Abstract
General moduli of smoothness generated by arbitrary periodic multipliers are introduced in the one-dimensional case. Their properties are studied in the spaces L p of 2π-periodic functions for all admissible parameters 0 < p ≤ +∞. A direct Jackson-type estimate, an inverse Bernstein-type estimate, and the equivalence to the polynomial K-functional generated by an associated homogeneous function are shown. Some special cases, in particular, the moduli related to the Weyl and Riesz derivatives and their (fractional) powers, are considered. The quality of approximation by families of linear polynomial operators is described in terms of the above mentioned quantities.
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Runovski, K., Schmeisser, HJ. (2014). General Moduli of Smoothness and Approximation by Families of Linear Polynomial Operators. In: Zayed, A., Schmeisser, G. (eds) New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-08801-3_11
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