Let D be a bounded symmetric domain of tube type and Σ be the Shilov boundary of D. Denote by H2(D) and A2(D) the Hardy and Bergman spaces, respectively, of holomorphic functions on D; and let B(H2(D)) and B(A2(D)) denote the closed unit balls in these spaces. For an integer l≥0 we define the notion Rlf of the lth radial derivative of a holomorphic function f on D, and we prove the following results: Let 0<ρ<1. Denote by W the class of holomorphic functions f on D for which Rlf∈B(H2(D)) and set X=C(ρΣ). 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 Rlf∈B(A2(D)), and X=C(ρΣ). Next, let X=Lp(ρΣ) (respectively, Lp(ρD)) for 1≤p≤∞, and let W be a class of holomorphic functions f on D for which Rlf∈B(Hp(D)) (respectively, B(Ap(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. Theory75, 183-197) and K. Yu. Osipenko (1995, J. Approx. Theory82, 135-155), and initiate the study of N-widths of spaces of holomorphic functions on bounded symmetric domains.
All Science Journal Classification (ASJC) codes
- Numerical Analysis
- Applied Mathematics