The publication introduces a brand new method of measuring smoothness. the necessity for this new notion arises from the failure of the classical moduli of smoothness to unravel a few simple difficulties, similar to characterizing the behaviour of top polynomial approximation in Lp -1,1 . the recent modulus, which has an easy shape, is also defined as a Peetre ok useful among an Lp house and a weighted Sobolev house. Connections among interpolation of areas and approximation concept are used in employing the modulus of smoothness. The purposes comprise top (weighted) polynomial approximation on a finite period, characterization of the speed of approximation given via classical operator methods equivalent to Bernstein, Kantorovich, Szasz-Mirakjan, and Post-Widder operators, Freud-type weighted polynomial approximation on countless periods with exponentially reducing weights and polynomial approximation in numerous variables. particular emphasis is put on the computability element of the moduli. the consequences are new, and whole proofs are given. it's was hoping that the publication can be of curiosity and worthwhile for mathematicians operating in approximation conception, interpolation of areas, numerical research and actual research.
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