Displaying similar documents to “Complete f -moment convergence for weighted sums of WOD arrays with statistical applications”

Orthogonal series regression estimation under long-range dependent errors

Waldemar Popiński (2001)

Applicationes Mathematicae

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This paper is concerned with general conditions for convergence rates of nonparametric orthogonal series estimators of the regression function. The estimators are obtained by the least squares method on the basis of an observation sample Y i = f ( X i ) + η i , i=1,...,n, where X i A d are independently chosen from a distribution with density ϱ ∈ L¹(A) and η i are zero mean stationary errors with long-range dependence. Convergence rates of the error n - 1 i = 1 n ( f ( X i ) - f ̂ N ( X i ) ) ² for the estimator f ̂ N ( x ) = k = 1 N c ̂ k e k ( x ) , constructed using an orthonormal system...

Orthogonal series estimation of band-limited regression functions

Waldemar Popiński (2014)

Applicationes Mathematicae

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The problem of nonparametric function fitting using the complete orthogonal system of Whittaker cardinal functions s k , k = 0,±1,..., for the observation model y j = f ( u j ) + η j , j = 1,...,n, is considered, where f ∈ L²(ℝ) ∩ BL(Ω) for Ω > 0 is a band-limited function, u j are independent random variables uniformly distributed in the observation interval [-T,T], η j are uncorrelated or correlated random variables with zero mean value and finite variance, independent of the observation points. Conditions...

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 n = 1 N U ( f · R ( g ) ) , where U and R are positive operators. We also study the pointwise convergence of the averages 1 / N n = 1 N f ( S x ) g ( R x ) when T and S are measure preserving transformations.

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 = n = 0 ( ( 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 = n = 0 ( 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...

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 ) 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 is statistically convergent to some ξ j for each fixed j ∈ ℕ ∖ ₁, (iii) the single sequence x j k : j is statistically convergent...

On orthogonal series estimation of bounded regression functions

Waldemar Popiński (2001)

Applicationes Mathematicae

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The problem of nonparametric estimation of a bounded regression function f L ² ( [ a , b ] d ) , [a,b] ⊂ ℝ, d ≥ 1, using an orthonormal system of functions e k , k=1,2,..., is considered in the case when the observations follow the model Y i = f ( X i ) + η i , i=1,...,n, where X i and η i are i.i.d. copies of independent random variables X and η, respectively, the distribution of X has density ϱ, and η has mean zero and finite variance. The estimators are constructed by proper truncation of the function f ̂ ( x ) = k = 1 N ( n ) c ̂ k e k ( x ) , where the coefficients c ̂ , . . . , c ̂ N ( n ) ...

On the uniform convergence of double sine series

Péter Kórus, Ferenc Móricz (2009)

Studia Mathematica

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Let a single sine series (*) k = 1 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 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 (**) k = 1 l = 1 c k l s i n k x s i n l y , even with complex coefficients c k l . We also...

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 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...

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 ) : = j D ε ( x ) e * j ( x ) e j , where D ε ( x ) : = j : | e * j ( x ) | 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 | e * j ( x ) | < ε and consider...

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 ( n ) . This criterion consists of comparing the value of a functional 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 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 i j 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...

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”.

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...

Some generalizations of Olivier's theorem

Alain Faisant, Georges Grekos, Ladislav Mišík (2016)

Mathematica Bohemica

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Let n = 1 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 n a n = 0 . In the present paper: (a) We formulate and prove a necessary and sufficient condition for having lim n 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 -convergence, that is a convergence according to an ideal of subsets of . Again, Olivier’s theorem is a consequence...

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 .

Note on duality of weighted multi-parameter Triebel-Lizorkin spaces

Wei Ding, Jiao Chen, Yaoming Niu (2019)

Czechoslovak Mathematical Journal

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We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces F ˙ p α , q ( ω ; n 1 × n 2 ) . This space has been introduced and the result ( F ˙ p α , q ( ω ; n 1 × n 2 ) ) * = CMO p - α , q ' ( ω ; n 1 × n 2 ) for 0 < p 1 has been proved in Ding, Zhu (2017). In this paper, for 1 < p < , 0 < q < we establish its dual space H ˙ p α , q ( ω ; n 1 × n 2 ) .

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.