Uniform approximation by a non-convex cone of continuous functions

AbstractWe provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone K with non-empty interior. We also show that the set of nowhere K-monotone functions has a σ-porous complement in the space of continuous functions endowed with the uniform metric.

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The Role of Nonpolynomiality in Uniform Approximation by RBF Networks of Hankel Translates

Journal of Function Spaces ◽

10.1155/2019/1845491 ◽

2019 ◽

Vol 2019 ◽

pp. 1-10

Author(s):

Isabel Marrero

Keyword(s):

Uniform Approximation ◽

Linear Span ◽

Uniform Norm ◽

Continuous Functions ◽

Rbf Networks

Givenμ>-1/2andc∈I=]0,∞[, let the spaceCμ,c(respectively,Cμ) consist of all those continuous functionsuon]0,c](respectively,I) such that the limitlimz→0+⁡z-μ-1/2u(z)exists and is finite;Cμ,cis endowed with the uniform normuμ,∞,c=supz∈[0,c]⁡z-μ-1/2u(z) (u∈Cμ,c).Assumeϕ∈Cμdefines an absolutely regular Hankel-transformable distribution. Then, the linear span of dilates and Hankel translates ofϕis dense inCμ,cfor allc∈Iif, and only if,ϕ∉πμ, whereπμ=span{t2n+μ+1/2:n∈Z+}.

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Best Simultaneous approximation of quasi-continuous functions by monotone functions

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700032997 ◽

1991 ◽

Vol 50 (3) ◽

pp. 391-408 ◽

Cited By ~ 1

Author(s):

Salem M. A. Sahab

Keyword(s):

Banach Space ◽

Convex Cone ◽

Simultaneous Approximation ◽

Continuous Functions ◽

Unit Interval ◽

Closed Convex Cone ◽

Monotone Functions ◽

Measurable Functions ◽

Best Simultaneous Approximation ◽

Nondecreasing Functions

AbstractLet Q denote the Banach space (under the sup norm) of quasi-continuous functions on the unit interval [0, 1]. Let ℳ denote the closed convex cone comprised of monotone nondecreasing functions on [0, 1]. For f and g in Q and 1 < p < ∞, let hp denote the best Lp-simultaneous approximant of f and g by elements of ℳ. It is shown that hp converges uniformly as p → ∞ to a best L∞-simultaneous approximant of f and g by elements of ℳ. However, this convergence is not true in general for any pair of bounded Lebesgue measurable functions. If f and g are continuous, then each hp is continuous; so is limp→∞ hp = h∞.

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