New moduli of smoothness: weighted DT
نویسندگان
چکیده
We introduce new moduli of smoothness for functions f ∈ Lp[−1, 1]∩Cr−1(−1, 1), 1 ≤ p ≤ ∞, r ≥ 1, that have an (r − 1)st locally absolutely continuous derivative in (−1, 1), and such that φrf (r) is in Lp[−1, 1], where φ(x) = (1 − x2)1/2. These moduli are equivalent to certain weighted DT moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original DT moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in Lp[−1, 1] (no weight), by means of algebraic polynomials. Moreover, we also prove matching inverse theorems thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials.
منابع مشابه
Coconvex pointwise approximation
There are two kinds of estimates of the degree of approximation of continuous functions on [−1, 1] by algebraic polynomials, Nikolskii-type pointwise estimates and Jackson-type uniform estimates, involving either ordinary moduli of smoothness, or the DitzianTotik (DT) ones, or the recent estimates involving the weighted DTmoduli of smoothness. The purpose of this paper is to complete the table ...
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We introduce weighted moduli of smoothness for functions f ∈ Lp[−1, 1]∩Cr−1(−1, 1), r ≥ 1, that have an (r−1)st absolutely continuous derivative in (−1, 1) and such that φf (r) is in Lp[−1, 1], where φ(x) = (1−x). These moduli are equivalent to certain weighted D-T moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted appr...
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