On Functions in the Ball Algebra

Abstract

ABSTRACT. We show that there exists a function in a ball algebra such that almost every slice function has a series of Taylor coefficients divergent with every power p < 2.In §7.2 of [3] W. Rudin gives some examples of boundary behavior of holomorphic functions in complex balls of dimension 2 and 3. It was observed in [4, Remark 1.10], that using Theorem 1.2 of [4] such examples can be constructed in arbitrary dimension. In the present note we further pursue this idea. It is well known that in one variable there exists a function f(z) = Yln°=o anzn analytic for \z\ < 1 such that £ |an|p = oo for all p < 2. Our theorem generalizes this fact to functions of several variables.In this note, B will always denote the unit ball in the complex d-dimensional space Cd, S will stand for the unit sphere and a for the rotation invariant probability measure on S. A(B) will denote the ball algebra of all functions analytic in B and continuous in B. For 0 < p < oo, ||/||p denotes (/s |/(c)|pder(<))l/p. If / is a holomorphic function in B then it has a unique homogeneous expansion as / = 2^£Lo fnt fn i8 an analytic polynomial homogeneous of degree n.It was shown in [4, Theorem 1.2], that there exist polynomials (pn) homogeneous of degree nonfl such that (*) l|Pr,|| = 1 and ||Pn||oo<X.

Radon Inversion Problem for Holomorphic Functions on Circular, Strictly Convex Domains

“…Moreover, if p ≤ 1, then S p (g) ∈ C( ) and for p ∈ (1, 2], function S p (g) is square-integrable on all circles z∂D, z ∈ ∂ . This is a similar result to the ones obtained for functions in the ball algebra in [12] and for inner functions in [2] and [3]. In some sense these two approaches give us flexibility and diversity in solving Radon inversion problem and might be useful in view of constructing holomorphic functions with prescribed boundary behaviour.…”

“…Observe that by (12), the polynomial F satisfies condition (c2). For each z ∈ ∂ one may choose an index…”

Precise inclusion relations among Bergman–Besov and Bloch–Lipschitz spaces and on the unit ball of

“…Proof (i) This is just [, Proposition] written in different words. There is a constructive proof for in [, Theorem].…”