Paley-wiener approximations and multiscale approximations in Sobolev and besov spaces on manifolds

Journal of Geometric Analysis

Volumn 19, Issue 2, 2009, Pages 390-419

  Pesenson, Isaac ^a  

^a Temple University   (United States)

Author keywords

Bernstein inequality; Discrete spectral Fourier transform; Multiscale approximations; Paley Wiener functions on Riemannian manifolds; Quadrature formulas on manifolds; Sampling of non bandlimited functions; Schrodinger semigroup; Sobolev and Besov spaces

Indexed keywords

EID: 84867944400     PISSN: 10506926     EISSN: None     Source Type: Journal    
DOI: 10.1007/s12220-008-9059-2     Document Type: Article

References (40)

1. Akhiezer, J.: Theory of Approximation. Ungar, New York (1956)
2. Bergh, J., Lofstrom, J.: Interpolation Spaces. Springer, Berlin (1976)
3. Bertalmío, M., Mémoli, F., Cheng, L.-T., Sapiro, G., Osher, S.: Variational problems and partial differential equations on implicit surfaces: bye bye triangulated surfaces? In: Geometric Level Set Methods in Imaging, Vision, and Graphics, pp. 381-397. Springer, New York (2003)
4. Birman, M., Solomyak, M.: Spectral Theory of Selfadjoint Operators in Hilbert Space. D. Reidel, Dordrecht (1987)
5. G. Brown F. Dai 2005 Approximation of smooth functions on compact two-point homogeneous spaces J. Funct. Anal. 220 2 401 423 1076.41012 10.1016/j.jfa.2004.10.005 2119285
6. Butzer, P., Berens, H.: Semi-groups of Operators and Approximation. Springer, Berlin (1967)
7. J. Cheeger M. Gromov M. Taylor 1982 Finite propagation speed, kernel estimates for functions of the Laplace operator and the geometry of complete Riemannian manifolds J. Differ. Geom. 17 15 53 0493.53035 658471
8. R. Duffin A. Schaeffer 1952 A class of nonharmonic Fourier series Trans. AMS 72 341 366 0049.32401 10.2307/1990760 47179
9. M. Geisler 1988 Besov spaces on compact Lie groups Math. Nachr. 139 193 205 0703.43007 10.1002/mana.19881390118 978119
10. K. Hesse H.N. Mhaskar I.H. Sloan 2007 Quadrature in Besov spaces on the Euclidean sphere J. Complex. 23 4-6 528 552 1134.41013 10.1016/j.jco.2006.10.004 2372012
11. Kimmel, R.: Numerical Geometry of Images. Theory, Algorithms, and Applications. Springer, New York (2004)
12. Krein, S., Pesenson, I.: Interpolation Spaces and Approximation on Lie Groups. Voronezh State University, Voronezh (1990)
13. Krein, S., Petunin, Y., Semenov, E.: Interpolation of Linear Operators. Translations of Mathematical Monographs, vol. 54. AMS, Providence (1982)
14. H.N. Mhaskar 2006 Weighted quadrature formulas and approximation by zonal function networks on the sphere J. Complex. 22 3 348 370 1103.65028 10.1016/j.jco.2005.10.003 2229898
15. H. Mhaskar F. Narcowich J. Ward 1999 Approximation properties of zonal function networks using scattered data on the sphere. Radial basis functions and their applications Adv. Comput. Math. 11 2-3 121 137 0939.41012 10.1023/A:1018967708053 1731693
16. F. Mémoli G. Sapiro S. Osher 2004 Solving variational problems and partial differential equations mapping into general target manifolds J. Comput. Phys. 195 1 263 292 1044.65060 10.1016/j.jcp.2003.10.007 2043136
17. Minakshisundaram Pleijel 1949 Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds Can. J. Math. 1 242 256 0041.42701 31145
18. Nikolskii, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems. Springer, Berlin (1975)
19. I. Pesenson 1979 Interpolation spaces on Lie groups Dokl. Akad. Nauk SSSR 246 6 1298 1303 546322 (in Russian)
20. I. Pesenson 1983 Nikolskii-Besov spaces connected with representations of Lie groups Dokl. Akad. Nauk SSSR 273 1 45 49 722886 (in Russian)
21. I. Pesenson 1988 The best approximation in a representation space of a Lie group Dokl. Acad. Nauk USSR 302 5 1055 1059 Engl. transl. in Soviet Math. Dokl. 38(2), 384-388 (1989)
22. I. Pesenson 1990 The Bernstein inequality in the space of representation of Lie group Dokl. Acad. Nauk SSSR 313 86 90 English transl. in Soviet Math. Dokl. 42 (1991)
23. I. Pesenson 2000 A sampling theorem on homogeneous manifolds Trans. AMS 352 9 4257 4270 0976.43004 10.1090/S0002-9947-00-02592-7 1707201
24. I. Pesenson 2001 Sampling of band limited vectors J. Fourier Anal. Appl. 7 1 93 100 0998.42025 10.1007/s00041-001-0007-9 1812998
25. I. Pesenson 2004 An approach to spectral problems on Riemannian manifolds Pac. J. Math. 215 1 183 199 1069.58018 2060498
26. I. Pesenson 2004 Poincaré-type inequalities and reconstruction of Paley-Wiener functions on manifolds J. Geom. Anal. 4 1 101 121 2030577
27. I. Pesenson 2006 Deconvolution of band limited functions on symmetric spaces Houston J. Math. 32 1 183 204 1097.43007 2202361
28. Pesenson, I.: Frames in Paley-Wiener spaces on Riemannian manifolds. In: Integral Geometry and Tomography. Contemporary Mathematics, vol. 405, pp. 137-153. AMS, Providence (2006)
29. I. Pesenson 2008 Bernstein-Nikolskii inequalities and Riesz interpolation formula on compact homogeneous manifolds J. Approx. Theory 150 2 175 198 1144.43006 10.1016/j.jat.2007.06.001 2388855
30. Pesenson, I.: A discrete Helgason-Fourier transform for Sobolev and Besov functions on noncompact symmetric spaces. In: Contemporary Mathematics, vol. 464, pp. 231-249. AMS, Providence (2008)
31. Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2006)
32. Schmeisser, H., Sickel, W.: Sampling theory and function spaces. In: Applied Mathematics Reviews, vol. 1, pp. 205-284. World Scientific, River Edge (2000)
33. L. Skrzypczak 1991 Function spaces of Sobolev type on Riemannian symmetric manifolds Forum Math. 3 4 339 353 0738.46018 1115951 10.1515/form.1991.3.339
34. L. Skrzypczak 1996 Some equivalent norms in Sobolev-Besov spaces on symmetric Riemannian manifolds J. Lond. Math. Soc. (2) 53 3 569 581 1396720
35. L. Skrzypczak 1999 Heat and harmonic extensions for function spaces of Hardy-Sobolev-Besov type on symmetric spaces and Lie groups J. Approx. Theory 96 1 149 170 0918.43007 10.1006/jath.1998.3223 1659396
36. E. Stein 1957 Functions of exponential type Ann. Math. (2) 65 582 592 10.2307/1970066
37. Titchmarsh, E.: Theory of Fourier Integrals. Oxford University Press, Oxford (1948)
38. H. Triebel 1986 Spaces of Hardy-Sobolev-Besov type on complete Riemannian manifolds Ark. Mat. 24 299 337 0664.46026 10.1007/BF02384402 884191
39. H. Triebel 1987 Function spaces on Lie groups J. Lond. Math. Soc. 35 327 338 0587.46036 10.1112/jlms/s2-35.2.327 881521
40. Triebel, H.: Theory of Function Spaces II. Monographs in Mathematics, vol. 84. Birkhäuser, Basel (1992)