Power means and the reverse Hölder inequality

Victor D. Didenko; Anatolii A. Korenovskyi

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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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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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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...

On the behaviour close to the unit circle of the power series with Möbius function coefficients

Oleg Petrushov (2014)

Acta Arithmetica

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Let $(z) = \sum_{n = 1}^{\infty} μ (n) z^{n}$ . We prove that for each root of unity $e (β) = e^{2 π i β}$ there is an a > 0 such that $(e (β) r) = Ω ({(1 - r)}^{- a})$ as r → 1-. For roots of unity e(l/q) with q ≤ 100 we prove that these omega-estimates are true with a = 1/2. From omega-estimates for (z) we obtain omega-estimates for some finite sums.

On convergence sets of divergent power series

Buma L. Fridman, Daowei Ma, Tejinder S. Neelon (2012)

Annales Polonici Mathematici

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A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family $y = φ_{s} (t, x) = s b ₁ (x) t + b ₂ (x) t ² + \dots$ of analytic curves in ℂ × ℂⁿ passing through the origin, $C o n v_{φ} (f)$ of a formal power series f(y,t,x) ∈ ℂ[[y,t,x]] is defined to be the set of all s ∈ ℂ for which the power series $f (φ_{s} (t, x), t, x)$ converges as a series in (t,x). We prove that for a subset E ⊂ ℂ there exists a divergent formal power series f(y,t,x) ∈ ℂ[[y,t,x]] such that $E = C o n v_{φ} (f)$ if...

A note on the $[τ, f (x)]$ means

V. Swaminathan (1979)

Matematički Vesnik

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On the behavior close to the unit circle of the power series whose coefficients are squared Möbius function values

Oleg Petrushov (2015)

Acta Arithmetica

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We consider the behavior of the power series $_{0} (z) = \sum_{n = 1}^{\infty} μ^{2} (n) z^{n}$ as z tends to $e (β) = e^{2 π i β}$ along a radius of the unit circle. If β is irrational with irrationality exponent 2 then $_{0} (e (β) r) = O ({(1 - r)}^{- 1 / 2 - ε})$ . Also we consider the cases of higher irrationality exponent. We prove that for each δ there exist irrational numbers β such that $_{0} (e (β) r) = Ω ({(1 - r)}^{- 1 + δ})$ .

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 $L ₁ (ℝ^{d})$ .

Lacunary formal power series and the Stern-Brocot sequence

Jean-Paul Allouche, Michel Mendès France (2013)

Acta Arithmetica

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Let $F (X) = \sum_{n \geq 0} {(- 1)}^{ε ₙ} X^{- λ ₙ}$ be a real lacunary formal power series, where εₙ = 0,1 and $λ_{n + 1} / λ ₙ > 2$ . It is known that the denominators Qₙ(X) of the convergents of its continued fraction expansion are polynomials with coefficients 0, ±1, and that the number of nonzero terms in Qₙ(X) is the nth term of the Stern-Brocot sequence. We show that replacing the index n by any 2-adic integer ω makes sense. We prove that $Q_{ω} (X)$ is a polynomial if and only if ω ∈ ℤ. In all the other cases $Q_{ω} (X)$ is an infinite formal power series; we discuss...

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 some ordered continua of power $2^{a l e p h_{0}}$ containing a dense subset of power $ℵ_{1}$

Josef Novák (1951)

Czechoslovak Mathematical Journal

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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.

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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Estimates of $\sum_{k = 1}^{N} k^{- s} 〈 k x 〉^{- t}$

A. Kruse (1967)

Acta Arithmetica

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