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In this study we deal with the weighted uniform convergence of the Meyer-König and Zeller type operators with endpoint or inner singularities.

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Abstract.

We prove some new direct and converse results on simultaneous approximation by the combinations of Bernstein–Kantorovich operators using the Ditzian–Totik modulus of smoothness.

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We consider Steklov operators in weighted spaces of continuous functions on the whole real line and on a bounded interval. We study the connections of these operators with some second order degenerate parabolic problems establishing a general Voronovskaja type formula.

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We study the L p-saturation for the linear combination of Bernstein-Kantorovich operators. As a result we obtain the saturation class by using K-functional as well as some modulus of smoothness.

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We construct a new kind of rational operator which can be used to approximate functions with endpoints singularities by algebric weights in [−1,1], and establish new direct and converse results involving higher modulus of smoothness and a very general class of step functions, which cannot be obtained by weighted polynomial approximation. Our results also improve related results of Della Vecchia [5].

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In this paper, using the concept of statistical σ-convergence which is stronger than the statistical convergence, we obtain a statistical σ-approximation theorem for sequences of positive linear operators defined on the space of all real valued B-continuous functions on a compact subset of the real line. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. Also we compute the rate of statistical σ-convergence of sequence of positive linear operators.

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A general class of linear and positive operators dened by nite sum is constructed. Some of their approximation properties, including a convergence theorem and a Voronovskaja-type theorem are established. Next, the operators of the considered class which preserve exactly two test functions from the set {e 0, e 1, e 2} are determined. It is proved that the test functions e 0 and e 1 are preserved only by the Bernstein operators, the test functions e 0 and e 2 only by the King operators while the test functions e 1 and e 2 only by the operators recently introduced by P. I. Braica, O. T. Pop and A. D. Indrea in [4].

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In this paper we give the direct, inverse and equivalence theorems for generalization Meyer-König and Zeller type operators in the space L p (1 ≦ p ) with Ditzian-Totik modulus.

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We give a weighted Hermite-Fejr-type interpolatory method on the real line, which is a positive operator on “good” matrices. We give an example on “good” interpolatory matrix by weighted Fekete points. To prove the convergence theorem we need the generalization of “Rodrigues’ property”.

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Self-similarity of Bernstein polynomials, embodied in their subdivision property is used for construction of an Iterative (hyperbolic) Function System (IFS) whose attractor is the graph of a given algebraic polynomial of arbitrary degree. It is shown that such IFS is of just-touching type, and that it is peculiar to algebraic polynomials. Such IFS is then applied to faster evaluation of Bzier curves and to introduce interactive free-form modeling component into fractal sets.

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