Central limit theorem for square error of multivariate nonparametric box spline density estimators
We prove the central limit theorem for the integrated square error of multivariate box-spline density estimators.
Construction of Stancu-Hurwitz operator for two variables.
Convergence and Voronovskaja-type theorems for derivatives of generalized Baskakov operators
In the present paper our aim is to establish convergence and Voronovskaja-type theorems for first derivatives of generalized Baskakov operators for functions of one and two variables in exponential and polynomial weight spaces.
Convergence of approximation processes on convex cones.
Convergence of iterates of linear operators and the Kelisky-Rivlin type theorems
Let X be a Banach space and T ∈ L(X), the space of all bounded linear operators on X. We give a list of necessary and sufficient conditions for the uniform stability of T, that is, for the convergence of the sequence of iterates of T in the uniform topology of L(X). In particular, T is uniformly stable iff for some p ∈ ℕ, the restriction of the pth iterate of T to the range of I-T is a Banach contraction. Our proof is elementary: It uses simple facts from linear algebra, and the Banach Contraction...
Convexity and variation diminishing property for Bernstein polynomials in higher dimensions
Degenerate evolution problems and Beta-type operators
The present paper is concerned with the study of the differential operator Au(x):=α(x)u”(x)+β(x)u’(x) in the space C([0,1)] and of its adjoint Bv(x):=((αv)’(x)-β(x)v(x))’ in the space , where α(x):=x(1-x)/2 (0≤x≤1). A careful analysis of their main properties is carried out in view of some generation results available in [6, 12, 20] and [25]. In addition, we introduce and study two different kinds of Beta-type operators as a generalization of similar operators defined in [18]. Among the corresponding...
Density methods and results in approximation theory
Approximation theory and functional analysis share many common problems and points of contact. One of the areas of mutual interest is that of density results. In this paper we briefly survey various methods and results in this area starting from work of Weierstrass and Riesz, and extending to more recent times.
Direct and Converse Theorems for Generalized Bernstein-Type Operators
2000 Mathematics Subject Classification: 41A25, 41A27, 41A36.We establish direct and converse theorems for generalized parameter dependent Bernstein-type operators. The direct estimate is given using a K-functional and the inverse result is a strong converse inequality of type A, in the terminology of [2].
Direct approximation theorems for discrete type operators.
Direct results for mixed Beta-Szász type operators.
Discrete linear interpolatory operators.
Equivalence Between K-functionals Based on Continuous Linear Transforms
2000 Mathematics Subject Classification: 46B70, 41A10, 41A25, 41A27, 41A35, 41A36, 42A10.The paper presents a method of relating two K-functionals by means of a continuous linear transform of the function. In particular, a characterization of various weighted K-functionals by unweighted fixed-step moduli of smoothness is derived. This is applied in estimating the rate of convergence of several approximation processes.Partially supported by grant No. 103/2007 of the National Science Fund of the Sofia University....
Equivalent theorem on Lupaş operators.
Error estimates and the Voronovskaja theorem for modified Szász-Mirakyan operators
Exact bounds for some basis functions of approximation operators.
Fractional Korovkin Theory Based on Statistical Convergence
2000 Mathematics Subject Classification: 41A25, 41A36, 40G15.In this paper, we obtain some statistical Korovkin-type approximation theorems including fractional derivatives of functions. We also show that our new results are more applicable than the classical ones.
Fractional Trigonometric Korovkin Theory in Statistical Sense
2000 Mathematics Subject Classification: 41A25, 41A36.In the present paper, we improve the classical trigonometric Korovkin theory by using the concept of statistical convergence from the summability theory and also by considering the fractional derivatives of functions. We also show that our new results are more applicable than the classical ones.
General uniform approximation theory by multivariate singular integral operators
We study the uniform approximation properties of general multivariate singular integral operators on , N ≥ 1. We establish their convergence to the unit operator with rates. The estimates are pointwise and uniform. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators to which this theory can be applied directly.
Global approximation by modified Baskakov type operators.
In the present paper, we prove a global direct theorem for the modified Baskakov type operators in terms of so called Ditzian-Totik modulus of smoothness.