Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊

Péter Kórus; Ferenc Móricz

Displaying similar documents to “Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊”

On the uniform convergence of double sine series

Péter Kórus, Ferenc Móricz (2009)

Studia Mathematica

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Let a single sine series (*) $\sum_{k = 1}^{\infty} a_{k} s i n k x$ be given with nonnegative coefficients $a_{k}$ . If $a_{k}$ is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that $k a_{k} \to 0$ as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) $\sum_{k = 1}^{\infty} \sum_{l = 1}^{\infty} c_{k l} s i n k x s i n l y$ , even with complex coefficients $c_{k l}$ . We also...

On the double Lusin condition and convergence theorem for Kurzweil-Henstock type integrals

Abraham Racca, Emmanuel Cabral (2016)

Mathematica Bohemica

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Equiintegrability in a compact interval $E$ may be defined as a uniform integrability property that involves both the integrand $f_{n}$ and the corresponding primitive $F_{n}$ . The pointwise convergence of the integrands $f_{n}$ to some $f$ and the equiintegrability of the functions $f_{n}$ together imply that $f$ is also integrable with primitive $F$ and that the primitives $F_{n}$ converge uniformly to $F$ . In this paper, another uniform integrability property called uniform double Lusin condition introduced in the papers...

Regular statistical convergence of double sequences

Ferenc Móricz (2005)

Colloquium Mathematicae

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The concepts of statistical convergence of single and double sequences of complex numbers were introduced in [1] and [7], respectively. In this paper, we introduce the concept indicated in the title. A double sequence $x_{j k} : (j, k) \in ℕ ²$ is said to be regularly statistically convergent if (i) the double sequence $x_{j k}$ is statistically convergent to some ξ ∈ ℂ, (ii) the single sequence $x_{j k} : k \in ℕ$ is statistically convergent to some $ξ_{j} \in ℂ$ for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence $x_{j k} : j \in ℕ$ is statistically convergent...

On the completeness of $ℒ_{p}$ -spaces over a charge

Sreela Gangopadhyay (1990)

Colloquium Mathematicae

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Pointwise convergence of nonconventional averages

I. Assani (2005)

Colloquium Mathematicae

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We answer a question of H. Furstenberg on the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} U ⁿ (f \cdot R ⁿ (g))$ , where U and R are positive operators. We also study the pointwise convergence of the averages $1 / N \sum_{n = 1}^{N} f (S ⁿ x) g (R ⁿ x)$ when T and S are measure preserving transformations.

Necessary and sufficient conditions for the $L^{1}$ -convergence of double Fourier series

Péter Kórus (2018)

Czechoslovak Mathematical Journal

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We extend the results of paper of F. Móricz (2010), where necessary conditions were given for the $L^{1}$ -convergence of double Fourier series. We also give necessary and sufficient conditions for the $L^{1}$ -convergence under appropriate assumptions.

Convergence in the dual of certain $K M_{p}$ -spaces

Larry Kitchens, Charles Swartz (1974)

Colloquium Mathematicae

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On the convergence theory of double $K$ -weak splittings of type II

Vaibhav Shekhar, Nachiketa Mishra, Debasisha Mishra (2022)

Applications of Mathematics

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Recently, Wang (2017) has introduced the $K$ -nonnegative double splitting using the notion of matrices that leave a cone $K \subseteq ℝ^{n}$ invariant and studied its convergence theory by generalizing the corresponding results for the nonnegative double splitting by Song and Song (2011). However, the convergence theory for $K$ -weak regular and $K$ -nonnegative double splittings of type II is not yet studied. In this article, we first introduce this class of splittings and then discuss the convergence theory...

A condition equivalent to uniform ergodicity

Maria Elena Becker (2005)

Studia Mathematica

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Let T be a linear operator on a Banach space X with $s u p ₙ | | T ⁿ / n^{w} | | < \infty$ for some 0 ≤ w < 1. We show that the following conditions are equivalent: (i) $n^{- 1} \sum_{k = 0}^{n - 1} T^{k}$ converges uniformly; (ii) $c l (I - T) X = z \in X : l i m_{n} \sum_{k = 1}^{n} T^{k} z / k e x i s t s$ .

Nilakantha's accelerated series for π

David Brink (2015)

Acta Arithmetica

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We show how the idea behind a formula for π discovered by the Indian mathematician and astronomer Nilakantha (1445-1545) can be developed into a general series acceleration technique which, when applied to the Gregory-Leibniz series, gives the formula $π = \sum_{n = 0}^{\infty} ((5 n + 3) n! (2 n)!) / (2^{n - 1} (3 n + 2)!)$ with convergence as $13 . 5^{- n}$ , in much the same way as the Euler transformation gives $π = \sum_{n = 0}^{\infty} (2^{n + 1} n! n!) / (2 n + 1)!$ with convergence as $2^{- n}$ . Similar transformations lead to other accelerated series for π, including three “BBP-like” formulas, all of which are collected in...

Convergence of greedy approximation I. General systems

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

Studia Mathematica

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We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as $T_{ε} (x) : = \sum_{j \in D_{ε} (x)} e *_{j} (x) e_{j}$ , where $D_{ε} (x) : = j : | e *_{j} (x) | \geq ε$ . We study a generalized version of $T_{ε}$ that we call the weak thresholding approximation. We modify the $T_{ε} (x)$ in the following way. For ε > 0, t ∈ (0,1) we set $D_{t, ε} (x) : = j : t ε \leq | e *_{j} (x) | < ε$ and consider...

Hereditarily Hurewicz spaces and Arhangel'skii sheaf amalgamations

Boaz Tsaban, Lubomyr Zdomsky (2012)

Journal of the European Mathematical Society

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A classical theorem of Hurewicz characterizes spaces with the Hurewicz covering property as those having bounded continuous images in the Baire space. We give a similar characterization for spaces $X$ which have the Hurewicz property hereditarily. We proceed to consider the class of Arhangel’skii $α_{1}$ spaces, for which every sheaf at a point can be amalgamated in a natural way. Let $C_{p} (X)$ denote the space of continuous real-valued functions on $X$ with the topology of pointwise convergence. Our...

Maximally convergent rational approximants of meromorphic functions

Hans-Peter Blatt (2015)

Banach Center Publications

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Let f be meromorphic on the compact set E ⊂ C with maximal Green domain of meromorphy $E_{ρ (f)}$ , ρ(f) < ∞. We investigate rational approximants $r_{n, m ₙ}$ of f on E with numerator degree ≤ n and denominator degree ≤ mₙ. We show that a geometric convergence rate of order $ρ {(f)}^{- n}$ on E implies uniform maximal convergence in m₁-measure inside $E_{ρ (f)}$ if mₙ = o(n/log n) as n → ∞. If mₙ = o(n), n → ∞, then maximal convergence in capacity inside $E_{ρ (f)}$ can be proved at least for a subsequence Λ ⊂ ℕ. Moreover, an analogue...

On ultra-weak convergence in $L^{p}$

Donald E. Myers (1976)

Annales Polonici Mathematici

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Some generalizations of Olivier's theorem

Alain Faisant, Georges Grekos, Ladislav Mišík (2016)

Mathematica Bohemica

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Let $\sum_{n = 1}^{\infty} a_{n}$ be a convergent series of positive real numbers. L. Olivier proved that if the sequence $(a_{n})$ is non-increasing, then $lim_{n \to \infty} n a_{n} = 0$ . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having $lim_{n \to \infty} n a_{n} = 0$ ; 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 $ℐ$ -convergence, that is a convergence according to an ideal $ℐ$ of subsets of $ℕ$ . Again, Olivier’s theorem is a consequence...

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

J. A. Nitsche (1975)

Publications mathématiques et informatique de Rennes

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The gradient lemma

Urban Cegrell (2007)

Annales Polonici Mathematici

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We show that if a decreasing sequence of subharmonic functions converges to a function in $W_{l o c}^{1, 2}$ then the convergence is in $W_{l o c}^{1, 2}$ .

Convergence of Taylor series in Fock spaces

Haiying Li (2014)

Studia Mathematica

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It is well known that the Taylor series of every function in the Fock space $F_{α}^{p}$ converges in norm when 1 < p < ∞. It is also known that this is no longer true when p = 1. In this note we consider the case 0 < p < 1 and show that the Taylor series of functions in $F_{α}^{p}$ do not necessarily converge “in norm”.

The weak convergence of regenerative processes using some excursion path decompositions

Amaury Lambert, Florian Simatos (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider regenerative processes with values in some general Polish space. We define their $ε$ -big excursions as excursions $e$ such that $ϕ (e) g t; ε$ , where $ϕ$ is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of $e$ . We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of $ε$ -big excursions and of their endpoints, for all $ε$ in a set whose closure contains $0$ . Finally,...

Sobolev regularity via the convergence rate of convolutions and Jensen’s inequality

Mark A. Peletier, Robert Planqué, Matthias Röger (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We derive a new criterion for a real-valued function $u$ to be in the Sobolev space $W^{1, 2} (ℝ^{n})$ . This criterion consists of comparing the value of a functional $\int f (u)$ with the values of the same functional applied to convolutions of $u$ with a Dirac sequence. The difference of these values converges to zero as the convolutions approach $u$ , and we prove that the rate of convergence to zero is connected to regularity: $u \in W^{1, 2}$ if and only if the convergence is sufficiently fast. We finally apply our criterium to...

A-Statistical Convergence of Subsequence of Double Sequences

Harry I. Miller (2007)

Bollettino dell'Unione Matematica Italiana

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The concept of statistical convergence of a sequence was first introduced by H. Fast [7] in 1951. Recently, in the literature, the concept of statistical convergence of double sequences has been studied. The main result in this paper is a theorem that gives meaning to the statement: $s={s_{ij}}$ converges statistically $A$ to $L$ if and only if "most" of the "subsequences" of $s$ converge to $L$ in the ordinary sense. The results presented here are analogue of theorems in [12], [13] and [6] and are concerned...

On the rate of convergence of the Bézier-type operators

Grażyna Anioł (2006)

Bollettino dell'Unione Matematica Italiana

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For bounded functions $f$ on an interval $I$ , in particular, for functions of bounded p-th power variation on $I$ there is estimated the rate of pointwise convergence of the Bezier-type modification of the discrete Feller operators. In the main theorem the Chanturiya modulus of variation is used.

Existence and upper semicontinuity of uniform attractors in $H ¹ (ℝ^{N})$ for nonautonomous nonclassical diffusion equations

Cung The Anh, Nguyen Duong Toan (2014)

Annales Polonici Mathematici

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We prove the existence of uniform attractors $_{ε}$ in the space $H ¹ (ℝ^{N})$ for the nonautonomous nonclassical diffusion equation $u_{t} - ε Δ u_{t} - Δ u + f (x, u) + λ u = g (x, t)$ , ε ∈ [0,1]. The upper semicontinuity of the uniform attractors ${_{ε}}_{ε \in [0, 1]}$ at ε = 0 is also studied.

Convergence of greedy approximation II. The trigonometric system

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

Studia Mathematica

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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...

Some remarks on universality properties of $ℓ_{\infty} / c ₀$

Mikołaj Krupski, Witold Marciszewski (2012)

Colloquium Mathematicae

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We prove that if is not a Kunen cardinal, then there is a uniform Eberlein compact space K such that the Banach space C(K) does not embed isometrically into $ℓ_{\infty} / c ₀$ . We prove a similar result for isomorphic embeddings. Our arguments are minor modifications of the proofs of analogous results for Corson compacta obtained by S. Todorčević. We also construct a consistent example of a uniform Eberlein compactum whose space of continuous functions embeds isomorphically into $ℓ_{\infty} / c ₀$ , but fails to embed...