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Some generalizations of Olivier's theorem

Alain Faisant, Georges Grekos, Ladislav Mišík (2016)

Mathematica Bohemica

Let n = 1 a n be a convergent series of positive real numbers. L. Olivier proved that if the sequence ( a n ) is non-increasing, then lim n n a n = 0 . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having lim n n a n = 0 ; Olivier’s theorem is a consequence of our Theorem . (b) We prove properties analogous to Olivier’s property when the usual convergence is replaced by the -convergence, that is a convergence according to an ideal of subsets of . Again, Olivier’s theorem is a consequence of...

Some new facts about group 𝒢 generated by the family of convergent permutations

Roman Wituła, Edyta Hetmaniok, Damian Słota (2017)

Open Mathematics

The aim of this paper is to present some new and essential facts about group 𝒢 generated by the family of convergent permutations, i.e. the permutations on ℕ preserving the convergence of series of real terms. We prove that there exist permutations preserving the sum of series which do not belong to 𝒢. Additionally, we show that there exists a family G (possessing the cardinality equal to continuum) of groups of permutations on ℕ such that each one of these groups is different than 𝒢 and is composed...

Some results on convergence acceleration for the E-algorithm

A. Fdil (1997)

Applicationes Mathematicae

Some new results on convergence acceleration for the E-algorithm which is a general extrapolation method are obtained. A technique for avoiding numerical instability is proposed. Some applications are given. Theoretical results are illustrated by numerical experiments

Spherical summation : a problem of E.M. Stein

Antonio Cordoba, B. Lopez-Melero (1981)

Annales de l'institut Fourier

Writing ( T R λ f ) ^ ( ξ ) = ( 1 - | ξ | 2 / R 2 ) + λ f ^ ( ξ ) . E. Stein conjectured j | T R j λ f i | 2 1 / 2 p C j | f j | 2 1 / 2 p for λ > 0 , 4 3 p 4 and C = C λ , p . We prove this conjecture. We prove also f ( x ) = lim j T 2 j λ f ( x ) a.e. We only assume 4 3 + 2 λ < p < 4 1 - 2 λ .

Statistical cluster points of sequences in finite dimensional spaces

Serpil Pehlivan, A. Güncan, M. A. Mamedov (2004)

Czechoslovak Mathematical Journal

In this paper we study the set of statistical cluster points of sequences in m -dimensional spaces. We show that some properties of the set of statistical cluster points of the real number sequences remain in force for the sequences in m -dimensional spaces too. We also define a notion of Γ -statistical convergence. A sequence x is Γ -statistically convergent to a set C if C is a minimal closed set such that for every ϵ > 0 the set { k ρ ( C , x k ) ϵ } has density zero. It is shown that every statistically bounded sequence...

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