Voronovskaya estimates for convolution operators

We present a general method for establishing quantitative Voronovskaya-type estimates of convolution operators on homogeneous Banach spaces of periodic functions of one real variable or of functions on the real line. The method is based on properties of the Fourier transform of the kernel of the operator. We illustrate the elegance and the efficiency of this approach on two convolution operators-the Riesz typical means, and, in particular, the Fejer operator, and the generalized singular integral of Picard. A noteworthy feature of the former is the fact that, though the operator itself is saturated, the convergence in its Voronovskaya-type estimate can be of an arbitrary fast power-type provided that the function is smooth enough in a certain sense.