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In this paper we present and prove some new results concerning approximation properties of T means with respect to the Vilenkin system in Lebesgue spaces for any 1 <= p < infinity. As applications, we obtain extensions of some known approximation inequalities.

We characterize subsequences {S_nk} of partial sums with respect to (bounded or unbounded) Vilenkin systems of f ∈ L¹(G_m) for which almost everywhere convergence holds. Moreover, we construct an explicit f ∈ L^p(G_m), 1 ≤ p < ∞ whose partial sums (satisfying the same conditions which guarantee almost everywhere convergence) diverges on any set of measure zero. We also prove a similar divergence result for Vilenkin-Féjer means.

There is a lot of information available concerning Hardy-Hilbert type inequalities in one or more dimensions. In this paper we introduce the development of such inequalities on homogeneous groups. Moreover, we point out a unification of several of the Hardy-Hilbert type inequalities in the classical case to a general kernel case. Finally, we generalize these results to the homogeneous group case.

In this paper, we derive integer pseudo-parametric solutions to two sets of Diophantine equations. Moreover, we describe the so-called Double Crossed Ladder (DCL) and show how these results can be used to calculate an infinite number of integer solutions of its sides. In addition, we describe the fact that these results can be used to derive some corresponding sets of integer sides of more complex geometric structures.

The Peetre K-functional is a key object in the development of the real method of interpolation. In this paper we point out a less known relation to wavelet theory and its applications to approximation theory and engineering applications. As a new basis for further development of these studies we present some known properties in the form appropriate for further applications and then derive new information and prove some new results concerning the K-functional and its close relation to (almost) quasi-monotone functions, various indices and interpolation theory. In particular, we extend and unify some known function parameter generalizations of the standard real interpolation spaces (A0,A1)θ,q.

In this paper Rothe's classical method is extended so that it can be used to solve some linear parabolic boundary value problems in non-cylindrical domains. The corresponding existence and uniqueness theorems are proved and some further results and generalizations are discussed and applied.

In this manuscript a number of general inequalities for isotonic subadditive functionals on a set of positive mappings are proved and applied. In particular, it is pointed out that these inequalities both unify and generalize some general forms of the Hö̈lder, Popoviciu, Minkowski, Bellman and Power mean inequalities. Also some refinements of some of these results are proved.