Inequalities for the Extended Best Polynomial Approximation Operator in Orlicz Spaces
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In this paper we pursue the study of the best approximation operator extended from LΦ to Lφ, where φ denotes the derivative of the function Φ. We get pointwise convergence for the coefficients of the extended best approximation polynomials for a wide class of function f, closely related to the Calderón–Zygmund class t m p (x) which had been introduced in 1961. We also obtain weak and strong type inequalities for a maximal operator related to the extended best polynomial approximation and a norm convergence result for the coefficients is derived. In most of these results, we have to consider Matuszewska–Orlicz indices for the function φ.
KeywordsOrlicz spaces extended best polynomial approximation pointwise and norm convergence weak and strong type inequalities Orlicz indices
MR(2010) Subject Classification05B05 05B25 20B25
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The authors would like to thank the referee for such valuable comments and suggestions.
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