With randomly generated weights between input and hidden layers, a random vector functional link network is a universal approximator for continuous functions on compact sets with fast learning property.
This fact may be used to prove minimization results for continuous convex functionals, in the same way that the Bolzano Weierstrass theorem is used for continuous functions on Rd.
For continuous functions f the value of f(t), for t in the interval, changes only slightly, so it must be very close to f(t).
The literature on inequalities for continuous functions and their applications is vast (see [1 8]).
We study the approximation properties of the considered operators for continuous functions in weighted spaces.
For continuous functions and for convex subsets of Hausdorff topological vector spaces, Theorem 9.6 was due to Ma [12, Theorem 4].
In [1], Gao and Meng established a generalized Gronwall-Bellman type integral inequality, named Mate-Nevai type nonlinear integral inequality for continuous functions, which is one case of inequalities containing integration on infinite intervals for continuous functions.
We give a necessary and sufficient condition for convergence of this iteration for continuous functions on an arbitrary interval.
The following result that provides some bounds for continuous functions of bounded variation may be stated as well.
Summing up the above observations, we can present in Figure 2 Venn's diagram for continuous functions under our investigation.
