Best Approximation-Preserving Operators over Hardy Space

Abstract

Let Tn be the linear Hadamard convolution operator acting over Hardy space H q , 1 ≤ q ≤ ∞. We call Tn a best approximation-preserving operator (BAP operator) if Tn(en) = en, where en(z) := z n , and if Tn(f ) q ≤ En(f )q for all f ∈ H q , where En(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 Tn to be a BAP operator over H ∞ . 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 fn + c f N ≤ En(f )∞, where c > 0 and n < N , holds for every f ∈ H ∞ iff c ≤ 1 2 and N ≥ 2n + 1.