Abstract
Recently people proved that every f∈C[0,1] can be uniformly approximated by polynomial sequences {Pn}, {Qn} such for any x∈[0,1] and n=1,2,… that {fx98-1}. For example, Xie and Zhou[2] showed that one can construct such monotone polynomial sequences which do achieve the best uniform approximation rate for a continuous function. Actually they obtained a result as {fx98-2}, which essentially improved a conclusion in Gal and Szabados[1]. The present paper, by optimal procedure, improves this inequality to {fx98-3}, where ɛ is any positive real number.
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References
Gal, S. G. and Szabados, J., On monotone and doubly monotone polynomial approximation. Acta Math. Hugar., 59(1992), 395–399.
Xie, T. F. and Zhou, S. P., A remark on approximation by monotone sequences of polynomials. Acta Math. Hungar., 67(1995), 199–121.
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Xiaoqing, W. A development on approximation by monotone sequences of polynomials. Approx. Theory & its Appl. 14, 98–101 (1998). https://doi.org/10.1007/BF02856152
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DOI: https://doi.org/10.1007/BF02856152