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Periodica Mathematica Hungarica, 1993
A net \((f_ n)\) of functions on a topological space \(X\) to a uniform space \((Y,{\mathcal U})\) converges almost uniformly to a function \(f\) at \(x_ 0\in X\) if for each \(U\in{\mathcal U}\) there exists a neighborhood \(W\) of \(x_ 0\) such that eventually \((f_ n(x),f(x))\in U\) for each \(x\in W\).
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A net \((f_ n)\) of functions on a topological space \(X\) to a uniform space \((Y,{\mathcal U})\) converges almost uniformly to a function \(f\) at \(x_ 0\in X\) if for each \(U\in{\mathcal U}\) there exists a neighborhood \(W\) of \(x_ 0\) such that eventually \((f_ n(x),f(x))\in U\) for each \(x\in W\).
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2002
Uniform continuity, completeness and equicontinuity are the most important features of uniformities and uniform spaces. Uniform convergence spaces, the convergence generalization of uniform spaces, are not as strong as their topological counterparts. In particular uniform continuity is not a very strong property.
R. Beattie, H.-P. Butzmann
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Uniform continuity, completeness and equicontinuity are the most important features of uniformities and uniform spaces. Uniform convergence spaces, the convergence generalization of uniform spaces, are not as strong as their topological counterparts. In particular uniform continuity is not a very strong property.
R. Beattie, H.-P. Butzmann
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Polynomial Uniform Convergence
2012For a sequence of polynomials that approximate a function \( f \in C\left[ {a,b} \right] \), an important issue for investigation is if the corresponding sequence of approximation errors converges uniformly to zero on \( \left[ {a,b} \right] \). In this chapter, we first show that such convergence is not guaranteed in the case that the polynomials are ...
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A Theorem in Uniform Convergence
The Mathematical Gazette, 1946One of the most striking results in the theory of uniform convergence asserts that for a monotonic sequence of continuous functions, continuity of the limit function is a necessary and sufficient condition for uniform convergence.
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Uniform Stochastic Convergence
1994Abstract This chapter concerns random sequences of functions on metric spaces. The main issue is the distinction between convergence at all points of the space (pointwise) and uniform convergence, where limit points are also taken into account. The role of the stochastic equicontinuity property is highlighted. Generic uniform convergence
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