Ordinary convergence follows from statistical summability (C,1) in the case of slowly decreasing or oscillating sequences

Ferenc Móricz

Displaying similar documents to “Ordinary convergence follows from statistical summability (C,1) in the case of slowly decreasing or oscillating sequences”

Statistical extensions of some classical Tauberian theorems in nondiscrete setting

Ferenc Móricz (2007)

Colloquium Mathematicae

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Schmidt’s classical Tauberian theorem says that if a sequence $(s_{k} : k = 0, 1, . . .)$ of real numbers is summable (C,1) to a finite limit and slowly decreasing, then it converges to the same limit. In this paper, we prove a nondiscrete version of Schmidt’s theorem in the setting of statistical summability (C,1) of real-valued functions that are slowly decreasing on ℝ ₊. We prove another Tauberian theorem in the case of complex-valued functions that are slowly oscillating on ℝ ₊. In the proofs we make use...

Necessary and sufficient Tauberian conditions for the logarithmic summability of functions and sequences

Ferenc Móricz (2013)

Studia Mathematica

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Let s: [1,∞) → ℂ be a locally Lebesgue integrable function. We say that s is summable (L,1) if there exists some A ∈ ℂ such that $l i m_{t \to \infty} τ (t) = A$ , where $τ (t) : = 1 / (l o g t) \int_{1}^{t} s (u) / u d u$ . (*) It is clear that if the ordinary limit s(t) → A exists, then also τ(t) → A as t → ∞. We present sufficient conditions, which are also necessary, in order that the converse implication hold true. As corollaries, we obtain so-called Tauberian theorems which are analogous to those known in the case of summability (C,1). For example, if the function...

Some Tauberian theorems on $A_{λ}$ summability methods of integrals

B. P. Mishra, D. Singh (1976)

Matematički Vesnik

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Tauberian theorems for Cesàro summable double integrals over $ℝ_{+}^{2}$

Ferenc Móricz (2000)

Studia Mathematica

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Given ⨍ ∈ $L_{l}^{1} o c (ℝ_{+}^{2})$ , denote by s(w,z) its integral over the rectangle [0,w]× [0,z] and by σ(u,v) its (C,1,1) mean, that is, the average value of s(w,z) over [0,u] × [0,v], where u,v,w,z>0. Our permanent assumption is that (*) σ(u,v) → A as u,v → ∞, where A is a finite number. First, we consider real-valued functions ⨍ and give one-sided Tauberian conditions which are necessary and sufficient in order that the convergence (**) s(u,v) → A as u,v → ∞ follow from (*). Corollaries allow these...

On the $C^{α} | C^{β}$ convergence

L. Jeśmanowicz (1962)

Annales Polonici Mathematici

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On $[F, d_{n}]$ summability of Fourier series

P. L. Sharma, V. Prakash, S. K. Varma (1985)

Matematički Vesnik

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Relations among K $S (λ)$ and certain other methods for evaluations of sequences and series

B. Martić (1964)

Matematički Vesnik

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On the inclusion relation between strong $(J, p_{n})$ and strong $(\bar{N}, p_{n})$ summability methods

R. K. Jain, A. Ganguly (1978)

Matematički Vesnik

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Translativity of absolute weighted mean summability

Cihan Orhan (1998)

Czechoslovak Mathematical Journal

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In this paper, we give necessary and sufficient conditions on $(p_{n})$ for $| R, p_{n} |_{k}$ , $k \geq 1$ , to be translative. So we extend the known results of Al-Madi [1] and Cesco $[4]$ to the case $k > 1$ .

On generalized absolute Cesaro summability factors

A. Nihal Tuncer (2002)

Annales Polonici Mathematici

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Using δ-quasi-monotone and any almost increasing sequences we prove a theorem on ${| C, α, β; δ |}_{k}$ summability factors of infinite series, which generalizes a theorem of Mazhar [7] on ${| C, 1 |}_{k}$ summability factors.

A note on the $[τ, f (x)]$ means

V. Swaminathan (1979)

Matematički Vesnik

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On $| V, λ |$ summability of a factored Fourier series

P. L. Sharma, R. K. Jain (1970)

Matematički Vesnik

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Summability of first integrals of a $C^{ω}$ -non-integrable resonant Hamiltonian system

Masafumi Yoshino (2012)

Banach Center Publications

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This article studies the summability of first integrals of a $C^{ω}$ -non-integrable resonant Hamiltonian system. The first integrals are expressed in terms of formal exponential transseries and their Borel sums. Smooth Liouville integrability and a relation to the Birkhoff transformation are discussed from the point of view of the summability.

Correction to “Summability in $ł (p_{1}, p_{2}, . . .)$ spaces” Studia Mathematica 25 (1965), p. 277-280)

Victor Klee (1969)

Studia Mathematica

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Multi-dimensional Fejér summability and local Hardy spaces

Ferenc Weisz (2009)

Studia Mathematica

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It is proved that the multi-dimensional maximal Fejér operator defined in a cone is bounded from the amalgam Hardy space $W (h_{p}, ℓ_{\infty})$ to $W (L_{p}, ℓ_{\infty})$ . This implies the almost everywhere convergence of the Fejér means in a cone for all $f \in W (L ₁, ℓ_{\infty})$ , which is larger than $L ₁ (ℝ^{d})$ .