1.1. In some recent work [3], [4], the author has used the convolution structure for Jacobi series, introduced by Askey and Wainger [2], in order to study the summation of Jacobi series by classical summability methods. Many of these summability methods, in fact, can be interpreted as convolution operators and it is possible to investigate the order of approximation of these operators by the same techniques as are used for trigonometric convolution operators. In this paper some new summability kernels are introduced, which can be written in a simple closed form by means of Jacobi polynomials. Even in the case of Fourier series (a=s=—½) these kernels induce new approximation processes. The saturation order and the saturation class of these processes are obtained.