Abstract
The rate of uniform convergence for Hermite-Fejer polynomials to any continuous function f(x) in each closed sub-interval of (-1,1) has been given by Schonhage in 1971 by means of estimating the rate of convergence. The present paper deals with the acceleration of convergence and the rate of convergence by improving the estimate given by Schonhage, throughout two parallel ways, firstly, by use of the averaged moduli of smoothness or -moduli that gives much better estimation than that of the moduli of continuity or -moduli. Secondly, by make use of the necessary and sufficient conditions that we borrow from Szego in 1959 together with the well-known Fejer's identity (3.8) and the properties of -moduli in addition to some known results that have been given by Murray Spiegel in 1981 pp299-345.
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