Statistical convergence of sequences and series of complex numbers with applications in Fourier Analysis and Summability

This is a survey paper on recent results indicated in the title. In contrast to the famous examples of Kolmogorov and Fejér on the pointwise divergence of Fourier series, the statistical convergence of the Fourier series of any integrable function takes place at almost every point; and the statistical convergencr of the Fourier series of any continuous function is uniform. Furthermore, Tauberian conditions are also presented, under which ordinary convergence of any sequence of real or complex numbers follows from its statistical summability.     

Usefulness of Transgenic Luminescent Bacteria in Direct Bioautographic Investigation of Chamomile Extracts

JPC – Journal of Planar Chromatography – Modern TLC - Direct bioautography performed with luminescence gene-tagged bacteria enables almost real-time detection of antimicrobial compounds...     

Absolute convergence of multiple Fourier series revisited

In a recent paper [4], Gogoladze and Meskhia generalized the classical results of Bernstein, Szász, Zygmund and others related to absolute convergence of single trigonometric Fourier series. Our aim is to extend these results from single to multiple Fourier series. To this effect, we introduce the notions of multiplicative moduli of continuity and that of smoothness. Multiplicative Lipschitz classes of functions in several variables, and functions of bounded s-variation in the sense of Vitali are also considered.     

Absolutely convergent Fourier series, classical function classes and Paley’s theorem

This is a survey paper on the recent progress in the study of the continuity and smoothness properties of a function f with absolutely convergent Fourier series. We give best possible sufficient conditions in terms of the Fourier coefficients of f which ensure the belonging of f either to one of the Lipschitz classes Lip(α) and lip(α) for some 0 < α ≤ 1, or to one of the Zygmund classes Zyg(α) and zyg(α) for some 0 < α ≤ 2. We also discuss the termwise differentiation of Fourier series. Our theorems generalize those by R. P. Boas Jr., J. Németh and R. E. A. C. Paley, and a number of them are first published in this paper or proved in a simpler way.     

New Approach to Mechanism of Action of Paclitaxel by Means of BioArena Studies

JPC – Journal of Planar Chromatography – Modern TLC - The accumulation of hydrogen peroxide (H2O2) is an early and crucial step in paclitaxel-induced cancer cell death before commitment...     

Isolation of Some Antimicrobial Compounds of Red Wine by OPLC Flowing Eluent Wall Technique

The first results of the combination of BioArena and OPLC with flowing eluent wall technique (FEW) show an especially attractive solution of the detection of antimicrobial activity and compound isolation based on peak collection. Because of the different separation mechanisms applied for on-line FEW–OPLC isolation of red wine constituents and for further off-line OPLC investigation of fractions, the combined system yields high selectivity.     

On the harmonic averages of numerical sequences

In recent years, the almost sure central limit theorem attracted widespread attention in Probability Theory. It involves the harmonic (also called logarithmic) averages of a certain numerical sequence formed from a sequence of independent, identically distributed random variables. Our primary aim is to study the convergence behavior of the sequence of harmonic averages of a given numerical sequence from the viewpoint of Summability Theory. Received: 12 May 2005; revised: 1 July 2005