Convergence and high order of approximation by Steklov sampling operators

In this paper we introduce a new class of sampling-type operators, named Steklovsampling operators. The idea is to consider a sampling series based on a kernel functionthat is a discrete approximate identity, and which constitutes a reconstruction processof a given signalf, based on a family of sample values which are Steklov integrals oforderrevaluated at the nodesk/w,k is an element of Z,w>0. The convergence properties of theintroduced sampling operators in continuous functions spaces and in theL(p)-settinghave been studied. Moreover, the main properties of the Steklov-type functions havebeen exploited in order to establish results concerning the high order of approximation.Such results have been obtained in a quantitative version thanks to the use of thewell-known modulus of smoothness of the approximated functions, and assumingsuitable Strang-Fix type conditions, which are very typical assumptions in applicationsinvolving Fourier and Harmonic analysis. Concerning the quantitative estimates, weproposed two different approaches; the first one holds in the case of Steklov samplingoperators defined with kernels with compact support, its proof is substantially basedon the application of the generalized Minkowski inequality, and it is valid with respectto thep-norm, with 1 <= p <=+infinity. In the second case, the restriction on the supportof the kernel is removed and the corresponding estimates are valid only for 1<=+infinity. Here, the key point of the proof is the application of the well-known Hardy-Littlewood maximal inequality. Finally, a deep comparison between the proposedSteklov sampling series and the already existing sampling-type operators has beengiven, in order to show the effectiveness of the proposed constructive method ofapproximation. Examples of kernel functions satisfying the required assumptions havebeen provided.