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

B. P. Mishra; D. Singh

Displaying similar documents to “Some Tauberian theorems on $A_{λ}$ summability methods of integrals”

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...

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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On the $C^{α} | C^{β}$ convergence

L. Jeśmanowicz (1962)

Annales Polonici Mathematici

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

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

Matematički Vesnik

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A note on the $[τ, f (x)]$ means

V. Swaminathan (1979)

Matematički Vesnik

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

Ferenc Móricz (2004)

Colloquium Mathematicae

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Schmidt’s Tauberian theorem says that if a sequence (xk) of real numbers is slowly decreasing and $l i m_{n \to \infty} (1 / n) \sum_{k = 1}^{n} x_{k} = L$ , then $l i m_{k \to \infty} x_{k} = L$ . The notion of slow decrease includes Hardy’s two-sided as well as Landau’s one-sided Tauberian conditions as special cases. We show that ordinary summability (C,1) can be replaced by the weaker assumption of statistical summability (C,1) in Schmidt’s theorem. Two recent theorems of Fridy and Khan are also corollaries of our Theorems 1 and 2. In the Appendix, we present a new proof...

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 $L ₁ (ℝ^{d})$ .

On the Nörlund means of Vilenkin-Fourier series

István Blahota, Lars-Erik Persson, Giorgi Tephnadze (2015)

Czechoslovak Mathematical Journal

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We prove and discuss some new $(H_{p}, L_{p})$ -type inequalities of weighted maximal operators of Vilenkin-Nörlund means with non-increasing coefficients ${q_{k} : k \geq 0}$ . These results are the best possible in a special sense. As applications, some well-known as well as new results are pointed out in the theory of strong convergence of such Vilenkin-Nörlund means. To fulfil our main aims we also prove some new estimates of independent interest for the kernels of these summability results. In the special cases of...

A note on a critical problem with natural growth in the gradient

Boumediene Abdellaoui, Ireneo Peral (2006)

Journal of the European Mathematical Society

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The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes $- Δ u - Λ_{N} \frac{u}{{| x |}^{2}} = {|\nabla u + \frac{N - 2}{2} \frac{u}{{| x |}^{2}} x|}^{2} {| x |}^{(N - 2) / 2} + λ f (x)$ in $Ω$ , $u = 0$ on $\partial Ω$ , $Λ_{N} = {((N - 2) / 2)}^{2}$ . This problem is a particular case of problem (2). Notice that $(N - 2) / 2$ is optimal as coefficient and exponent on the right hand side.

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.

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

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

Matematički Vesnik

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Estimates with global range for oscillatory integrals with concave phase

Bjorn Gabriel Walther (2002)

Colloquium Mathematicae

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We consider the maximal function $| | (S^{a} f) [x] {| |}_{L^{\infty} [- 1, 1]}$ where $(S^{a} f) {(t)}^{\land} (ξ) = e^{{i t | ξ |}^{a}} f ̂ (ξ)$ and 0 < a < 1. We prove the global estimate $| | S^{a} f {| |}_{L ² (ℝ, L^{\infty} [- 1, 1])} {\leq C | | f | |}_{H^{s} (ℝ)}$ , s > a/4, with C independent of f. This is known to be almost sharp with respect to the Sobolev regularity s.

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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Extensions of weak type multipliers

P. Mohanty, S. Madan (2003)

Studia Mathematica

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We prove that if $Λ \in M_{p} (ℝ^{N})$ and has compact support then Λ is a weak summability kernel for 1 < p < ∞, where $M_{p} (ℝ^{N})$ is the space of multipliers of $L^{p} (ℝ^{N})$ .

Existence of discontinuous absolute minima for certain multiple integrals without growth properties

Lamberto Cesari (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In the present paper the author discusses certain multiple integrals $I(u)$ of the calculus of variations satisfying convexity conditions, and no growth property, and the corresponding Serrin integrals $\mathfrak{I}(u)$ , to which the existence theorems in [3,4,5] do not apply. However, in the present paper, the integrals $I(u)$ and $\mathfrak{I}(u)$ are reduced to simpler form $H(v)$ and $\mathcal{H}(v)$ to which the existence theorems above apply. Thus, we derive that $I(u)\leq\mathfrak{I}(u)$ , $H(v)\leq\mathcal{H}(v)$ , we obtain the existence of the absolute minimum for the Serrin forms $\mathfrak{I}(u)$ ...

Addendum to “On the $p^{λ}$ problem”

Stephan Baier (2005)

Acta Arithmetica

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Marcinkiewicz integrals on product spaces

H. Al-Qassem, A. Al-Salman, L. C. Cheng, Y. Pan (2005)

Studia Mathematica

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We prove the $L^{p}$ boundedness of the Marcinkiewicz integral operators $μ_{Ω}$ on $ℝ^{n ₁} \times \dots \times ℝ^{n_{k}}$ under the condition that $Ω \in L {(l o g L)}^{k / 2} (^{n ₁ - 1} \times \dots \times^{n_{k} - 1})$ . The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.

On the solutions of the functional equation $ϕ (x) + ϕ^{2} (x) = F (x)$

M. Malenica (1982)

Matematički Vesnik

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A uniqueness theorem for viscous flows on exterior domains with summability assumptions on the gradient of pressure.

Giovanni P. Galdi, Paolo Maremonti (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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In questa Nota si fornisce un teorema di unicità per soluzioni regolari delle equazioni di Navier-Stokes in domini esterni. Tale teorema non richiede che le velocità tendano ad un prefissato limite all'infinito, mentre il gradiente di pressione è supposto essere di $q$ -ma potenza sommabile nel cilindro spazio-temporale $(q\in(1,\infty))$ . Questo risultato non può essere ulteriormente generalizzato al caso $q=\infty$ , a causa di noti controesempi.

Some new Formulas involving $Γ_{q}$ Functions

Thomas Ernst (2007)

Rendiconti del Seminario Matematico della Università di Padova

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A complete characterization of R-sets in the theory of differentiation of integrals

G. A. Karagulyan (2007)

Studia Mathematica

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Let $_{s}$ be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis $_{s}$ differentiates the integral of f if s ∉ S, and $D ̅_{s} f (x) = l i m s u p_{d i a m (R) \to 0, x \in R \in_{s}} {| R |}^{- 1} \int_{R} f = \infty$ almost everywhere if s ∈ S. If the condition $D ̅_{s} f (x) = \infty$ holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a $G_{δ}$ (resp. a $G_{δ σ}$ ).

Displaying similar documents to “Some Tauberian theorems on A λ summability methods of integrals”

Displaying similar documents to “Some Tauberian theorems on $A_{λ}$ summability methods of integrals”