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

JOURNAL OF APPROXIMATION THEORY

卷:104

Article

Ding, HM ; Gross, KI ; Richards, DS

关键词: Jordan algebra;    symmetric cone;    bounded symmetric domain;    Shilov boundary;    Bergman space;    Hardy space;    Sobolev space;    N-width;    radial derivative;    reproducing kernel;    spherical function;   

DOI  :  10.1006/jath.1999.3445

来源: Elsevier

【 摘 要 】

Let D be a bounded symmetric domain of tube type and Sigma be the Shilov boundary of D. Denote by H-2(D) and A(2)(D) the Hardy and Bergman spaces, respectively. 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 < rho < 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 the 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 holomorphic 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)) (respectively, B(A(p)(D))). We show that the Kolmogorov, linear, Gelfand. and Bernstein N-widths all coincide, we calculate the exact value, and we identify optimal subspaces or optimal linear operators. These results extend work of Yu. A. Farkov (1993, J. Approx. Theory 75, 183-197) and K. Yu. Osipenko (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.

【 授权许可】

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