A lethargy result for real analytic functions

We prove that, if ( C [ a , b ] , { A n } ) is an approximation scheme and { A n } satisfies the de La Vallée Poussin Theorem, there are examples of real-valued continuous functions on [ a , b ] , analytic on ( a , b ] , which are “poorly approximable” by the elements of { A n } . This illustrates the thesis that the smoothness conditions guaranteeing that a function is “well approximable” must be “global”. The failure of smoothness at endpoints may result in an arbitrarily slow rate of approximation.