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We deal with a new type of statistical convergence for double sequences, called

The classical Bohman-Korovkin theorem (see [

This paper continues and develops the theory introduced in [

Our general approach enables one to unify various extensions of the Korovkin theorem, by choosing suitably the function

Let

Let

It is not difficult to check that the

if

Let

Given a double sequence

If we take

The

We denote by

(a) It is easy to see that if

Note that, in general, the converse implication is not true. Indeed, let

(b) Observe that if

Let

We now recall the notion of modular space (see also [

A functional

Let

The

A modular

A modular

A modular

A modular

for each

for every

Let

For every

We now define the modular and strong convergences in the context of the

A double sequence

For

Given a subset

We recall the following.

Let

Throughout this paper, we will consider some kinds of rates of approximation associated with the Korovkin theorem in the context of

Observe that the Euclidean multidimensional spaces endowed with the usual metric fulfil

For

We now prove some Korovkin-type theorems with respect to an abstract finite set of test functions

Let

We say that the double sequence

Some examples of operators satisfying property

Set

for every neighborhood

From now on, we will assume that

We now prove the following.

Let

If

Let

For each

As

The proof of the last part of the theorem is analogous.

We now give the main Korovkin-type theorem.

Let

Let

Fix arbitrarily

(a) Note that, in Theorem

(b) By a similar technique, we can prove an analogous result in the space

We present some estimates of rates of triangular

Let

Let

We now recall the following results (see also [

Let

Now, we are in position to prove the following.

Let

Then for every

Furthermore, similar results hold when the symbol

Let

Let now

One can ask whether it is possible in the Korovkin-type theorems to relax the positivity condition on the linear operators involved. We now give some positive answers also in the context of

Let

Let

there is a positive real constant

Some examples in which properties (P1), (P2), and (P3) are satisfied can be found in [

We now prove the following Korovkin-type theorem for not necessarily positive linear operators, in the setting of

Let

If

Furthermore, if

Let

Let

In this section, we give some application of Theorem

For every

For every

For every

For every

Let

For each

By virtue of the Jensen inequality and the Fubini theorem, we have

Fix now

We now consider a direct extension to the bivariate case of the classical one-dimensional moment kernel (see also [

For every