THE N-WIDTHS OF HARDY–SOBOLEV AND BERGMAN–SOBOLEV SPACES OF COMPLEX VARIABLES.

Let D be a bounded symmetric domain and Σ be the Shilov boundary of D. For R≥1, l∈ℤ+ and 1≤p≤∞, let DR=RD, Hp,l(DR) and Ap,l(DR) be the Hardy–Sobolev and Bergman–Sobolev spaces on DR, respectively. In this paper we show that the Kolmogorov, linear, Gel'fand, and Bernstein N-widths of Hp,l(DR) in Lp(Σ) all coincide, calculate the exact value, and identify optimal subspaces or optimal linear operators. We also do the same for N-widths of Ap,l(DR) in Lp(D). Moreover, we obtain new asymptotic estimates for the linear and Gel'fand N-widths of $H_{p,l}(B_R^n)$ and $A_{p,l}(B_R^n)$ in Lq(Sn) and Lq(Bn), where R>1, l∈ℤ+, 2≤p≤q≤∞, $B^n=\{z\in {\mathbb C}^n{:}\ |z|=\big(\sum_{j=1}^n|z_j|^2\big)^{1/2} < 1\}$, $S^n=\{z\in {\mathbb C}^n{:}\ |z|=1\}$ are the unit ball and unit sphere in ${\mathbb C}^n$, respectively, and $B_R^n=RB^n$. Furthermore, we obtain asymptotic estimates for the linear and Gel'fand N-widths of Hp,l(DR) in Lq(Σ), where R>1, l∈ℤ+ and 2≤p≤q≤∞. [ABSTRACT FROM AUTHOR]