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Linear approximation
About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.
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TL;DR: In this paper, two versions of the approximate relations of the deformation theory of continuous media, known as the complete version and the incomplete version of the quadratic approximation of the non-linear theory are analysed.
32 citations
06 Sep 1993
TL;DR: It is demonstrated that, using standard ideas from the theory of spline approximation, it is possible to construct such networks to provide localized approximation and can be used for the simultaneous approximation of a function and its derivative.
Abstract: The problem of constructing universal networks capable of approximating all functions having bounded derivatives is discussed. It is demonstrated that, using standard ideas from the theory of spline approximation, it is possible to construct such networks to provide localized approximation. The networks can be used to implement multivariate analogues of the Chui-Wang wavelets (1990) and also for the simultaneous approximation of a function and its derivative. The number of neurons required to yield the desired approximation at any point does not depend upon the degree of accuracy desired. >
32 citations
TL;DR: In this article, the analysis of Part I is extended to the case in which both even and odd variables are needed to describe the macroscopic state of a system, and the usual phenomenological equations, obeying reciprocal relations in the form given by Casimir, are described.
32 citations
TL;DR: In this paper, it was shown that a shift-invariant space can be described by a system of linear partial difference equations with constant coefficients, whose solvability is characterized by an old theorem of Toeplitz.
Abstract: We take an algebraic approach to the problem of approximation by dilated shifts of basis functions. Given a finite collection D of compactly supported functions in Lp(RIS) (1 0), where Sh is the hdilate of S. We prove that (Sh: h > 0) provides Lp-approximation order r only if S contains all the polynomials of total degree less than r . In particular, in the case where D consists of a single function (p with its moment f (p # 0, we characterize the approximation order of (Sh: h > 0) by showing that the above condition on polynomial containment is also sufficient. The above results on approximation order are obtained through a careful analysis of the structure of shift-invariant spaces. It is demonstrated that a shiftinvariant space can be described by a certain system of linear partial difference equations with constant coefficients. Such a system then can be reduced to an infinite system of linear equations, whose solvability is characterized by an old theorem of Toeplitz. Thus, the Toeplitz theorem sheds light into approximation theory. It is also used to give a very simple proof for the well-known Ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients.
32 citations
TL;DR: In this paper, the existence and uniqueness of eigenvalues were studied and three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method (SLAM), were investigated.
Abstract: Nonlinear rank-one modiflcation of the symmetric eigenvalue problem arises from eigenvibrations of mechanical structures with elastically attached loads and calculation of the propagation modes in optical flber. In this paper, we flrst study the existence and uniqueness of eigenvalues, and then investigate three numerical algorithms, namely Picard iteration, nonlinear Rayleigh quotient iteration and successive linear approximation method (SLAM). The global convergence of the SLAM is proven under some mild assumptions. Numerical examples illustrate that the SLAM is the most robust method. Mathematics subject classiflcation: 65F15, 65H17, 15A18, 35P30, 65Y20.
32 citations