Lower bounds of some singular integrals and their applications

摘要

Let beta > 1 and f is an element of C( R). We are interested in the lower bounds of the integral: [GRAPHICS] where h > 0 and Delta(2)(t) f(x) = f(x+ t)+ f(x-t)-2f(x). Using the lower bounds for these integrals we obtain in particular for the so-called Fejer operator sigma(n)(f) of f is an element of C-2 pi the following asymptotic expression [GRAPHICS] which essentially improves the results concerning the approximation behavior of this operator.