### A conjecture related to the approximation operators of binomial type.

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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.

We prove some versions of abstract Korovkin-type theorems in modular function spaces, with respect to filter convergence for linear positive operators, by considering several kinds of test functions. We give some results with respect to an axiomatic convergence, including almost convergence. An extension to non positive operators is also studied. Finally, we give some examples and applications to moment and bivariate Kantorovich-type operators, showing that our results are proper extensions of the...

An approximation error and an asymptotic formula are given for shift invariant operators of polynomial order ϱ. Density estimators based on shift invariant operators are introduced and AMISE is calculated.

In this paper we study the approximation of continuous functions F, defined on a compact Hausdorff space S, whose values F(t), for each t in S, are convex subsets of a normed space E. Both quantitative estimates (in the Hausdorff semimetric) and Bohman-Korovkin type approximation theorems for sequences of monotone operators are obtained.