Recently people proved that every f∈C[0, 1] can be uniformly approximated by polynomial sequences {P_n}, {Q_n} such for any x∈[0,1] and n=1,2,…that Q_n(x)<Q_(n+1)(x)<f(x)<P_(n+1)(x)<P_n(x). For example...Recently people proved that every f∈C[0, 1] can be uniformly approximated by polynomial sequences {P_n}, {Q_n} such for any x∈[0,1] and n=1,2,…that Q_n(x)<Q_(n+1)(x)<f(x)<P_(n+1)(x)<P_n(x). For example, Xie and Zhou showed that one can construct such monotone polynomial sequences which do achieve the best uniform approximation rate for a continuous func- tion. Actually they obtained a result as ‖P_n(x)-Q_n(x)‖≤42E_n (f), (1) which essentially improved a conclusion in Gal and Szabados. The present paper, by optimal procedure, improves this inequality to ‖[P_n(x)-Q_n(x)‖≤(18+ε)E_n(f), where εis any positive real number.展开更多
文摘Recently people proved that every f∈C[0, 1] can be uniformly approximated by polynomial sequences {P_n}, {Q_n} such for any x∈[0,1] and n=1,2,…that Q_n(x)<Q_(n+1)(x)<f(x)<P_(n+1)(x)<P_n(x). For example, Xie and Zhou showed that one can construct such monotone polynomial sequences which do achieve the best uniform approximation rate for a continuous func- tion. Actually they obtained a result as ‖P_n(x)-Q_n(x)‖≤42E_n (f), (1) which essentially improved a conclusion in Gal and Szabados. The present paper, by optimal procedure, improves this inequality to ‖[P_n(x)-Q_n(x)‖≤(18+ε)E_n(f), where εis any positive real number.
