On a joint approximation question in $$H^p$$ H p spaces
- PDF / 246,557 Bytes
- 4 Pages / 439.37 x 666.142 pts Page_size
- 27 Downloads / 209 Views
Archiv der Mathematik
On a joint approximation question in H p spaces Arthur A. Danielyan Abstract. We consider Mergelyan sets and Farrell sets for H p (1 ≤ p < ∞) spaces in the unit disc for both the weak topology and the norm topology, and give a short proof of a theorem of P´erez-Gonz´ alez which answers a question proposed by Rubel and Stray (J Approx Theory 37:44–50, 1983). Mathematics Subject Classification. Primary 30E10, 46E15. Keywords. Mergelyan set, Farrell set, H p spaces, Joint approximation by polynomials.
The concepts of Mergelyan sets and Farrell sets have been introduced by Rubel [9] for his joint approximation problem and studied by many authors (for some early references, see, e.g., [2–7,9–13], and references therein). In this paper, we consider Mergelyan sets and Farrell sets for H p (1 ≤ p < ∞) spaces in the unit disc and give a short proof of a result of P´erez-Gonz´alez [7] which answers a question proposed by Rubel and Stray [11]. For the H 1 space, our method provides a result which has been assumed to be proved but in fact there is a gap in the proof. Mergelyan sets and Farrell sets have been considered for H p spaces both for the weak topology τp and the · p -norm topology (cf. [7,11]). Following the notations of [7,11], denote by D the open unit disc and by T the unit circle. Let m be the normalized Lebesgue measure on T . Consider a space A of analytic functions defined on D. Let τ be a given topology on A such that the set P of all polynomials is τ -dense in A. Mergelyan sets and Farrell sets for the pair (A, τ ) are defined as follows. Definition 1. A relatively closed subset F ⊂ D is a Mergelyan set for (A, τ ) if for any f ∈ A with a uniformly continuous restriction f |F to F , there exists a sequence {pn } ⊂ P such that (1) pn → f in the topology τ, (2) pn → f uniformly on F .
A. A. Danielyan
Arch. Math.
As usual, we denote by gF the sup norm of a (bounded) function g on a set F. Definition 2. A relatively closed subset F ⊂ D is a Farrell set for (A, τ ) if for any f ∈ A with a bounded restriction f |F to F , there exists a sequence {pn } ⊂ P such that (1) pn → f in the topology τ, (2) pn → f pointwise on F and limn→∞ pn F = f F . The theorem of P´erez-Gonz´alez [7] states that Mergelyan sets and Farrell sets for H p (1 ≤ p < ∞) spaces remain the same if either the weak topology τp or the · p -norm topology is considered. This result extends a theorem of Rubel and Stray and answers to a question proposed by them [11]. Theorem A (P´erez-Gonz´alez). Let 1 ≤ p < ∞, and let F be a relatively closed subset of D. The following seven statements are equivalent to each other. (i) F is a Farrell set for (H p , τp ). (ii) F is a Mergelyan set for (H p , τp ). (iii) There is a set E ⊂ F¯ ∩T with m(E) = 0 such that if ζ belongs to F¯ ∩T \E, then a sequence {ζn } exists in F converging nontangentially to ζ. (iv) If g is uniformly continuous on F and f is any function of H p with the restriction f |F bounded, then there exists a sequence {pn } in P satisfying: (α) pn → f weakly, a
Data Loading...
