A Bibliography on the Approximation of Functions by Operators of Class S2m or Sm Involving Kernels of Finite Oscillations

TL;DRAbstract

Concerning research on the approximation of functions by means of positive linear operators (PLO), great progress has been made, in particular due to the famous test function theorem of H. BOHMAN — P.P. KOROVKIN. On the other hand, all these investigations are limited in their efficiency by a second theorem of P.P. KOROVKIN stating e.g. that for singular convolution integrals with positive polynomial kernel the optimal (critical) order of approximation cannot exceed O(n-2), n → ∞. Many ways have been tried to bypass this dilemma, thus e.g. by taking different linear combinations of positive kernels, iterates of kernels, m-singular integrals, etc. The most attractive approach (at least for the author) seems to be by introducing kernels of finite (limited) oscillations, this being so since the graphical behaviour and the peaking property of these kernels is, in comparison with positive ones, very clear and intuitive.

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