By Rita A. Hibschweiler, Thomas H. MacGregor

This quantity is concentrated on Banach areas of capabilities analytic within the open unit disc, corresponding to the classical Hardy and Bergman areas, and weighted types of those areas. different areas into account right here comprise the Bloch house, the households of Cauchy transforms and fractional Cauchy transforms, BMO, VMO, and the Fock area. many of the paintings offers with questions on features in different complicated variables.

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Kolmogorov in the 1950's. Kolmogorov was motivated by Vitushkin's work on Hilbert's 13th problem and Shannon's information theory. Note that Kolmogorov's original definition (see [12]) is slightly different from ours (he used the logarithm to base 2). Here we follow [14]. We will need some basic properties of the c-entropy. 1. 2. Let { (Ej, pj) : j = 1,2, ... , k} be a family of totally bounded metric spaces. e. E = E1 X E2 X ... X Ek , P((X1' X2,···, Xk), (Yl, Y2,"" Yk)) = maxjpj(xj, Yj) . OLEG EROSHKIN 26 Then :Hg(E) :::; L :Hg(Ej) j PROOF.

2005. P. 6128, Succ. CENTRE-VILLE, MONTREAL, QUEBEC H3C 3J7, CANADA E-mail address: fournier~dms. wnontreal. cl Contemporary Mathematics Volume 454, 2008 Approximating z in Hardy and Bergman Norms Zdeilka Guadarrama and Dmitry Khavinson ABSTRACT. We consider the problem of finding the best analytic approximation in Smirnov and Bergman norm to general monomials of the type znzm. We show that in the case of approximation to z in the annulus (and the disk) the best approximation is the same for all values of p.

17] M. J. Martin, D. Vukotic, Isometries of the Bloch space among the composition operators, Bull. London Math. Soc. (to appear). [18] W. Rudin, FUnction Theory in Polydiscs, W. A. , New York, 1969. [19] J. H. Shapiro, Composition operators and classical function theory, Springer, New York, 1993. [20] R. M. Timoney, Bloch functions in several complex variables, I, Bull. London Math. Soc. 12 (1980),241-267. [21] R. M. Timoney, Bloch functions in several complex variables, II, J. Reine Angew. Math.