Boolean algebras and uniform convergence of series

We continue our studies (2, 3, 4, 5) of the algebraic, geometric, and analytical similarities and contrasts between Boolean algebras and the real field. In this note we contrast the convergence of series in set algebras with that in the real field.

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Matrix summability and uniform convergence of series

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-07-08882-x ◽

2007 ◽

Vol 135 (11) ◽

pp. 3571-3580 ◽

Cited By ~ 3

Author(s):

Antonio Aizpuru ◽

Francisco J. García-Pacheco ◽

Consuelo Pérez-Eslava

Keyword(s):

Uniform Convergence ◽

Matrix Summability ◽

Convergence Of Series

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Sigma-compactness of metric boolean algebras and uniform convergence of frequencies to probabilities

Proceedings of the Steklov Institute of Mathematics ◽

10.1134/s0081543811020118 ◽

2011 ◽

Vol 272 (S1) ◽

pp. 138-151

Author(s):

E. G. Pytkeev ◽

M. Yu. Khachai

Keyword(s):

Uniform Convergence ◽

Boolean Algebras

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On the Uniform Convergence of Sine Series with Square Root

Journal of Function Spaces ◽

10.1155/2019/1342189 ◽

2019 ◽

Vol 2019 ◽

pp. 1-11

Author(s):

Sergiusz Kęska

Keyword(s):

Uniform Convergence ◽

Real Sequence ◽

Sufficient Condition ◽

Square Root ◽

Necessary And Sufficient Condition ◽

Sine Series ◽

Convergence Of Series ◽

Necessary And Sufficient

Chaundy and Jolliffe proved that if {ck}k=1∞ is a nonincreasing real sequence with limk→∞ck=0, then the series ∑k=1∞‍cksin⁡kx converges uniformly if and only if kck→0. The purpose of this paper is to show that kck→0 is a necessary and sufficient condition for the uniform convergence of series ∑k=1∞‍cksin⁡kθ in θ∈[0,π]. However for ∑k=1∞‍cksin⁡k2θ it is not true in θ∈[0,π].

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ON ABEL CONVERGENT SERIES OF FUNCTIONS

JOURNAL OF ADVANCES IN MATHEMATICS ◽

10.24297/jam.v11i9.826 ◽

2016 ◽

Vol 11 (9) ◽

pp. 5639-5644

Author(s):

Erdal Gul ◽

Mehmet Albayrak

Keyword(s):

Uniform Convergence ◽

Pointwise Convergence ◽

Convergent Series ◽

Convergence Of Series ◽

Real Functions ◽

Uniformly Convergent ◽

Series Of Functions

In this paper, we are concerned with Abel uniform convergence and Abel pointwise convergence of series of real functions where a series of functions Î£ fn is called Abel uniformly convergent to a function f if for each " > 0 there is a _ > 0 such that jfx(t) ô€€€ f(t)j < " for 1 ô€€€ _ < x < 1 and 8t 2 X, and a series of functions Î£ fn is called Abel pointwisely convergent to f if for each t 2 X and 8" > 0 there is a _("; t) such that for 1 ô€€€ _ < x < 1 jfx(t) ô€€€ f(t)j < ":

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Schur Lemma and Uniform Convergence of Series through Convergence Methods

Mathematics ◽

10.3390/math8101744 ◽

2020 ◽

Vol 8 (10) ◽

pp. 1744

Author(s):

Fernando León-Saavedra ◽

María del Pilar Romero de la Rosa ◽

Antonio Sala

Keyword(s):

Uniform Convergence ◽

Summability Method ◽

Underlying Structure ◽

Convergence Of Series ◽

Schur’S Lemma ◽

Schur Lemma ◽

Schur's Lemma

In this note, we prove a Schur-type lemma for bounded multiplier series. This result allows us to obtain a unified vision of several previous results, focusing on the underlying structure and the properties that a summability method must satisfy in order to establish a result of Schur’s lemma type.

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UNIFORM CONVERGENCE OF SERIES ESTIMATORS OVER FUNCTION SPACES

Econometric Theory ◽

10.1017/s0266466608080584 ◽

2008 ◽

Vol 24 (6) ◽

pp. 1463-1499 ◽

Cited By ~ 12

Author(s):

Kyungchul Song

Keyword(s):

Convergence Rate ◽

Uniform Convergence ◽

Rate Of Convergence ◽

Basis Function ◽

Function Spaces ◽

Smallest Eigenvalue ◽

Convergence Of Series ◽

Uniform Convergence Rate ◽

Finite Integral ◽

Function Vector

This paper considers a series estimator of E[α(Y)|λ(X) = λ̄], (α,λ) ∈ 𝛢 × Λ, indexed by function spaces, and establishes the estimator's uniform convergence rate over λ̄ ∈ R, α ∈ 𝛢, and λ ∈ Λ, when 𝛢 and Λ have a finite integral bracketing entropy. The rate of convergence depends on the bracketing entropies of 𝛢 and Λ in general. In particular, we demonstrate that when each λ ∈ Λ is locally uniformly ℒ2-continuous in a parameter from a space of polynomial discrimination and the basis function vector pK in the series estimator keeps the smallest eigenvalue of E[pK(λ(X))pK(λ(X))‼] above zero uniformly over λ ∈ Λ, we can obtain the same convergence rate as that established by de Jong (2002, Journal of Econometrics 111, 1–9).

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