Appendix F: Uniformly Convergent Sequence of Functions

2015 ◽  
pp. 247-252
Keyword(s):  
Convergent Sequence ◽  
Uniformly Convergent ◽  
Sequence Of Functions

scholarly journals Hyers-Ulam Stability of Iterative Equation in the Class of Lipschitz Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Chao Xia ◽  
Wei Song
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Functional Equations ◽  
Convergent Sequence ◽  
Lipschitz Functions ◽  
Ulam Stability ◽  
Iterative Equation ◽  
Iterative Equations ◽  
Uniformly Convergent ◽  
Sequence Of Functions

Hyers-Ulam stability is a basic sense of stability for functional equations. In the present paper we discuss the Hyers-Ulam stability of a kind of iterative equations in the class of Lipschitz functions. By the construction of a uniformly convergent sequence of functions we prove that, for every approximate solution of such an equation, there exists an exact solution near it.

scholarly journals Asymptotic Behavior of Resolvents of a Convergent Sequence of Convex Functions on Complete Geodesic Spaces

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 21
Author(s):  
Yasunori Kimura ◽  
Keisuke Shindo
Keyword(s):  
Asymptotic Behavior ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Convergent Sequence ◽  
Lower Semicontinuous Function ◽  
Semicontinuous Function ◽  
Geodesic Space ◽  
Proper Convex ◽  
Convex Lower Semicontinuous Function ◽  
Sequence Of Functions

The asymptotic behavior of resolvents of a proper convex lower semicontinuous function is studied in the various settings of spaces. In this paper, we consider the asymptotic behavior of the resolvents of a sequence of functions defined in a complete geodesic space. To obtain the result, we assume the Mosco convergence of the sets of minimizers of these functions.

scholarly journals A cauchy criterion and a convergence theorem for Riemann-complete integral

1972 ◽  
Vol 13 (1) ◽  
pp. 21-24
Author(s):  
H. W. Pu
Keyword(s):  
Convergence Theorem ◽  
Convergent Sequence ◽  
Monotone Convergence ◽  
Lebesgue Integral ◽  
Necessary And Sufficient Condition ◽  
Integrable Functions ◽  
Type Integral ◽  
Uniformly Convergent ◽  
Necessary And Sufficient ◽  
Generalized Type

In 1957 Kurzweil [1] proved some theorems concerning a generalized type of differential equations by defining a Riemann-type integral, but he did not study its properties beyond the needs of that research. This was done by R. Henstock [2, 3], who named it a Riemann-complete integral. He showed that the Riemann-complete integral includes the Lebesgue integral and that the Levi monotone convergence theorem holds. The purpose of the present paper is to give a necessary and sufficient condition for a function to be Riemann-complete integrable and to establish a termwise integration theorem for a uniformly convergent sequence of Riemann-complete integrable functions.

scholarly journals A characterization of the uniform convergence points set of some convergent sequence of functions

2021 ◽  
Vol 71 (2) ◽  
pp. 423-428
Author(s):  
Olena Karlova
Keyword(s):  
Uniform Convergence ◽  
Convergent Sequence ◽  
Normal Space ◽  
Disjoint Sequence ◽  
Isolated Points ◽  
Perfectly Normal Space ◽  
Set Of Points ◽  
Sequence Of Functions ◽  
Perfectly Normal

Abstract We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if X is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and A ⊆ X, then A is the set of points of the uniform convergence for some convergent sequence (fn ) n∈ω of functions fn : X → ℝ if and only if A is Gδ -set which contains all isolated points of X. This result generalizes a theorem of Ján Borsík published in 2019.

A generalized Bernstein approximation theorem

1988 ◽  
Vol 104 (2) ◽  
pp. 317-330 ◽  
Author(s):  
João B. Prolla
Keyword(s):  
Convex Cone ◽  
Hausdorff Space ◽  
Bernstein Polynomials ◽  
Convergent Sequence ◽  
Vector Subspace ◽  
Weierstrass Theorem ◽  
Classic Result ◽  
Bernstein Approximation ◽  
Uniformly Convergent ◽  
Image Position

A celebrated theorem of Weierstrass states that any continuous real-valued function f defined on the closed interval [0, 1] ⊂ ℝ is the limit of a uniformly convergent sequence of polynomials. One of the most elegant and elementary proofs of this classic result is that which uses the Bernstein polynomials of fone for each integer n ≥ 1. Bernstein's Theorem states that Bn(f) → f uniformly on [0, 1] and, since each Bn(f) is a polynomial of degree at most n, we have as a consequence Weierstrass' theorem. See for example Lorentz [9]. The operator Bn, defined on the space C([0, 1]; ℝ) with values in the vector subspace of all polynomials of degree at most n has the property that Bn(f) ≥ 0 whenever f ≥ 0. Thus Bernstein's Theorem also establishes the fact that each positive continuous real-valued function on [0, 1] is the limit of a uniformly convergent sequence of positive polynomials. This raises the following natural question: consider a compact Hausdorff space X and the convex cone C+(X):= {f ∈ C(X; ℝ); f ≥ 0}. Now the analogue of Bernstein's Theorem would be a theorem stating when a convex cone contained in C+(X) is dense in it. More generally, one raises the question of describing the closure of a convex cone contained in C(X; ℝ), and, in particular, the closure of A+:= {f ∈ A; f ≥ 0}, where A is a subalgebra of C(X; ℝ).

scholarly journals (Strong) weak exhaustiveness and (strong uniform) continuity

Filomat ◽  
2010 ◽  
Vol 24 (4) ◽  
pp. 63-75 ◽  
Author(s):  
Agata Caserta ◽  
Maio Di ◽  
L'ubica Holá
Keyword(s):  
Uniform Convergence ◽  
Metric Spaces ◽  
Hausdorff Metric ◽  
Convergent Sequence ◽  
Uniform Continuity ◽  
Direct Image ◽  
Continuity Property ◽  
Property A ◽  
Sequence Of Functions

In this paper we continue, in the realm of metric spaces, the study of exhaustiveness and weak exhaustiveness at a point of a net of functions initiated by Gregoriades and Papanastassiou in 2008. We prove that exhaustiveness at every point of a net of pointwise convergent functions is equivalent to uniform convergence on compacta. We extend exhaustiveness-type properties to subsets. First, we introduce the notion of strong exhaustiveness at a subset B for sequences of functions and prove its equivalence with strong exhaustiveness at P0 (B) of the sequence of the direct image maps, where the hypersets are equipped with the Hausdorff metric. Furthermore, we show that the notion of strong-weak exhaustiveness at a subset is the proper tool to investigate when the limit of a pointwise convergent sequence of functions fulfills the strong uniform continuity property, a new pregnant form of uniform continuity discovered by Beer and Levi in 2009.

Sequentially Relatively Uniformly Complete Riesz Spaces and Vulikh Algebras

1972 ◽  
Vol 24 (6) ◽  
pp. 1110-1113 ◽  
Author(s):  
C. T. Tucker
Keyword(s):  
Positive Integer ◽  
Cauchy Sequence ◽  
Convergent Sequence ◽  
Zero Element ◽  
Riesz Space ◽  
Riesz Spaces ◽  
Weak Unit ◽  
Uniformly Convergent ◽  
Positive Integers ◽  
Unique Limit

Throughout this paper V will denote an Archimedean Riesz space with a weak unit e and a zero element θ. A sequence f1,f2,f3, … of points of V is said to converge relatively uniformly to a point f (with regulator the point g of V) if, for each ∈ > 0, there is a number N such that, if n is a positive integer and n > N, then |f — fn| < ∈g. In an Archimedean Riesz space a relatively uniformly convergent sequence has a unique limit. The sequence f1, f2, f3, … is called a relatively uniform Cauchy sequence (with regulator g) if, for each ∈ > 0, there is a number N such that if n and m are positive integers and n, m > N, then |fn — fm| < eg. A subset M of V is said to be sequentially relatively uniformly complete, written s.r.u.-complete, whenever every relatively uniform Cauchy sequence of points of M (with regulator in V) converges to a point of M. This property was defined by Luxemburg and Moore in [4] and some related conditions were derived.

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