Approximation by linear combinations of positive convolution integrals
Consider the singular convolution integral of \(f\in C[-\pi,\pi]\), namely, \(I_ n(P,f;x)=(1/\pi)\int^{\pi}_{-\pi}f(x-t)p_ n(t)dt\), where the kernel \(P=\{p_ n:\) \(n\in {\mathbb{N}}\}\) is a sequence of even, normalized trigonometric polynomials. If P is positive, a well-known theorem of P. P. Korovkin states that there exists an arbitrarily often differentiable function \(f\in C[-\pi,\pi]\) such that \(\limsup_{n\to \infty}n^ 2\| I_ n(P,f;\cdot)-f(\cdot)\| >0.\) This result led to many investigations concerning the construction as well as the fine structure of kernels P for which the associated convolution integral \(I_ n(P,f;x)\) has a better rate of approximation than \(O(n^{-2})\). The author considers linear combinations of positive kernels (thus of convolution integrals) in order to improve this rate. The main aim is to introduce a rather general method leading to the construction of linear combinations having high rates of convergence. For this purpose the underlying kernel P needs first to be classified in terms of the so- called convergence factors, namely the Fourier coefficients of the positive P. The special case of Fejér-type kernel is treated in Section 3. For the classes of P considered the author shows in Section 4 that a linear combination of P may be constructed giving the rate \(O(n^{-\tau s})\). Section 5 deals with two quantiative approximation theorems, those of Jackson and Voronovskaja type for P. In Section 6 the construction is applied to several classical examples of positive P. Finally, Section 7 is devoted to a brief outline of the construction of linear combinations that are not saturated, a well-known example being the means of de La Vallée Poussin.
scientific article; zbMATH DE number 4120718
scientific article