Shannon sampling and function reconstruction from point values

Bulletin of the American Mathematical Society

Volumn 41, Issue 3, 2004, Pages 279-305

  Smale, Steve ^a   Zhou, Ding Xuan ^b  

^a TOYOTA TECHNOLOGICAL INSTITUTE AT CHICAGO   (United States)

^b CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Learning theory; Regularization; Rich data; Sampling theory

Indexed keywords

EID: 3042850649     PISSN: 02730979     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0273-0979-04-01025-0     Document Type: Article

References (33)

1. A. Aldroubi, Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces, Appl. Comput. Harmon. Anal. 13 (2002), 151-161. MR 20031:42045
2. A. Aldroubi and K. Gröchenig, Non-uniform sampling and reconstruction in shift-invariant spaces, SIAM Review 43 (2001), 585-620. MR 2003e:94040
3. N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337-404. MR 14:479c
4. R. F. Bass and K. Gröchenig, Random sampling of multivariate trigonometric polynomials, preprint, 2003.
5. G. Bennett, Probability inequalities for the sum of independent random variables, J. Amer. Statis. Assoc. 57 (1962), 33-45.
6. S. Bochner, Hilbert distances and positive definite functions, Ann. of Math. (2) 42 (1941), 647-656. MR 3:206d
7. T. Chan, J. Shen, and L. Vese, Variational PDE models in image processing, Notices Amer. Math. Soc. 50 (2003), 14-26. MR 2003m:94008
8. F. Cucker and S. Smale, On the mathematical foundations of learning, Bull. Amer. Math. Soc. (N.S.) 30 (2002), 1-49. MR 2003a:68118
9. F. Cucker and S. Smale, Best choices for regularization parameters in learning theory, Found. Comput. Math. 2 (2002), 413-428. MR 2003k:68089
10. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. MR 93e:42045
11. J. P. Dedieu, Newton's method and some complexity aspects of the zero-finding problem, in Foundations of Computational Mathematics, R. A. DeVore, A. Iserles, and E. Süli (eds.), Cambridge University Press, 2001, pp. 45-67. MR 2002d:65050
12. L. Devroye, L. Györfi, and G. Lugosi, A Probabilistic Theory of Pattern Recognition, Springer-Verlag, New York, 1996. MR 97d:68196
13. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Mathematics and Its Applications, 375, Kluwer, Dordrecht, 1996. MR 97k:65145
14. T. Evgeniou, M. Pontil, and T. Poggio, Regularization networks and support vector machines, Adv. Comput. Math. 13 (2000), 1-50. MR 2001f:68053
15. H. G. Feichtinger, Banach convolution algebras of Wiener type, in Functions, Series, and Operators, Vol. I, II, North-Holland, Amsterdam, 1983, pp. 509-524. MR 85j:43005
16. V. Ivanov, The Theory of Approximate Methods and Their Application to the Numerical Solution of Singular Integral Equations, Nordhoff International, 1976. MR 53:8841
17. C. McDiarmid, Concentration, in Probabilistic Methods for Algorithmic Discrete Mathematics, Springer-Verlag, Berlin, 1998, pp. 195-248. MR 2000d:60032
18. P. Niyogi, The Informational Complexity of Learning, Kluwer, Dordrecht, 1998.
19. T. Poggio and S. Smale, The mathematics of learning: dealing with data, Notices Amer. Math. Soc. 50 (2003), 537-544.
20. D. Pollard, Convergence of Stochastic Processes, Springer-Verlag, New York, 1984. MR 881:60074
21. A. N. Shiryaev, Probability, Springer-Verlag, New York, 1996. MR 97c:60003
22. S. Smale and D. X. Zhou, Estimating the approximation error in learning theory, Anal. Appl. 1 (2003), 17-41. MR 2003m:68111
23. G. Strang and G. Fix, A Fourier analysis of the finite element variational method, in Constructive Aspects of Functional Analysis, G. Geymonat (ed.), C. I. M. E., 1971, pp. 796-830.
24. W. C. Sun and X. W. Zhou, On the stability of multivariate trigonometric systems, J. Math. Anal. Appl. 235 (1999), 159-167. MR 2001k:42040
25. A. Tikhonov and V. Arsenin, Solutions of Ill-posed Problems, V. H. Winston, Washington, DC, 1977. MR 58:13604
26. M. Unser, Sampling-50 years after Shannon, Proc. IEEE 88 (2000), 569-587.
27. V. Vapnik, Statistical Learning Theory, John Wiley & Sons, New York, 1998. MR 99h:62052
28. G. Voronoi, Recherches sur les parallelodres primitives, J. Reine Angew. Math. 134 (1908), 198-287.
29. G. Wahba, Spline Models for Observational Data, SIAM, Philadelphia, PA, 1990. MR 91g:62028
30. R. Young, An Introduction to Non-Harmonic Fourier Series, Academic Press, New York-London, 1980. MR 81m:42027
31. A. I. Zayed, Advances in Shannon's Sampling Theory, CRC Press, Boca Raton, FL, 1993. MR 95f:94008
32. D. X. Zhou, The covering number in learning theory, J. Complexity 18 (2002), 739-767. MR 2003k:68095
33. D. X. Zhou, The capacity of reproducing kernel spaces in learning theory, IEEE Trans. Inform. Theory 49 (2003), 1743-1752. MR 2004c:62095