BEST LINEAR APPROXIMATIONS OF FUNCTIONS ANALYTICALLY CONTINUABLE FROM A GIVEN CONTINUUM INTO A GIVEN REGION
V D Erokhin · Russian Mathematical Surveys · 1968
Let be a continuum (other than a single point) in the -plane not disconnecting the plane, a simply-connected domain containing . The class consists of those functions that are analytic in and satisfy the inequality The author proves the following theorem: Here is the -entropy of , and the -dimensional linear diameter of in the space of all functions continuous on . The norm on is For the proof a basis is constructed in the space of functions holomorphic in ; it coincides with the Faber basis if is a level curve of . A fundamental part in this construction is played by a lemma which states that the domain can be mapped conformally into a domain , where is a level curve of . In the appendix, which is written by A. L. Levin and V. M. Tikhomirov, a similar theorem is proved (under additional assumptions) for the case when is multiply-connected and may consist of several continua. CONTENTS Preface (V. M. Tikhomirov) Introduction § 1. The problem of bases. The main lemma § 2. Fundamental properties of the bases constructed § 3. Asymptotic theory of ε-entropy § 4. Estimates for n -dimensional diameters § 5. Some “extraneous” results Appendix. A. L. Levin and V. M. Tikhomirov, On a theorem of Erokhin References