On $|V,\lambda |$ summability of a factored Fourier series

P. L. Sharma; R. K. Jain

Displaying similar documents to “On $| V, λ |$ summability of a factored Fourier series”

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

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

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

B. Martić (1964)

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On the $C^{α} | C^{β}$ convergence

L. Jeśmanowicz (1962)

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

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

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Multi-dimensional Fejér summability and local Hardy spaces

Ferenc Weisz (2009)

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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 pointwise strong approximation by (C,1)(E,1) means

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We present an estimate of the (C,1)(E,1)-strong means with mixed powers of the Fourier series of a function $f \in L_{2 π}$ as a generalization of the result obtained by M. Yildrim and F. Karakus. Some corollaries on the norm approximation are also given.

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A note on the $[τ, f (x)]$ means

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Convergence of greedy approximation II. The trigonometric system

S. V. Konyagin, V. N. Temlyakov (2003)

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We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form $G ₘ (f) : = \sum_{k \in Λ} f ̂ (k) e^{i (k, x)}$ , where $Λ \subset ℤ^{d}$ is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in...

Lebesgue type points in strong (C,α) approximation of Fourier series

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We present an estimation of the $H_{k ₀, k_{r}}^{q, α} f$ and $H_{λ, u}^{ϕ, α} f$ means as approximation versions of the Totik type generalization (see [5], [6]) of the result of G. H. Hardy, J. E. Littlewood. Some corollaries on the norm approximation are also given.

Extensions of weak type multipliers

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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 $L^{p} (ℝ^{N})$ .

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

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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.

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

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

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Estimates with global range for oscillatory integrals with concave phase

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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.

$L_{\infty}$ -Convergence of Finite Element Approximation

J. A. Nitsche (1975)

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On the approximation of real continuous functions by series of solutions of a single system of partial differential equations

Carsten Elsner (2006)

Colloquium Mathematicae

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We prove the existence of an effectively computable integer polynomial P(x,t₀,...,t₅) having the following property. Every continuous function $f : ℝ^{s} \to ℝ$ can be approximated with arbitrary accuracy by an infinite sum $\sum_{r = 1}^{\infty} H_{r} (x ₁, . . ., x_{s}) \in C^{\infty} (ℝ^{s})$ of analytic functions $H_{r}$ , each solving the same system of universal partial differential equations, namely $P (x_{σ}; H_{r}, \partial H_{r} / \partial x_{σ}, . . ., \partial ⁵ H_{r} / \partial x ⁵_{σ} ⁵) = 0$ (σ = 1,..., s).

Approximation of functions from $L^{p} {(ω ̃)}_{β}$ by linear operators of conjugate Fourier series

WŁodzimierz Łenski, Bogdan Szal (2011)

Banach Center Publications

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We show the results corresponding to some theorems of S. Lal and H. K. Nigam [Int. J. Math. Math. Sci. 27 (2001), 555-563] on the norm and pointwise approximation of conjugate functions and to the results of the authors [Acta Comment. Univ. Tartu. Math. 13 (2009), 11-24] also on such approximations.

On $L^{p}$ integrability and convergence of trigonometric series

Dansheng Yu, Ping Zhou, Songping Zhou (2007)

Studia Mathematica

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We first give a necessary and sufficient condition for $x^{- γ} ϕ (x) \in L^{p}$ , 1 < p < ∞, 1/p - 1 < γ < 1/p, where ϕ(x) is the sum of either $\sum_{k = 1}^{\infty} a_{k} c o s k x$ or $\sum_{k = 1}^{\infty} b_{k} s i n k x$ , under the condition that λₙ (where λₙ is aₙ or bₙ respectively) belongs to the class of so called Mean Value Bounded Variation Sequences (MVBVS). Then we discuss the relations among the Fourier coefficients λₙ and the sum function ϕ(x) under the condition that λₙ ∈ MVBVS, and deduce a sharp estimate for the weighted modulus of continuity of ϕ(x)...

Un ensemble remarquable du point de vue de l’analyse de Fourier sur $Q_{p}$

J. P. Schreiber (1971)

Colloquium Mathematicae

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Analytical theory of nonlinear oscillations $X$ : some classes of equations $\ddot{x}+g(x)=0$ with no finite Fourier series solution

Chike Obi (1979)

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Si dimostra che l'equazione considerata nel titolo ammette soluzioni periodiche rappresentabili da una serie di Fourier con un numero finito di termini solo se $g(x)$ è lineare.

The degree of approximation by Hausdorff means of a conjugate Fourier series

Sergiusz Kęska (2016)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The purpose of this paper is to analyze the degree of approximation of a function $\bar{f}$ that is a conjugate of a function $f$ belonging to the Lipschitz class by Hausdorff means of a conjugate series of the Fourier series.

A transplantation theorem for ultraspherical polynomials at critical index

J. J. Guadalupe, V. I. Kolyada (2001)

Studia Mathematica

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We investigate the behaviour of Fourier coefficients with respect to the system of ultraspherical polynomials. This leads us to the study of the “boundary” Lorentz space $ℒ_{λ}$ corresponding to the left endpoint of the mean convergence interval. The ultraspherical coefficients $c ₙ^{(λ)} (f)$ of $ℒ_{λ}$ -functions turn out to behave like the Fourier coefficients of functions in the real Hardy space ReH¹. Namely, we prove that for any $f \in ℒ_{λ}$ the series $\sum_{n = 1}^{\infty} c ₙ^{(λ)} (f) c o s n θ$ is the Fourier series of some function φ ∈ ReH¹ with ${| | φ | |}_{R e H ¹} {\leq c | | f | |}_{ℒ_{λ}}$ . ...

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 further note on a class of $I_{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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Some duality results on bounded approximation properties of pairs

Eve Oja, Silja Treialt (2013)

Studia Mathematica

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The main result is as follows. Let X be a Banach space and let Y be a closed subspace of X. Assume that the pair $(X *, Y^{⊥})$ has the λ-bounded approximation property. Then there exists a net $(S_{α})$ of finite-rank operators on X such that $S_{α} (Y) \subset Y$ and $| | S_{α} | | \leq λ$ for all α, and $(S_{α})$ and $(S *_{α})$ converge pointwise to the identity operators on X and X*, respectively. This means that the pair (X,Y) has the λ-bounded duality approximation property.

Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series

Earl Berkson (2014)

Studia Mathematica

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Let $f \in V_{r} () \cup_{r} ()$ , where, for 1 ≤ r < ∞, $V_{r} ()$ (resp., $_{r} ()$ ) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values,...

Displaying similar documents to “On | V , λ | summability of a factored Fourier series”

Displaying similar documents to “On $| V, λ |$ summability of a factored Fourier series”