TY - JOUR

T1 - New Moduli of Smoothness

T2 - Weighted DT Moduli Revisited and Applied

AU - Kopotun, K. A.

AU - Leviatan, D.

AU - Shevchuk, I. A.

N1 - Publisher Copyright:
© 2014, Springer Science+Business Media New York.

PY - 2015/8/1

Y1 - 2015/8/1

N2 - We introduce new moduli of smoothness for functions (Formula Presented), that have an (r-1)st locally absolutely continuous derivative in (-1,1), and such that φrf(r) is in Lp[-1,1], where (Formula Presented). These moduli are equivalent to certain weighted Ditzian–Totik (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 $$L_p[-1,1]$$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.

AB - We introduce new moduli of smoothness for functions (Formula Presented), that have an (r-1)st locally absolutely continuous derivative in (-1,1), and such that φrf(r) is in Lp[-1,1], where (Formula Presented). These moduli are equivalent to certain weighted Ditzian–Totik (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 $$L_p[-1,1]$$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.

KW - Approximation by polynomials in the L-norm

KW - Degree of approximation

KW - Jackson-type estimates

KW - Moduli of smoothness

UR - http://www.scopus.com/inward/record.url?scp=84933671197&partnerID=8YFLogxK

U2 - 10.1007/s00365-014-9270-2

DO - 10.1007/s00365-014-9270-2

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AN - SCOPUS:84933671197

SN - 0176-4276

VL - 42

SP - 129

EP - 159

JO - Constructive Approximation

JF - Constructive Approximation

IS - 1

ER -