Shannon sampling and function reconstruction from point values
Authors:
Steve Smale and Ding-Xuan Zhou
Journal:
Bull. Amer. Math. Soc. 41 (2004), 279-305
MSC (2000):
Primary 68T05, 94A20; Secondary 68P05, 42B10
DOI:
https://doi.org/10.1090/S0273-0979-04-01025-0
Published electronically:
April 13, 2004
MathSciNet review:
2058288
Full-text PDF
References | Similar Articles | Additional Information
- [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 2003i: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.) 39 (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 86i: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 56:13604
- [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
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Additional Information
Steve Smale
Affiliation:
Toyota Technological Institute at Chicago, 1427 East 60th Street, Chicago, Illinois 60637
Email:
smale@math.berkeley.edu
Ding-Xuan Zhou
Affiliation:
Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China
Email:
mazhou@math.cityu.edu.hk
DOI:
https://doi.org/10.1090/S0273-0979-04-01025-0
Keywords:
Learning theory,
sampling theory,
regularization,
rich data
Received by editor(s):
October 28, 2003
Published electronically:
April 13, 2004
Additional Notes:
The first author is partially supported by NSF grant 0325113.
The second author is supported partially by the Research Grants Council of Hong Kong [Project No. CityU 103303] and by City University of Hong Kong [Project No. 7001442].
Dedicated:
Dedicated to the memory of René Thom
Article copyright:
© Copyright 2004
American Mathematical Society
