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Characterization of Best Approximations
2001We give a characterization theorem for best approximations from convex sets. This result will prove useful over and over again throughout the book. Indeed, it will be the basis for every characterization theorem that we give. The notion of a dual cone plays an essential role in this characterization.
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Existence and characterization of best φ-approximations by linear subspaces
Advances in Pure and Applied Mathematics, 2017AbstractGiven an Orlicz ...
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Characterizations of Best Complex Chebyshev Approximate Solutions of $Av = b$
SIAM Journal on Numerical Analysis, 1976Let $Av = b$ be an overdetermined system of m complex equations in n unknowns with rank $(A) = k$ and $m \geqq 2k + 1$. It is shown that if $x_\infty $ is a Chebyshev solution of $Av = b$, then $x_\infty $ is also a Chebyshev solution of a $2k + 1 \times n$ subsystem of $Av = b,A_{2k + 1} v = b_{2k + 1} $. Furthermore, $\| {b - Ax_\infty } \|_\infty = \
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The Characterization of Best Nonlinear Tchebycheff Approximations
1959NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. Consider a continuous function, F([...]) of n parameters and [...]. Such a function is said to have Property NS if the following theorem is valid for every continuous function, f(x): THEOREM: F([...]) is a best approximation to f(x) if ...
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A Characterization of an Element of Best Simultaneous Approximation.
Numerical Functional Analysis and Optimization, 1983Deutsch [4] has suggested that some problems of best simultaneous approximation might profitably be viewed as problems of best approximation in an appropriate product space. A few authors have touched upon this approach; none, however, have pursued it consistently or developed a complete problem along such a line, even in the simplest of cases. In this
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A GENERAL CHARACTERIZATION OF BEST RESTRICTED RANGE APPROXIMATIONS
Quaestiones Mathematicae, 1997Abstract A general characterization of best restricted range approximations is established under an arbitrary norm on C [a, b]. Consequently, many previous characterizations of various best constrained approximations can be treated as special cases of this general characterization theorem.
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Characterization of Generalized Lipschitz Classes by Best Approximation with Splines
SIAM Journal on Numerical Analysis, 1974Equivalence theorems for best approximation by splines with equidistant nodes on a finite interval are established. They yield characterizations of the generalized Lipschitz spaces $(\omega _r (t,f)_p $ being the rth modulus of continuity) \[ Lip(\psi ,q,r;p) = \{ f \in L_p (a,b):\left\{ {\begin{array}{*{20}c} {\int_0^1 {[\psi ^{ - 1} (t)\omega _r (t,f)
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Characterization of best uniform approximation with restricted ranges of derivatives
Approximation Theory and its Applications, 1997Let \(K\) be a compact subset of the interval \([a,b]\) contained at least \(n+1\) points and let \(C(K)\) be the Banach space (with respect to the sup-norm) of all continuous real-valued functions defined on \(K\). Let also \(\{\varphi_1, \dots, \varphi_n\}\) be a linearly independent system of continuous functions on \([a,b]\) having derivatives of ...
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Approximation Theory and its Applications, 1991
Let \(R\) be a normed linear space, \(\{\phi_ i: i-1,2,\dots,n\}\) be \(n\) linearly independent elements of \(R\), \(\Phi_ n=\text{span}\{\phi_ 1,\dots,\phi_ n\}\) be the \(n\)-dimensional subspace of \(R\) (called the set of generalized polynomials) and \(K\) be an arbitrary convex subset of \(\Phi_ n\).
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Let \(R\) be a normed linear space, \(\{\phi_ i: i-1,2,\dots,n\}\) be \(n\) linearly independent elements of \(R\), \(\Phi_ n=\text{span}\{\phi_ 1,\dots,\phi_ n\}\) be the \(n\)-dimensional subspace of \(R\) (called the set of generalized polynomials) and \(K\) be an arbitrary convex subset of \(\Phi_ n\).
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