Abstract
We obtain asymptotic values for the integraln-widths of Sobolev classesW r2 equipped with Gaussian measure in theL q -norm.
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Yu. A. Brudnyi, A. F. Timan (1959):Constructive characteristics of the compact sets in Banach spaces and ε-entropy. Dokl. Akad. Nauk SSSR,277: 1048–1072.
A. P. Buslaev (1988):On best approximation of the random functions and functionals Third Saratov Winter School,2: 14–17.
A. Yu. Garnaev, E. D. Gluskin (1984):About widths of Euclidean ball. Dokl. Akad. Nauk SSSR,277: 1048–1052.
E. D. Gluskin (1983):Norms of random matrices and diameters of finite dimensional sets. Math. Sb.,120: 180–189.
S. Heinrich (1990):Probabilistic complexity analysis for linear problems in bounded domains J. Comlexity,6: 231–255.
R. S. Ismagilov (1974):Diameters of sets in normed spaces and the approximation of functions by trigonometric polynomials. Uspehi Mat. Nauk,29: 161–178.
B. S. Kashin (1977):Diameters of some finite-dimensional sets and classes of smooth functions. Math. USSR-Izv.,11: 317–333.
H. H. Kuo (1975): Gaussian Measures in Banach Spaces. Lecture Notes in Mathematics, Vol. 463, Berlin: Springer-Verlag.
F. M. Larkin (1970):Gaussian measures in Hilbert spaces with reproducing kernel functions. Math. Comput.24: 911–921.
V. E. Maiorov (1992):Integral and Kolmogorov widths of classes of differentiable functions. Soviet Math. Dokl.,43: 866–870.
V. E. Maiorov (1992):Widths of spaces endowed a Gaussian measure. Russian Akad. Sci.,45: 305–309.
V. E. Maiorov (1993):Average n-widths of the Wiener space in the L ∞ -norm. J. Complexity,9: 222–230.
V. E. Maiorov (1993):Kolmogorov (n, δ)-widths of the spaces of the smooth functions. Math. Sb.,184: 49–70.
V. E. Maiorov (to appear):About n-widths of Wiener Space in L q -norm. J. Complexity.
Yu. Makovoz (1988):A simple proof of an inequality in the theory of n-widths. Constructive Theory of Functions, Vol. 87. Sofia.
P. Mathe (1991):Random approximation of Sobolev embeddings J. Complexity,7: 261–281.
P. Mathe (1993):A minimax principle for the optimal error of Monte Carlo methods. Constr. Approx.,9: 23–29.
A. Papageorgiou, G. W. Wasilkowski (1990):Average complexity of multivariate problems. J. Complexity,6: 1–23.
A. Pietsch (1980): Operator Ideals. Amsterdam: North-Holland.
A. Pinkus (1985):n-Widths in Approximation Theory. Berlin: Springer-Verlag.
G. Pisier (1989) The Volume of Convex Bodies and Banach Space Geometry. Cambridge: Cambridge University Press.
G. E. Shilov, Fan Dyk Tin (1967): Integral, Measure and Derivative on Linear Spaces. Moscow: Nauka.
M. I. Stesin (1975):Alexandrov diameters of finite-dimensional sets and classes of smooth functions. Dokl. Akad. Nauk SSSR,220: 1278–1281.
Sun Yongsheng, Wang Chengyong (preprint): μ-Average n-widths on the Wiener space.
V. M. Tikhomirov (1976): Some Problems in Approximation Theory (in Russian). Moscow: Moscow State University.
J. F. Traub, H. Wozniakowski (1980): A General Theory of Optimal Algorithm. New York: Academic Press.
J. F. Traub, G. W. Wasilkowski, H. Wozniakowski (1988): Information-Based Complexity. New York: Academic Press.
S. M. Voronin, N. Temirgaliev (1984):On some applications of Banach measure. Izv. Akad. Nauk Kaz. SSR, ser. ph.-math.,5: 8–11.
A. Zygmund (1968): Trigonometric Series. Cambridge: Cambridge University Press.
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Communicated by Vladimir Telmyakov.
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Mairrov, V.E. Widths and distributions of values of the approximation functional on the sobolev spaces with measure. Constr. Approx 12, 443–462 (1996). https://doi.org/10.1007/BF02437502
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DOI: https://doi.org/10.1007/BF02437502