Ideal relatively uniform convergence with Korovkin and Voronovskaya types approximation theorems

We introduce the notion of ideally relative uniform convergence of sequences of real valued functions. We then apply this notion to prove Korovkin-type approximation theorem, and then construct an illustrative example by taking (p,q)-Bernstein operators which proves that our Korovkin theorem is stronger than its classical version as well as statistical relative uniform convergence. The rate of ideal relatively uniform convergence of positive linear operators by means of modulus of continuity is calculated. Finally, the Voronovskaya-type approximation theorem is also investigated.

Statistical Order Convergence and Statistically Relatively Uniform Convergence in Riesz Spaces

A new concept of statistically e-uniform Cauchy sequences is introduced to study statistical order convergence, statistically relatively uniform convergence, and norm statistical convergence in Riesz spaces. We prove that, for statistically e-uniform Cauchy sequences, these three kinds of convergence for sequences coincide. Moreover, we show that the statistical order convergence and the statistically relatively uniform convergence need not be equivalent. Finally, we prove that, for monotone sequences in Banach lattices, the norm statistical convergence coincides with the weak statistical convergence.

Cantor extension of an abelian lattice ordered group equipped with a weak relatively uniform convergence

The notion of weak relatively uniform convergence (wru-convergence, for short) on an abelian lattice ordered group G has been investigated in a previous authors’ article. In the present paper we deal with Cantor extension of G and completion of G with respect to a wru-convergence on G.

Weak relatively uniform convergences on abelian lattice ordered groups

The notion of relatively uniform convergence has been applied in the theory of vector lattices and in the theory of archimedean lattice ordered groups. Let G be an abelian lattice ordered group. In the present paper we introduce the notion of weak relatively uniform convergence (wru-convergence, for short) on G generated by a system M of regulators. If G is archimedean and M = G +, then this type of convergence coincides with the relative uniform convergence on G. The relation of wru-convergence to the o-convergence is examined. If G has the diagonal property, then the system of all convex ℓ-subgroups of G closed with respect to wru-limits is a complete Brouwerian lattice. The Cauchy completeness with respect to wru-convergence is dealt with. Further, there is established that the system of all wru-convergences on an abelian divisible lattice ordered group G is a complete Brouwerian lattice.

Relatively uniform convergences in archimedean lattice ordered groups

For an archimedean lattice ordered group G let G d and G∧ be the divisible hull or the Dedekind completion of G, respectively. Put G d∧ = X. Then X is a vector lattice. In the present paper we deal with the relations between the relatively uniform convergence on X and the relatively uniform convergence on G. We also consider the relations between the o-convergence and the relatively uniform convergence on G. For any nonempty class τ of lattice ordered groups we introduce the notion of τ-radical class; we apply this notion by investigating relative uniform convergences.

On a relative uniform completion of an archimedean lattice ordered group

The notion of a relatively uniform convergence (ru-convergence) has been used first in vector lattices and then in Archimedean lattice ordered groups.Let G be an Archimedean lattice ordered group. In the present paper, a relative uniform completion (ru-completion) $$ G_{\omega _1 } $$ of G is dealt with. It is known that $$ G_{\omega _1 } $$ exists and it is uniquely determined up to isomorphisms over G. The ru-completion of a finite direct product and of a completely subdirect product are established. We examine also whether certain properties of G remain valid in $$ G_{\omega _1 } $$. Finally, we are interested in the existence of a greatest convex l-subgroup of G, which is complete with respect to ru-convergence.