Quasi-interpolants for genuine Baskakov-Durrmeyer-type operators (Q2845288)

This book (the author's PhD thesis) is devoted to the approximation of continuous functions defined on compact intervals or on the positive semi-axis, by using the genuine Baskakov-Durrmeyer type operators. In 1957, V. A. Baskakov introduced a class of discrete positive linear operators, containing as particular cases the classical operators of Bernstein, Szasz-Mirakjan and Baskakov. In the Durrmeyer variant of this class, the point evaluations are replaced by weighted integrals of the approximated function; see [\textit{M. Heilmann}, Approximation on [0,\(\infty)\) by the method of Baskakov-Durrmeyer type operators. Dortmund: Univ. Dortmund, Fachbereich Mathematik (Diss.) (1987; Zbl 0728.41032); \textit{E. E. Berdysheva}, J. Approximation Theory 149, No. 2, 131--150 (2007; Zbl 1132.41328)]. Briefly speaking, the genuine Baskakov-Durrmeyer operators are constructed by using evaluations at the endpoints of the interval, and weighted integrals instead of the other point evaluations. They inherit several nice properties of the discrete operators (e.g. preservation of linear functions, interpolation at endpoints) and of their Durrmeyer variants (e.g. existence of eigenpolynomials independent of the order of the operator).NEWLINENEWLINEAll these facts are presented in the first chapter of the book, with references to a rich literature.NEWLINENEWLINEIn order to get a better order of approximation, the genuine Baskakov-Durrmeyer operators are composed with suitable differential operators; the resulting operators, called quasi-interpolants, are investigated in Chapter 2.NEWLINENEWLINEIn Chapter 3, a Bernstein-type inequality is obtained; it will be essential in the last two chapters, which contain the main results.NEWLINENEWLINEIn Chapter 4, the order of approximation by quasi-interpolants is studied in relation with the degree of smoothness of the function. For sufficiently smooth functions an error estimate of Jackson-Favard type is given. The error of approximation in the sup-norm is estimated in terms of a suitable \(K\)-functional. Asymptotic formulas for the error of approximation and simultaneous approximation are given.NEWLINENEWLINEStrong inverse results of type A (for genuine Baskakov-Durrmeyer type operators) and of type B (for quasi-interpolants) are presented in the last chapter. They complement the direct results obtained in Chapter 4.NEWLINENEWLINESeveral known results, appearing for the first time in book form, are generalized and improved by the author. The book provides a detailed and careful presentation of the state of the art in this area of research.