The N-Widths of Spaces of Holomorphic Functions on Bounded Symmetric Domains of Tube Type

✍ Scribed by Hongming Ding; Kenneth I. Gross; Donald St.P. Richards

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 183 KB
• Volume: 104
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1999.3445

✦ Synopsis

Let D be a bounded symmetric domain of tube type and 7 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 balls in these spaces. For an integer l 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<<1. Denote by W the class of holomorphic functions f on D for which R l f # B(H 2 (D)) and set X=C( \7). 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 W is the class of holomorphic functions f for which R l f # B(A 2 (D)), and X=C( \7). Next, let X=L p ( \7) (respectively, L p (\D)) for 1 p

, and let W be a class of holomorphic functions f on D for which R l f # 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