On Best Uniform Approximation by Bounded Analytic Functions

This paper gives a counterexample to the conjecture that the continuity of the conjugate f of an f e C(T) implies the continuity of the best uniform approximation g e H??(T) of f. It also states two conditions which imply the continuity of g. Let L??(T) the space of bounded measurable functions on the unit circle T, H??(T) the subalgebra of L??(T) consisting of nontangential limits of bounded analytic functions in the unit disk and write jlfjIjo for the (essential supremum) norm of f E LI (T). Also, let C(T) be the space of all continuous functions on T. It is known that any f E LI (T) has at least one best approximation g E HI (T), in the sense that d =jlf -gllo inf ||f -h||o hEHand that, by duality d = sup{ f( ) F(O) : FH (T), F(O) =O1jF1 ? 1} where HP (T) (0 < p < x0) is the Hardy space of all nontangential limits of functions F analytic in the unit disc such that 27r .0dO JIFIIP = sup ] F(re') < +oo. Moreover, if f is continuous, then the best approximation g of f is unique and there is at least one F, for which the supremum (*) is attained. Also f, g and any of those maximizing F's are connected by (1) f (?) g() = jjf-911 gi0 F(O) a.e. (dO) f () -g(o = lf gIoojF(O)j which implies If(O) g(O)I = jjf g9l11 = d a.e. (dO). We need the following result (see [1 or 2]): THEOREM 1 (CARLESON-JACOBS). Iff CE C(T), g E H00(T), F E H1(T) are connected by (1), then (a) F e HP(T), for all p< +00, Received by the editors May 19, 1987. 1980 Mathematics Subject Classification (1985 Remvsion). Primary 30D55; Secondary 42A50. Partially supported by NSF Grant # DMS85-03780. (?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page