA, 19122, USA

    Isaac Z. Pesenson

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  1. Hans G. Feichtinger
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  2. Hartmut Führ
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  3. Isaac Z. Pesenson
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Correspondence to Hartmut Führ.

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Communicated by Stephan Dahlke.

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Feichtinger, H.G., Führ, H. & Pesenson, I.Z. Geometric Space–Frequency Analysis on Manifolds. J Fourier Anal Appl 22, 1294–1355 (2016). https://doi.org/10.1007/s00041-015-9457-3

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  • DOI: https://doi.org/10.1007/s00041-015-9457-3

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