Asymptotic Formulas for Convolution Operators with Spline Kernels Original Research Article

We derive asymptotic formulas for convolution operators with spline kernels for differentiable functions. These formulas are analogous to Bernstein′s extension of Voronovskaya′s results on Bernstein polynomials for functions with higher order derivatives. Two classes of operators are considered, viz., the de la Vallée Poussin-Schoenberg operators with trigonometric spline kernels and the singular integrals of Riemann-Lebesgue with periodic polynomial spline kernels. The former includes the de la Vallée means as a special case.