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Approximation of bounded continuous functions by linear combinations of Phillips operators

Tachev G.

Demonstratio Mathematica

2014

Vol. 47, nr 3

662--671

We study the approximation properties of linear combinations of the so-called Phillips operators, which can be considered as genuine Szász–Mirakjan–Durrmeyer operators. As main result, we prove a direct estimate for the rate of approximation of bounded continuous functions (…), measured in (…)-norm and thus generalizing the results, proved earlier by Gupta, Agrawal, and Gairola in [3]. Our estimates rely on the recent results, obtained in the joint works of M. Heilmann and the author [10, 11].

Linear combinations of right invertible operators in commutative algebras with logarithms

Przeworska-Rolewicz D.

Demonstratio Mathematica

1998

Vol. 31, nr 4

887-898

It is well known that a power of a right invertible operators is again right invertible, as well as a polynomial in a right invertible operator under appropriate assumptions. However , a linear combination of right invertible operators (in particular , their sum and/or difference) in general is not right invertible. It will be shown how to solve equations with linear combinations of right invertible operators in commutative algebras using properties of logarithmic and antilogarithmic mappings. The used method is, in a sense, a kind of the variables separation method. We shall obtain also an analogue of the classical Fourier method for partial differential equations. Note that results concerning the Fourier method are proved under weaker assumptions than those obtained in PR[l] (cf. also PR[2]). The extensive bibliography of the subject can be found in PR[2] and PR[4].