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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].

Edgeworth expansions for L-statistics

Alberink B., Pap G., van Zuijlen M. C. A.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 2

277--302

We study the approximation by a short Edgeworth expansion of the distribution function of normalized linear combinations [formula] of order statistics of n independent random variables with common distribution function F. Under the assumptions [formula] for some p1, q1, r1 ∈ R p2, q2, r2, C≥0, к ∈ [0.5/4) with an appropriate balance in these parameters, and under additional moment conditions, the rate of uniform convergence is shown to be of order n -1. Moreover, a special case is considered where the cjn are generated by a sequence of weight functions of a special structure.

Linear combinations of Szasz-Baskakov type operators

Agrawal P., Thamer K.

Demonstratio Mathematica

1999

Vol. 32, nr 3

575-580