scholarly journals Almost uniform convergence versus pointwise convergence

1960 ◽  
Vol 11 (6) ◽  
pp. 986-986 ◽  
Author(s):  
John W. Brace
Keyword(s):  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Almost Uniform Convergence

scholarly journals Noncommutative strong maximals and almost uniform convergence in several directions – Addendum

2021 ◽  
Vol 9 ◽  
Author(s):  
José M. Conde-Alonso ◽  
Adrián M. González-Pérez ◽  
Javier Parcet
Keyword(s):  
Uniform Convergence ◽  
Almost Uniform Convergence

scholarly journals Pointwise and Uniform Convergence of Fourier Extensions

2019 ◽  
Vol 52 (1) ◽  
pp. 139-175
Author(s):  
Marcus Webb ◽  
Vincent Coppé ◽  
Daan Huybrechs
Keyword(s):  
Fourier Series ◽  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Gibbs Phenomenon ◽  
Lebesgue Function ◽  
Bernstein Type ◽  
Legendre Series ◽  
The Best Approximation ◽  
Open Questions ◽  
Algebraic Order

AbstractFourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the $$L^2$$ L 2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the $$L^2$$ L 2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

scholarly journals A remark on convergence of test functions

1975 ◽  
Vol 20 (1) ◽  
pp. 73-76 ◽  
Author(s):  
W. F. Moss
Keyword(s):  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Generalized Functions ◽  
Test Functions ◽  
Definition Of ◽  
Theory Of Generalized Functions

In this note it is shown in the most frequently encountered spaces of test functions in the theory of generalized functions that the customary definitions of convergence are equivalent to apparently much weaker definitions. For example, in the space g the condition of uniform convergence of the functions together with all derivatives (which appears in the definition of convergence) is equivalent to the condition of pointwise convergence of the functions alone. Thus verification of convergence is simplified somewhat.

scholarly journals λ-Statistical Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Vatan Karakaya ◽  
Necip Şimşek ◽  
Müzeyyen Ertürk ◽  
Faik Gürsoy
Keyword(s):  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Statistical Convergence ◽  
Normed Spaces ◽  
Basic Properties ◽  
Intuitionistic Fuzzy ◽  
Convergence Of Sequences ◽  
Intuitionistic Fuzzy Normed Spaces ◽  
Convergent Sequences

We studyλ-statistically convergent sequences of functions in intuitionistic fuzzy normed spaces. We define concept ofλ-statistical pointwise convergence andλ-statistical uniform convergence in intuitionistic fuzzy normed spaces and we give some basic properties of these concepts.

scholarly journals Ideal convergent function sequences in random 2-normed spaces

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 557-567
Author(s):  
Ekrem Savaş ◽  
Mehmet Gürdal
Keyword(s):  
Uniform Convergence ◽  
Recent Work ◽  
Pointwise Convergence ◽  
Normed Spaces ◽  
Basic Properties ◽  
Convergence Of Sequences

In the present paper we are concerned with I-convergence of sequences of functions in random 2-normed spaces. Particularly, following the line of recent work of Karakaya et al. [23], we introduce the concepts of ideal uniform convergence and ideal pointwise convergence in the topology induced by random 2-normed spaces, and give some basic properties of these concepts.

On Almost Uniform Convergence of Families of Functions

1964 ◽  
Vol 7 (1) ◽  
pp. 45-48 ◽  
Author(s):  
Elias Zakon
Keyword(s):  
Uniform Convergence ◽  
Present Note ◽  
Measure Zero ◽  
Slight Modification ◽  
Real Parameter ◽  
General Proof ◽  
Almost Uniform Convergence

In [5] Tolstov showed by a counterexample that Egoroff' s theorem on almost uniform convergence cannot be extended to families of functions (ft(x)}, with t a continuous real parameter. However, Frumkin [2] proved that this is possible provided that some sets of measure zero (depending on t) are disregarded when each particular ft(x) is considered. This interesting result was obtained by using the rather involvec machinery of Kantorovitch' s semi-ordered spaces and Lp spaces. In the present note we intend to give a simpler and more general proof. Indeed, it will be seen that only a slight modification of the standard proof of Egoroff's theorem is necessary to obtain Frumkin' s theorem in a more general form. We shall establish the following result.

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