Generalized Jackson kernels in approximation theory

We study approximative identities In(f)(x) = 1 2 Z ? n(u)f(x ? u) du with positive even trigonometric polynomials n of degree n as kernels. It is well known that in general the approximants In(f) are norm convergent to f with order at most n ?2 and that convergence for the test function f(x) = (2 sin(x=2)) 2 at x = 0 implies norm convergence for a wide class of functions f. We discuss the order of convergence at x = 0 for (2 sin(x=2)) 2r where r 1 is an integer and characterize kernels n which produce best convergence. This leads to estimates of the type Z ? n(u)juj 2r du = O(n ?2r) which arise e.g. in the investigation of the degree of approximation of diierentiable functions. Generalized Jackson kernels will be shown to have the same (best possible) rate of convergence. All the kernels discussed are suitable for approximative identities. Moreover, their special properties nd applications, also in complex approximation theory 1], 2], 3], 7], 10]. x1 Introduction Let r 1 be a natural number. We consider the singular integral