Voronovskaya estimates for convolution operators

TitleVoronovskaya estimates for convolution operators
Publication TypeJournal Article
Year of Publication2023
AuthorsDraganov, BR
JournalDolomites Research Notes on Approximation
Date Published01/2023
PublisherPadova University Press
Place PublishedPadova, IT
ISSN Number 2035-6803

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 Fejér 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.