entropy of classes of holomorphic functions - Springer Link

Mathematical Notes, Vol. 68, No. ~, ~000

On the ~-Entropy

of Classes of Holomorphic

Functions

Yu. A. F a r k o v

UDC

ABSTRACT. Suppose t h a t BaR is a ball of radius R in C d and a is the standard measure on the unit sphere in C d. For R > 1, 1 < p < oo, and for the natural numbers l , d , by / ~ R ( l , p , d ) we denote the class of functions f holomorphic in BaR and such t h a t in the homogeneous polynomial expansion of t h e first l summands the zero and radial derivatives of order I belong to t h e closed unit ball of the H a r d y space HP(BaR). In this paper an asymptotic formula for the e-entropy of the class /~R(I, p, d) in the spaces LP(a), 1 < p < oo, and C(B'I~) is obtained. KEY WORDS: e-entropy, holomorphic function, Hardy space, n-width, Jordan measure.

1. S t a t e m e n t o f t h e r e s u l t a n d b a c k g r o u n d o f t h e p r o b l e m

Suppose that A is a precompact set in the metric space X and e > 0 is fixed. Let the minimum number of points in the e-net for A in X . The quantity

Ne(A ; X) denote

9/e(A; X ) : = l o g N , ( A ; X ) ,

e-entropy of the set A in X (see, for example, [1]). is its boundary, B d := R B a is a ball of radius R, is its closure, and a is the standard measure on S a a(Sa) = 1 (see [2]). The Hardy spaces HP(Bd),

where the logarithm is taken to the base 2, is called the Suppose that B a is the unit ball in C a , S a := OB d

1 < p 1, 1 < p < oo, l, d, N 9 N, 1 < N . Then the following estimates hold:

liPMn)llx, _