Volume 20, Issue 1

Abstract. H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded, closed, and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection. In the present paper we show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of closed and convex sets, which are not necessarily bounded.

Abstract. In 1954 M. Heins proved that, for every analytic set A containing the infinity, there exists an entire function whose set of asymptotic values at the infinity equals A . We obtain analogs of this result for functions analytic in planar domains of arbitrary connectivity.

Abstract. In previous work of the author, a convolution and multiplication product for the set of Associated Homogeneous Distributions (AHDs) with support in ℝ was defined and fully investigated. Here this definition is used to calculate the multiplication product of homogeneous distributions of the form ( x ± i 0) z , for all z∈ℂ${{z\in \mathbb {C}}}$. Multiplication products of AHDs generally contain an arbitrary constant if the resulting degree of homogeneity is a negative integer, i.e., if it is a critical product. However, critical products of the forms (x+i0)a.(x+i0)b$(x+i0) ^{a}\mathbin {\textbf {.}}(x+i0) ^{b}$ and (x-i0)a.(x-i0)b$(x-i0) ^{a}\mathbin {\textbf {.}}(x-i0) ^{b}$, with a+b∈ℤ-${a+b\in \mathbb {Z}_{-}}$, are exceptionally unique. This fact combined with Sokhotskii–Plemelj expressions then leads to linear dependencies of the arbitrary constants occurring in products like δ(k).δ(l)${{\delta ^{(k)}\mathbin {\textbf {.}}\delta ^{(l)}}}$, η(k).δ(l)${{\eta ^{(k)}\mathbin {\textbf {.}}\delta ^{(l)}}}$, δ(k).η(l)${{\delta ^{(k)}\mathbin {\textbf {.}}\eta ^{(l)}}}$ and η(k).η(l)${{\eta ^{(k)}\mathbin {\textbf {.}}\eta ^{(l)}}}$ for all k,l∈ℕ${{k,l\in \mathbb {N}}}$ (η≜1πx-1${{\eta \triangleq \frac{1}{\pi }x^{-1}}}$). This in turn gives a unique distribution for products like δ(k).η(l)+η(k).δ(l)${{\delta ^{(k)}\mathbin {\textbf {.}}\eta ^{(l)}+\eta ^{(k)}\mathbin {\textbf {.}}\delta ^{(l)}}}$ and δ(k).δ(l)-η(k).η(l)${{\delta ^{(k)}\mathbin {\textbf {.}}\delta ^{(l)}-\eta ^{(k)}\mathbin {\textbf {.}}\eta ^{(l)}}}$. The latter two products are of interest in quantum field theory and appear for instance in products of the partial derivatives of the zero-mass two-point Wightman distribution.

Abstract. In the paper, the authors obtain some Hermite–Hadamard type integral inequalities for extended s -convex functions on the co-ordinates in a rectangle.

Abstract. In this paper we present some different types of ideal convergence/divergence and of ideal continuity for Riesz space-valued functions, and prove some basic properties and comparison results. We investigate the relations among different modes of ideal continuity and present a characterization of the ( AP )-property for ideals of an abstract set Λ. Finally we pose some open problems.

Abstract. The complex Bloch varieties and the associated Fermi curves of two-dimensional periodic Schrödinger operators with quasi-periodic boundary conditions are defined as complex analytic varieties, the Schrödinger potentials being from the Lorentz–Fourier space ℱℓ∞,1$\mathcal {F}\ell ^{\infty ,1}$. Then, an asymptotic analysis of the Fermi curves is performed. The decomposition of a Fermi curve into a compact part, an asymptotically free part, and thin handles, is recovered as expected. Furthermore, it is shown that the set of potentials whose associated Fermi curve has finite geometric genus is a dense subset of ℱℓ∞,1$\mathcal {F}\ell ^{\infty ,1}$. Moreover, the Fourier transforms of the potentials are locally isomorphic to perturbed Fourier transforms induced by the handles. Finally, an asymptotic family of parameters describing the sizes of the handles is introduced. These parameters are good candidates for describing parts of the space of all Fermi curves.

Abstract. In the paper there are determined, for some classes defined by coefficient or analytic conditions, the sets of complex parameter γ, for which all the functions of the appropriate family have some geometrical properties. There are also provided the examples of the mappings showing that some inclusions between classes are impossible or confirming that sets of the parameter γ cannot be extended in some cases without loss of these geometric properties.

Abstract. The generalization of Gadjiev–Ibragimov operators by means of Kantorovich and Kirov–Popova is considered and the order of weighted approximation of unbounded functions are obtained.

Abstract. Aim of this short paper is to construct in any infinite-dimensional Hilbert space a series with terms tending to zero such that some of its rearrangements possess the discrete set of limit points.

Abstract. This is an erratum to [J. Appl. Anal. 15 (2009), no. 2, 247–267], http://dx.doi.org/10.1515/JAA.2009.247. In this note the author would like to make a correction in formulas (2.12) and (2.13) for the limit limϵ→0(uϵ(x,t),vϵ(x,t))${\lim _{\epsilon \rightarrow 0}(u^\epsilon (x,t),v^\epsilon (x,t))}$ stated in Theorem (2.1); there is no change in the proof.