I–Convergence of Arithmetical Functions

Let n > 1 be an integer with its canonical representation, n = p 1 α 1 p 2 α 2 ⋯ p k α k . Put H n = max α 1 … α k , h n = min α 1 … α k , ω n = k , Ω n = α 1 + ⋯ + α k , f n = ∏ d ∣ n d and f ∗ n = f n n . Many authors deal with the statistical convergence of these arithmetical functions. For instance, the notion of normal order is defined by means of statistical convergence. The statistical convergence is equivalent with I d –convergence, where I d is the ideal of all subsets of positive integers having the asymptotic density zero. In this part, we will study I –convergence of the well-known arithmetical functions, where I = I c q = A ⊂ N : ∑ a ∈ A a − q < + ∞ is an admissible ideal on N such that for q ∈ 0 1 we have I c q ⊊ I d , thus I c q –convergence is stronger than the statistical convergence ( I d –convergence)

(I^h)-convergence and convergence of positive series

In 1827 L. Olivier proved result about the speed of convergence to zero of the terms of convergent positive series with non-increasing terms so-called Olivier\u27s Theorem. T. Šalát and V. Toma made remark that the monotonicity condition in Olivier\u27s Theorem can be dropped if the convergence of the sequence (nan) is weakened by means of the notion of I-convergence for an appropriate ideal I. Results of this type are called a modified Olivier\u27s Theorem. In connection with this we will study the properties of summable ideals Ih where h: R+→R+ is a function such that Σn∈Nh(n)=+∞ and Ih={A⊊N : Σn∈Ah(n)<+∞}. We show that Ih-convergence and Ih*-convergence are equivalent. What does not valid in general. Further we also show that the modified Olivier\u27s Theorem is not valid for summable ideals Ih in generally. We find sufficient conditions for real function h: R+→R+ such that modified Olivier\u27s Theorem remains valid for ideal Ih

Some remarks concerning strongly separately continuous functions on spaces ℓ_p with p ∊ [1;+∞]

We give a sufficient condition on strongly separately continuousfunction f to be continuous on space ℓ_p for p ∊ 2 [1;+∞]. We prove theexistence of an ssc function f : ℓ_∞ → R which is not Baire measurable.We show that any open set in ℓ_p is the set of discontinuities of a stronglyseparately continuous real-valued function for p ∊ [1;+∞).</p

Ic(q)-konvergence aritmetických funkcí

The statistical convergence is equivalent with Id-convergence, where Id is the ideal of all subsets of positive integers having the asymptotic density zero. In this paper we will study I-convergence of well known arithmetical functions, where I=Ic(q) is an admissible ideal on N for q in ]0,1] such that Ic(q) is a proper subset of Id.Statistická konvergence je ekvivalentní s Id-konvergencí, kde Id je ideál všech podmnožin N, které mají asymptotickou hustotu rovnou 0. V našem článku se zabýváme studiem I-konvergence některých známých aritmetických funkcí, kde I=Ic(q) je přípustný ideál na N pro q z ]0,1], takový že Ic(q) je vlastní podmnožina Id