### A Characterization of Multivariate Quasi-interpolation Formulas and its Applications.

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2000 Mathematics Subject Classification: 41A25, 41A36.The purpose of this paper is to present a characterization of a certain Peetre K-functional in Lp[−1,1] norm, for 1 ≤ p ≤ 2 by means of a modulus of smoothness. This modulus is based on the classical one taken on a certain linear transform of the function.

Using the concept of $\mathcal{I}$-convergence we provide a Korovkin type approximation theorem by means of positive linear operators defined on an appropriate weighted space given with any interval of the real line. We also study rates of convergence by means of the modulus of continuity and the elements of the Lipschitz class.

In this paper, we present some theorems on weighted approximation by two dimensional nonlinear singular integral operators in the following form: T λ ( f ; x , y ) = ∬ R 2 ( t − x , s − y , f ( t , s ) ) d s d t , ( x , y ) ∈ R 2 , λ ∈ Λ , $${T}_{\lambda}(f;x,y)=\underset{{\mathbb{R}}^{2}}{\phantom{\rule{0.277778em}{0ex}}\int \int \phantom{\rule{0.277778em}{0ex}}}(t-x,s-y,f(t,s))dsdt,\phantom{\rule{0.277778em}{0ex}}(x,y)\in {\mathbb{R}}^{2},\lambda \in \Lambda ,$$ where Λ is a set of non-negative numbers with accumulation point λ0.

2000 Mathematics Subject Classification: 46B70, 41A25, 41A17, 26D10. ∗Part of the results were reported at the Conference “Pioneers of Bulgarian Mathematics”, Sofia, 2006.Certain types of weighted Peetre K-functionals are characterized by means of the classical moduli of smoothness taken on a proper linear transforms of the function. The weights with power-type asymptotic at the ends of the interval with arbitrary real exponents are considered. This paper extends the method and results presented...