Best approximation-preserving operators over Hardy space

Let T-n be the linear Hadamard convolution operator acting over Hardy space H-q, 1 <= q <= infinity. We call T-n a best approximation-preserving operator (BAP operator) if T-n(e(n)) = e(n), where e(n)(z) := z(n), and if parallel to T-n(f) parallel to(q) <= E-n(f)(q) for all f is an element of H-q, where E-n(f)(q) is the best approximation by algebraic polynomials of degree a most n - 1 in H-q space. We give necessary and sufficient conditions for T-n to be a BAP operator over H-infinity. We apply this result to establish an exact lower bound for the best approximation of bounded holomorphic functions. In particular, we show that the Landau-type inequality vertical bar(f) over capn vertical bar + c vertical bar(f) over capN vertical bar <= E-n (f)(infinity), where c > 0 and n < N, holds for every f is an element of H-infinity iff c <= 1/2 and N >= 2n + 1.