Lorentz, G. G. Metric entropy and approximation. (English) Zbl 0158.13603 Bull. Am. Math. Soc. 72, 903-937 (1966). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 59 Documents Keywords:functional analysis PDFBibTeX XMLCite \textit{G. G. Lorentz}, Bull. Am. Math. Soc. 72, 903--937 (1966; Zbl 0158.13603) Full Text: DOI References: [1] N. I. Achieser, Theory of approximation, Translated by Charles J. Hyman, Frederick Ungar Publishing Co., New York, 1956. · Zbl 0072.28403 [2] S. Ja. Al\(^{\prime}\)per, On \?-entropy of certain classes of functions, Soviet Math. Dokl. 1 (1960), 667 – 669. · Zbl 0108.07003 [3] T. Bonnesen and W. Fenchel, Theorie der konvexen Körper, Springer-Verlag, Berlin-New York, 1974 (German). Berichtigter Reprint. · Zbl 0008.07708 [4] Ju. A. Brudnyĭ and B. D. Kotljar, The order of growth of \?-entropy on certain classes of functions, Dokl. Akad. Nauk SSSR 148 (1963), 1001 – 1004 (Russian). [5] Ju. A. Brudnyĭ and A. F. Timan, Constructive charateristics of compact sets in Banach spaces and \?-entropty, Dokl. Akad. Nauk SSSR 126 (1959), 927 – 930 (Russian). · Zbl 0088.31804 [6] G. F. Clements, Entropies of sets of functions of bounded variation, Canad. J. Math. 15 (1963), 422 – 432. · Zbl 0178.05501 [7] G. F. Clements, Entropies of several sets of real valued functions, Pacific J. Math. 13 (1963), 1085 – 1095. · Zbl 0158.05002 [8] Philip J. Davis, Interpolation and approximation, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1963. [9] V. K. Dzyadyk, On a problem of S. M. Nikol\(^{\prime}\)skiĭ in a complex region, Izv. Akad. Nauk SSSR. Ser. Mat. 23 (1959), 697 – 736 (Russian). · Zbl 0143.29604 [10] V. K. Dzjadik, Theorems on the transformation and approximation of analytic functions, Dokl. Akad. Nauk SSSR 151 (1963), 269 – 272 (Russian). V. K. Dzjadik, On the theory of approximation of continuous functions in closed regions and on a problem of S. M. Nikol\(^{\prime}\)skiĭ, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1135 – 1164 (Russian). [11] V. D. Erohin, On conformal transformations of rings and the fundamental basis of the space of functions analytic in an elementary neighbourhood of an arbitrary continuum, Dokl. Akad. Nauk SSSR 120 (1958), 689 – 692 (Russian). · Zbl 0098.08603 [12] V. D. Erohin, Asymptotic theory of the \?-entropy of analytic functions, Dokl. Akad. Nauk SSSR 120 (1958), 949 – 952 (Russian). · Zbl 0098.08701 [13] K. K. Golovkin, The \?-entropy of certain compact sets of differentiable functions in spaces with monotone norm, Dokl. Akad. Nauk SSSR 158 (1964), 261 – 263 (Russian). [14] A. Ja. Helemskiĭ and G. M. Henkin, Embeddings of compacta into ellipsoids, Vestnik Moskov. Univ. Ser. I Mat. Meh. 1963 (1963), no. 2, 3 – 12 (Russian, with English summary). [15] G. M. Henkin, Linear superpositions of continuously differentiable functions, Dokl. Akad. Nauk SSSR 157 (1964), 288 – 290 (Russian). [16] A. N. Kolmogorov, On certain asymptotic characteristics of completely bounded metric spaces, Dokl. Akad. Nauk SSSR (N.S.) 108 (1956), 385 – 388 (Russian). · Zbl 0070.11501 [17] A. N. Kolmogorov, On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR 114 (1957), 953 – 956 (Russian). · Zbl 0090.27103 [18] A. N. Kolmogorov, On linear dimensionality of topological vector spaces, Dokl. Akad. Nauk SSSR 120 (1958), 239 – 241 (Russian). · Zbl 0080.31203 [19] A. N. Kolmogorov and V. M. Tihomirov, \?-entropy and \?-capacity of sets in function spaces, Uspehi Mat. Nauk 14 (1959), no. 2 (86), 3 – 86 (Russian). · Zbl 0090.33503 [20] B. D. Kotljar, The order of growth of \?-entropy on the class of quasi-smooth functions, Uspehi Mat. Nauk 18 (1963), no. 2 (110), 135 – 138 (Russian). [21] G. G. Lorentz, Lower bounds for the degree of approximation, Trans. Amer. Math. Soc. 97 (1960), 25 – 34. · Zbl 0128.06701 [22] G. G. Lorentz, Metric entropy, widths, and superpositions of functions, Amer. Math. Monthly 69 (1962), 469 – 485. · Zbl 0124.28402 [23] G. G. Lorentz, Entropy and its applications, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 1 (1964), 97 – 103. · Zbl 0204.13201 [24] G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. · Zbl 0153.38901 [25] B. S. Mitjagin, Approximate dimension and bases in nuclear spaces, Uspehi Mat. Nauk 16 (1961), no. 4 (100), 63 – 132 (Russian). [26] O. A. Oleĭnik, Estimates of the Betti numbers of real algebraic hypersurfaces, Mat. Sbornik N.S. 28(70) (1951), 635 – 640 (Russian). [27] A. M. Olevskiĭ, On a problem of P. L. Ul\(^{\prime}\)janov, Uspehi Mat. Nauk 20 (1965), no. 2 (122), 197 – 202 (Russian). · Zbl 0136.04503 [28] A. Pełczyński, On the approximation of \?-spaces by finite dimensional spaces, Bull. Acad. Polon. Sci. Cl. III. 5 (1957), 879 – 881, LXXV (English, with Russian summary). · Zbl 0078.28703 [29] B. Penkov and Bl. Sendov, Entropy of the set of continuous functions of several variables, C. R. Acad. Bulgare Sci. 17 (1964), 335 – 337 (Russian). [30] S. Rolewicz, On spaces of holomorphic functions, Studia Math. 21 (1961/1962), 135 – 160. · Zbl 0123.30304 [31] Harold S. Shapiro, Some negative theorems of approximation theory, Michigan Math. J. 11 (1964), 211 – 217. · Zbl 0123.09202 [32] S. A. Smoljak, \?-entropy of the classes \?_{\?^{\?,\?}}(\?) and \?_{\?^{\?}}(\?) in the metric of \?\(_{2}\), Soviet Math. Dokl. 1 (1960), 192 – 196. [33] V. M. Tihomirov, Diameters of sets in functional spaces and the theory of best approximations, Russian Math. Surveys 15 (1960), no. 3, 75 – 111. · Zbl 0097.09103 [34] V. M. Tihomirov, The \?-entropy of certain classes of periodic functions., Uspehi Mat. Nauk 17 (1962), no. 6 (108), 163 – 169 (Russian). [35] A. F. Timan, Theory of approximation of functions of a real variable, Translated from the Russian by J. Berry. English translation edited and editorial preface by J. Cossar. International Series of Monographs in Pure and Applied Mathematics, Vol. 34, A Pergamon Press Book. The Macmillan Co., New York, 1963. · Zbl 0117.29001 [36] A. F. Timan, The order of growth of \?-entropy of spaces of real continuous functionals defined on a connected compactum, Uspehi Mat. Nauk 19 (1964), no. 1 (115), 173 – 177 (Russian). · Zbl 0131.32703 [37] A. G. Vituškin, Theory of the transmission and processing of information, Translated from the Russian by Ruth Feinstein; translation editor A. D. Booth, Pergamon Press, New York-Oxford-London-Paris, 1961. [38] A. G. Vituškin, Some properties of linear superpositions of smooth functions, Dokl. Akad. Nauk SSSR 156 (1964), 1003 – 1006 (Russian). [39] A. G. Vituškin, A proof of the existence of analytic functions of several variables not representable by linear superpositions of continuously differentiable functions of fewer variables, Dokl. Akad. Nauk SSSR 156 (1964), 1258 – 1261 (Russian). [40] A. C. Vosburg, Metric entropy of certain classes of Lipschitz functions, Proc. Amer. Math. Soc. 17 (1966), 665 – 669. · Zbl 0151.17801 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.