CORRECTION TO “ON THE PROBLEM OF BEST CONVERGENCE RATES OF DENSITY ESTIMATES

The assertion of Th.1 in1]should be replaced bylimsup n→∞ a_nn~(k/(2k m)=∞.(A)Since the proof of Th.1 in1]is somewhat in error,we give here a sketch ofproof of(A).Choose f∈C_ka with f(x)≥a>0 for ‖x‖≤ε>0,and define h_δ(x)=f(x) e_(kδ)(x),where e_(kδ)(x),as well as d and C_(kα)~(n)(d) to appear in the following,are thesame as in1].Choose ■>0 so that h_δ∈C_(kα) for δ∈(0,■).For each δ in(0,■),thereexists an integer n such that h_δ∈C_(kα)~(n)(d).Hence an integer N can be found such that