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It is shown that a Korovkin-type theorem does not hold in the weighted space of Lp−locally integrable functions on the whole real axis.

In this paper, we introduce a new type modulus of continuity for function f belonging to a particular weighted subspace of C [0, ∞) and show that it has some properties of ordinary modulus of continuity. We obtain some estimates of approximation of functions with respect to a suitable weighted norm via the new type moduli of continuity. Finally, we give some examples.

We prove theorems on convergence of multidimensional nonlinear integrals in Lebesgue points of generated function, and show that the main results are applicable to a wide class of exponentially nonlinear integral operators, which may be constructed by using well known positive kernels in approximation theory.

Approximation properties of sequences of k-positive operators, i.e. linear operators acting in the space of analytical functions and preserving the cone of functions with non-negative Taylor coefficients, are studied. Some general theorems which are valid in the space of functions that are analytical in a bounded simply connected domain are proved.In this paper we study the problem of approximation of analytical functions in some bounded complex domain by sequences of linear operators. Some properties of classes of analytical functions and operators acting in these classes may be found in [4,12]. The problem of approximation of an analytical function in the unit disk |z| < 1 by sequences of linear operators was studied in [5]. A definition of so-called k-positive operators acting on analytical functions and preserving the subspace of functions with non-negative Taylor coefficients was introduced in [5] (see also [8]) to obtain Korovkin-type approximation theorems.Some results on the approximation of analytical functions by linear k-positive operators were also established in [1][2][3]6,[9][10][11]. The problem of statistical convergence of a sequence of linear k-positive operators was studied in [3], where the author used the method of [7] and the * Corresponding author. E-mail addresses: akif gadjiev@mail.az (A.D. Gadjiev), gurbanalizade@gmail.com (A.M. Ghorbanalizadeh). 1 A.D Gadžiev, A.D. Gadjiev and A. Haciyev are names of the same person.

We introduce the family of linear operatorsassociated to a certain "admissible bunch" of operators S t , t > 0, acting on L p (R n , dm, and investigate the approximation properties of this family as → 0 + . We give some applications to the Riesz and the Bessel potentials generated by the ordinary (Euclidean) and generalized translations. .tr (I.A. Aliev), frteb@aas.ab.az (A.D. Gadjiev), aral@science.ankara.edu.tr (A. Aral).