On statistical $\mathfrak{A}$-Cauchy and statistical $\mathfrak{A}$-summability via ideal

The notion of statistical convergence was extended to $\mathfrak{I}$ I -convergence by (Kostyrko et al. in Real Anal. Exch. 26(2):669–686, 2000). In this paper we use such technique and introduce the notion of statistically $\mathfrak{A}^{\mathfrak{I}}$ A I -Cauchy and statistically $\mathfrak{A}^{\mathfrak{I}^{\ast }}$ A I ∗ -Cauchy summability via the notion of ideal. We obtain some relations between them and prove that under certain conditions statistical $\mathfrak{A}^{\mathfrak{I}}$ A I -Cauchy and statistical $\mathfrak{A}^{\mathfrak{I}^{\ast }}$ A I ∗ -Cauchy summability are equivalent. Moreover, we give some Tauberian theorems for statistical $\mathfrak{A}^{\mathfrak{I}}$ A I -summability.

On I-deferred statistical convergence of order α

The idea of I-convergence of real sequences was introduced by Kostyrko et al. [Kostyrko, P., Sal?t, T. and Wilczy?ski, W. I-convergence, Real Anal. Exchange 26(2) (2000/2001), 669-686] and also independently by Nuray and Ruckle [Nuray, F. and Ruckle,W. H. Generalized statistical convergence and convergence free spaces. J. Math. Anal. Appl. 245(2) (2000), 513-527]. In this paper we introduce I-deferred statistical convergence of order ? and strong I-deferred Ces?ro convergence of order ? and investigated between their relationship.

On I-lacunary statistical convergence of order α of sequences of sets

The idea of I-convergence of real sequences was introduced by Kostyrko et al. [Kostyrko, P. ; Sal?t, T. and Wilczy?ski, W. I-convergence, Real Anal. Exchange 26(2) (2000/2001), 669-686] and also independently by Nuray and Ruckle [Nuray, F. and Ruckle,W. H. Generalized statistical convergence and convergence free spaces, J. Math. Anal. Appl. 245(2) (2000), 513-527]. In this paper we introduce the concepts of Wijsman I-lacunary statistical convergence of order ? and Wijsman strongly I-lacunary statistical convergence of order ?, and investigated between their relationship.

On wavelet induced isomorphisms for reducing subspaces

In this paper, we adapt the notion of a wavelet induced isomorphism of [Formula: see text] associated with a wavelet set, introduced in [E. J. Ionascu, A new construction of wavelet sets, Real Anal. Exchange 28(2) (2002/03) 593–610], to the case of an [Formula: see text]-wavelet set, where [Formula: see text] is a reducing subspace [X. Dai and S. Lu, Wavelets in subspaces, Michigan Math. J. 43 (1996) 81–98]. We characterize all these wavelet induced isomorphisms similar to those given in Ionascu paper and provide specific examples of this theory in the case of symmetric [Formula: see text]-wavelet sets. Examples when [Formula: see text] is the classical Hardy space are also considered.

On wavelet induced isomorphisms for joint (d, -d)-dilation wavelet and multiwavelet sets

For a dyadic wavelet set W, Ionascu [A new construction of wavelet sets, Real Anal. Exchange28 (2002) 593–610] obtained a measurable self-bijection on the interval [0, 1), called the wavelet induced isomorphism of [0, 1), denoted by [Formula: see text]. Extending the result for a d-dilation wavelet set, we characterize a joint (d, -d)-dilation wavelet set, where |d| is an integer greater than 1, in terms of wavelet induced isomorphisms. Its analogue for a joint (d, -d)-dilation multiwavelet set has also been provided. In addition, denoting by [Formula: see text], the wavelet induced isomorphism associated with a d-dilation wavelet set W, we show that for a joint (d, -d)-dilation wavelet set W, the measures of the fixed point sets of [Formula: see text] and [Formula: see text] are equal almost everywhere.

Random affine code tree fractals and Falconer–Sloan condition

We calculate the almost sure dimension for a general class of random affine code tree fractals in $\mathbb{R}^{d}$. The result is based on a probabilistic version of the Falconer–Sloan condition $C(s)$ introduced in Falconer and Sloan [Continuity of subadditive pressure for self-affine sets. Real Anal. Exchange 34 (2009), 413–427]. We verify that, in general, systems having a small number of maps do not satisfy condition $C(s)$. However, there exists a natural number $n$ such that for typical systems the family of all iterates up to level $n$ satisfies condition $C(s)$.

On the generalization of density topologies on the real line

The paper concerns the density points with respect to the sequences of intervals tending to zero in the family of Lebesgue measurable sets. It shows that for some sequences analogue of the Lebesgue density theorem holds. Simultaneously, this paper presents proof of theorem that for any sequence of intervals tending to zero a relevant operator ϕJ generates a topology. It is almost but not exactly the same result as in the category aspect presented in [WIERTELAK, R.: A generalization of density topology with respect to category, Real Anal. Exchange 32 (2006/2007), 273–286]. Therefore this paper is a continuation of the previous research concerning similarities and differences between measure and category.

On ideal convergence in probabilistic normed spaces

An interesting generalization of statistical convergence is I-convergence which was introduced by P.Kostyrko et al [KOSTYRKO,P.—ŠALÁT,T.—WILCZYŃSKI,W.: I-Convergence, Real Anal. Exchange 26 (2000–2001), 669–686]. In this paper, we define and study the concept of I-convergence, I*-convergence, I-limit points and I-cluster points in probabilistic normed space. We discuss the relationship between I-convergence and I*-convergence, i.e. we show that I*-convergence implies the I-convergence in probabilistic normed space. Furthermore, we have also demonstrated through an example that, in general, I-convergence does not imply I*-convergence in probabilistic normed space.

Some continuous operations on pairs of cliquish functions

The algebraic or lattice operations in the classes of cliquish or quasicontinuous functions are well known [Z. Grande: On the maximal multiplicativefamily for the class of quasicontinuous functions, Real Anal. Exchange 15 (1989-1990), 437-441, Z. Grande, L. Soltysik: Some remarks on quasicontinuousreal functions, Problemy Mat. 10 (1990), 79-86]. This also pertains to the symmetrical quasicontinuity or symmetrical cliquishness [Z. Grande: On the maximaladditive and multiplicative families for the quasicontinuities of Piotrowskiand Vallin, Real Anal. Exchange 32 (2007), 511-518]. In this article, we examine the superpositions F(f, g), where F is a continuous operation and f, g are cliquish (symmetrically cliquish) or f is continuous (f is symmetrically quasicontinuous with continuous sections) and g is quasicontinuous (symmetrically quasicontinuous).