It is shown that some well-known properties of the Sobolev space L(p)(l) (Omega) do not admit extension to the space L(p)(l) (Omega) of the functions with l-th order derivatives in L(p) (Omega), l > 1, without requ...
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It is shown that some well-known properties of the Sobolev space L(p)(l) (Omega) do not admit extension to the space L(p)(l) (Omega) of the functions with l-th order derivatives in L(p) (Omega), l > 1, without requirements to the domain Omega. Namely, we give examples of Omega such that (i) L(p)(l) (Omega) boolean AND L(infinity) (Omega) is not dense in L(p)(l) (Omega), (ii) L(p)(l) (Omega) boolean AND L(infinity) (Omega) is not a Banach algebra. (iii) the strong capacitary inequality for the norm in L(p)(l) (Omega) fails. In the Appendix necessary and sufficient conditions are given for the imbeddings L(p)(l) (Omega) subset of L(q) (Omega, mu) and H-p(l) (R(n)) subset of L(q) (R(n), mu), where p greater than or equal to 1, p > q > 0, mu is a measure and H-p(l) (Omega) is the Bessel potential space, 1 < p < infinity, l > 0.
This paper continues the investigation of representations of continuous functions f(x1, ..., x(n)) with n greater-than-or-equal-to in the form f(x1, ..., x(n)) = SIGMA(q=0)2n PHI(q)[SIGMA(p=1)n lambda(p)psi(x(p) + qe(...
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This paper continues the investigation of representations of continuous functions f(x1, ..., x(n)) with n greater-than-or-equal-to in the form f(x1, ..., x(n)) = SIGMA(q=0)2n PHI(q)[SIGMA(p=1)n lambda(p)psi(x(p) + qe(n))] with a predetermined function psi that is independent of n. The function psi is defined through its graph that is the limit point of iterated contraction mappings. The functions psi and PHI(q) are the uniform limits of sequences of computable functions constructed with a fixed mapping sigma, which itself can be approximated with sigmoid functions,
approximation properties of a class of artificial neural networks are established. It is shown that feedforward networks with one layer of sigmoidal nonlinearities achieve integrated squared error of order O(1/n), whe...
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approximation properties of a class of artificial neural networks are established. It is shown that feedforward networks with one layer of sigmoidal nonlinearities achieve integrated squared error of order O(1/n), where n is the number of nodes. The function approximated is assumed to have a bound on the first moment of the magnitude distribution of the Fourier transform. The nonlinear parameters associated with the sigmoidal nodes, as well as the parameters of linear combination, are adjusted in the approximation. In contrast, it is shown that for series expansions with n terms, in which only the parameters of linear combination are adjusted, the integrated squared approximation error cannot be made smaller than order 1/n2/d uniformly for functions satisfying the same smoothness assumption, where d is the dimension of the input to the function. For the class of functions examined here, the approximation rate and the parsimony of the parameterization of the networks are surprisingly advantageous in high-dimensional settings.
Abstract: In contrast to the behavior of best uniform polynomial approximants on $[0,1]$ we show that if $f \in C[0,1]$ there exists a sequence of polynomials $\{ {P_n}\}$ of respective degree $\leq n$ which c...
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Abstract: In contrast to the behavior of best uniform polynomial approximants on $[0,1]$ we show that if $f \in C[0,1]$ there exists a sequence of polynomials $\{ {P_n}\}$ of respective degree $\leq n$ which converges uniformly to $f$ on $[0,1]$ and geometrically fast at each point of $[0,1]$ where $f$ is analytic. Moreover we describe the best possible rates of convergence at all regular points for such a sequence.
A nonnegative, infinitely differentiable function φ defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ∫ 0 1 φ(t)dt=1...
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A nonnegative, infinitely differentiable function φ defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ∫
0
1
φ(t)dt=1. In this article, the following problem is considered. Determine Δ
k
=inf∫
0
1
|φ(k)(t)|dt,k=1, 2, ..., where φ(k) denotes thekth derivative of φ and the infimum is taken over the set of all mollifier functions φ, which is a convex set. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. The problem is reducible to three equivalent problems, a nonlinear programming problem, a problem on the functions of bounded variation, and an approximation problem involving Tchebycheff polynomials. One of the results of this article shows that Δ
k
=k!22k−1,k=1, 2, .... The numerical values of the optimal solutions of the three problems are obtained as a function ofk. Some inequalities of independent interest are also derived.
A possible mathematical formulation of the practical problem of computer-aided design of electrical circuits (for example) and systems and engineering designs in general, subject to tolerances on k independent paramet...
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A possible mathematical formulation of the practical problem of computer-aided design of electrical circuits (for example) and systems and engineering designs in general, subject to tolerances on k independent parameters, is proposed. An automated scheme is suggested, starting from arbitrary initial acceptable or unacceptable designs and culminating in designs which, under reasonable restrictions, are acceptable in the worst-case sense. It is proved, in particular, that, if the region of points in the parameter space for which designs are both feasible and acceptable satisfies a certain condition (less restrictive than convexity), then no more than 2(k) points, the vertices of the tolerance region, need to be considered during optimization.
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