The N-widths of spaces of holomorphic functions on bounded symmetric domains of tube type

Let D be a bounded symmetric domain of tube type and Sigma be the Shilov bo undary of D. Denote by H-2(D) and A(2)(D) the Hardy and Bergman spaces, res pectively. of holomorphic functions on D; and let B(H-2(D)) and B(A(2)(D)) denote the closed unit bulls in these spaces. For an integer l greater than or equal to 0 we define the notion R(l)f of the lth radial derivative of a holomorphic Function f on D, and we prove the following results: Let 0 < r ho < 1. Denote by W the class of holomorphic functions f on D for which R(l )f is an element of B(H-2(D)) and set X = C(rho Sigma). Then we show that t he linear and Gelfand N-widths of W in X coincide, and we compute the exact value. We do the same for the case in which I I I is the class of holomorp hic functions f for which R(l)f is an element of B(A(2)(D)). and X = C(rho Sigma). Next, let X = L-p(rho Sigma) (respectively, L-p(rho D)) for l less than or equal to p less than or equal to infinity, and let W be a class of holomorphic functions f on D for which R(l)f is an element of B(H-p(D)) (re spectively, B(A(p)(D))). We show that the Kolmogorov, linear, Gelfand. and Bernstein N-widths all coincide, we calculate the exact value, and we ident ify optimal subspaces or optimal linear operators. These results extend wor k of Yu. A. Farkov (1993, J. Approx. Theory 75, 183-197) and K. Yu. Osipenk o (1995, J. Approx. Theory 82, 135-155), and initiate the study of N-widths of spaces of holomorphic functions on bounded symmetric domains. (C) 2000 Academic Press.