Approximation of functions by means of a new generalized Bernstein operator
The paper presents some new contributions of the author to the theory of constructive approximation theory. First, one describes an urn model for deducing a discrete probability distribution which permits to construct a new positive linear operator of Bernstein type: \(L_{m,r}^{\alpha,\beta}\), depending on a non-negative integer parameter r and on two real parameters \(\alpha\) and \(\beta\), such that 0\(\leq \alpha \leq \beta\), and which is defined, for any \(f:[0,1]\to {\mathbb{R}},\) by \[ (L_{m,r}^{\alpha,\beta}f)(x):= \sum^{m-r}_{k=0} p_{m-r,k}(x)[(1-x)f(k+\alpha /m+\beta)+xf(k+r+\alpha /m+\beta)], \] where \(p_{m-r,k}(x):= \binom{m-r}{k} x^ k(1-x)^{m-r-k}\). The approximation properties of this operator are investigated in detail. A special attention is accorded to the case of the operator \(L_{m,r}=L^{o,o}_{m,r}\). One establishes several representations for the remainder of the approximation formula of the function f by means of this operator and one evaluates the orders of approximation, in terms of modulus of continuity of f, or of f'. The point spectrum of the operator \(L_{m,r}\) is determined and the quadrature formula which can be obtained by using this operator is given. Further new results and extensions of these operators can be found in the following recent papers of the author: 1) A note on a multiparameter Bernstein-type approximating operator, Mathematica (Cluj) 26(49), 153-157 (1984); 2) Generalized Bernstein operators, ''Babeş-Bolyai'' Univ., Faculty of Math., Research Seminaries, Preprint Nr. 6, 1984, Itinerant Seminar on functional equations, approximation and convexity, 185-192, Cluj-Napoca (1984); 3) Probabilistic approach to a class of generalized Bernstein approximating operators, Math.-Rev. Anal. Numér. Théor. Approximation. Anal. Numér. Théor. Approximation 14(1985) (in print).
scientific article; zbMATH DE number 3865904
scientific article