Shannon sampling and function reconstruction from point values

Smale, Steve; Zhou, Ding-Xuan

doi:10.1090/S0273-0979-04-01025-0

Shannon sampling and function reconstruction from point values
HTML articles powered by AMS MathViewer

by Steve Smale and Ding-Xuan Zhou PDF

Bull. Amer. Math. Soc. 41 (2004), 279-305 Request permission

References

Akram Aldroubi, Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces, Appl. Comput. Harmon. Anal. 13 (2002), no. 2, 151–161. MR 1942749, DOI 10.1016/S1063-5203(02)00503-1
Akram Aldroubi and Karlheinz Gröchenig, Nonuniform sampling and reconstruction in shift-invariant spaces, SIAM Rev. 43 (2001), no. 4, 585–620. MR 1882684, DOI 10.1137/S0036144501386986
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14

57

Sergio Sispanov, Generalización del teorema de Laguerre, Bol. Mat. 12 (1939), 113–117 (Spanish). MR 3
Tony F. Chan, Jianhong Shen, and Luminita Vese, Variational PDE models in image processing, Notices Amer. Math. Soc. 50 (2003), no. 1, 14–26. MR 1948832
Felipe Cucker and Steve Smale, On the mathematical foundations of learning, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 1–49. MR 1864085, DOI 10.1090/S0273-0979-01-00923-5
Felipe Cucker and Steve Smale, Best choices for regularization parameters in learning theory: on the bias-variance problem, Found. Comput. Math. 2 (2002), no. 4, 413–428. MR 1930945, DOI 10.1007/s102080010030
Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107, DOI 10.1137/1.9781611970104
Jean-Pierre Dedieu, Newton’s method and some complexity aspects of the zero-finding problem, Foundations of computational mathematics (Oxford, 1999) London Math. Soc. Lecture Note Ser., vol. 284, Cambridge Univ. Press, Cambridge, 2001, pp. 45–67. MR 1836614
Luc Devroye, László Györfi, and Gábor Lugosi, A probabilistic theory of pattern recognition, Applications of Mathematics (New York), vol. 31, Springer-Verlag, New York, 1996. MR 1383093, DOI 10.1007/978-1-4612-0711-5
Heinz W. Engl, Martin Hanke, and Andreas Neubauer, Regularization of inverse problems, Mathematics and its Applications, vol. 375, Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1408680, DOI 10.1007/978-94-009-1740-8
Theodoros Evgeniou, Massimiliano Pontil, and Tomaso Poggio, Regularization networks and support vector machines, Adv. Comput. Math. 13 (2000), no. 1, 1–50. MR 1759187, DOI 10.1023/A:1018946025316
H. G. Feichtinger, Banach convolution algebras of Wiener type, Functions, series, operators, Vol. I, II (Budapest, 1980) Colloq. Math. Soc. János Bolyai, vol. 35, North-Holland, Amsterdam, 1983, pp. 509–524. MR 751019
V. V. Ivanov, The theory of approximate methods and their application to the numerical solution of singular integral equations, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, No. 2, Noordhoff International Publishing, Leyden, 1976. Translated from the Russian by A. Ideh; Edited by R. S. Anderssen and D. Elliott. MR 0405045
Colin McDiarmid, Concentration, Probabilistic methods for algorithmic discrete mathematics, Algorithms Combin., vol. 16, Springer, Berlin, 1998, pp. 195–248. MR 1678578, DOI 10.1007/978-3-662-12788-9_{6}

50

David Pollard, Convergence of stochastic processes, Springer Series in Statistics, Springer-Verlag, New York, 1984. MR 762984, DOI 10.1007/978-1-4612-5254-2
A. N. Shiryaev, Probability, 2nd ed., Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1996. Translated from the first (1980) Russian edition by R. P. Boas. MR 1368405, DOI 10.1007/978-1-4757-2539-1
Steve Smale and Ding-Xuan Zhou, Estimating the approximation error in learning theory, Anal. Appl. (Singap.) 1 (2003), no. 1, 17–41. MR 1959283, DOI 10.1142/S0219530503000089

Constructive Aspects of Functional Analysis

Wenchang Sun and Xingwei Zhou, On the stability of multivariate trigonometric systems, J. Math. Anal. Appl. 235 (1999), no. 1, 159–167. MR 1758675, DOI 10.1006/jmaa.1999.6386
Andrey N. Tikhonov and Vasiliy Y. Arsenin, Solutions of ill-posed problems, Scripta Series in Mathematics, V. H. Winston & Sons, Washington, D.C.; John Wiley & Sons, New York-Toronto, Ont.-London, 1977. Translated from the Russian; Preface by translation editor Fritz John. MR 0455365

88

Vladimir N. Vapnik, Statistical learning theory, Adaptive and Learning Systems for Signal Processing, Communications, and Control, John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. MR 1641250

134

Grace Wahba, Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 59, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. MR 1045442, DOI 10.1137/1.9781611970128
Robert M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 591684
Ahmed I. Zayed, Advances in Shannon’s sampling theory, CRC Press, Boca Raton, FL, 1993. MR 1270907
Ding-Xuan Zhou, The covering number in learning theory, J. Complexity 18 (2002), no. 3, 739–767. MR 1928805, DOI 10.1006/jcom.2002.0635
Ding-Xuan Zhou, Capacity of reproducing kernel spaces in learning theory, IEEE Trans. Inform. Theory 49 (2003), no. 7, 1743–1752. MR 1985575, DOI 10.1109/TIT.2003.813564