On equivalence of moduli of smoothness of polynomials in , |
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Authors: | L |
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Affiliation: | aDepartment of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA;bSchool of Mathematical Sciences, Beijing Normal University, Beijing 100875, PR China |
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Abstract: | It is well known that for functions , 1p∞. For general functions fLp, it does not hold for 0<p<1, and its inverse is not true for any p in general. It has been shown in the literature, however, that for certain classes of functions the inverse is true, and the terms in the inequalities are all equivalent. Recently, Zhou and Zhou proved the equivalence for polynomials with p=∞. Using a technique by Ditzian, Hristov and Ivanov, we give a simpler proof to their result and extend it to the Lp space for 0<p∞. We then show its analogues for the Ditzian–Totik modulus of smoothness and the weighted Ditzian–Totik modulus of smoothness for polynomials with . |
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Keywords: | Equivalence Moduli of smoothness Weighted moduli of smoothness Polynomials |
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