Uniqueness Of Best Approximation Research Articles

Proximinality and co-proximinality in metric linear spaces

As a counterpart to best approximation, the concept of best coapproximation was introduced in normed linear spaces by C. Franchetti and M. Furi in 1972. Subsequently, this study was taken up by many researchers. In this paper, we discuss some results on the existence and uniqueness of best approximation and best coapproximation when the underlying spaces are metric linear spaces.

Proximinality and co-proximinality in metric linear spaces

AbstractAs a counterpart to best approximation, the concept of best coapproximation was introduced in normed linear spaces by C. Franchetti and M. Furi in 1972. Subsequently, this study was taken up by many researchers. In this paper, we discuss some results on the existence and uniqueness of best approximation and best coapproximation when the underlying spaces are metric linear spaces

The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors

In this paper we discuss the notion of singular vector tuples of a complex-valued $$d$$ d -mode tensor of dimension $$m_1\times \cdots \times m_d$$ m 1 × ? × m d . We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the corresponding Segre product. We give the formula for the number of singular vector tuples. We show similar results for tensors with partial symmetry. We give analogous results for the homogeneous pencil eigenvalue problem for cubic tensors, i.e., $$m_1=\cdots =m_d$$ m 1 = ? = m d . We show the uniqueness of best approximations for almost all real tensors in the following cases: rank-one approximation; rank-one approximation for partially symmetric tensors (this approximation is also partially symmetric); rank- $$(r_1,\ldots ,r_d)$$ ( r 1 , ? , r d ) approximation for $$d$$ d -mode tensors.

Manifolds of semi-negative curvature

The notion of nonpositive curvature in Alexandrov's sense is extended to include p-uniformly convex Banach spaces. Infinite dimensional manifolds of semi-negative curvature with a p-uniformly convex tangent norm fall in this class on nonpositively curved spaces, and several well-known results, such as existence and uniqueness of best approximations from convex closed sets, or the Bruhat-Tits fixed point theorem, are shown to hold in this setting, without dimension restrictions. Homogeneous spaces G/K of Banach-Lie groups of semi-negative curvature are also studied, explicit estimates on the geodesic distance and sectional curvature are obtained. A characterization of convex homogeneous submanifolds is given in terms of the Banach-Lie algebras. A splitting theorem via convex expansive submanifolds is proven, inducing the corresponding splitting of the Banach-Lie group G. Finally, these notions are used to study the structure of the classical Banach-Lie groups of bounded linear operators acting on a Hilbert space, and the splittings induced by conditional expectations in such setting.

Criteria of strong nearest-cross points and strong best approximation pairs

The concept of strong nearest-cross point (strong n.c. point) is introduced, which is the generalization of strong uniqueness of best approximation from a single point. The relation connecting to localization is discussed. Some criteria of strong n.c. points are given. The strong best approximation pairs are also studied.

Strong uniqueness of best approximations in spaces of bounded linear operators

The present paper is concerned with problems of the strong uniqueness of the best approximation and the characterization of a uniqueness element in operator spaces. Some results on the strong uniqueness of the best approximation operator from RS-sets are proved and the uniqueness element of a sun in the compact operator space from C0 to C0 is characterized by the strict Kolmogorov’s condition. Some recent results due to Lewicki and others are extended and improved.

Perfect spline approximation

Our study of perfect spline approximation reveals: (i) it is closely related to ΣΔ modulation used in one-bit quantization of bandlimited signals. In fact, they share the same recursive formulae, although in different contexts; (ii) the best rate of approximation by perfect splines of order r with equidistant knots of mesh size h is h r−1 . This rate is optimal in the sense that a function can be approximated with a better rate if and only if it is a polynomial of degree < r. The uniqueness of best approximation is studied, too. Along the way, we also give a result on an extremal problem, that is, among all perfect splines with integer knots on R , (multiples of) Euler splines have the smallest possible norms.

Existence and uniqueness results for neural network approximations

Some approximation theoretic questions concerning a certain class of neural networks are considered. The networks considered are single input, single output, single hidden layer, feedforward neural networks with continuous sigmoidal activation functions, no input weights but with hidden layer thresholds and output layer weights. Specifically, questions of existence and uniqueness of best approximations on a closed interval of the real line under mean-square and uniform approximation error measures are studied. A by-product of this study is a reparametrization of the class of networks considered in terms of rational functions of a single variable. This rational reparametrization is used to apply the theory of Pade approximation to the class of networks considered. In addition, a question related to the number of local minima arising in gradient algorithms for learning is examined.

Uniqueness of Best Approximation with Coefficient Constraints

Uniqueness of Best Approximation with Coefficient Constraints

Best Approximations by Vector-Valued Monotone Increasing or Convex Functions

Let ( E, || · ||) be an ordered Banach space over the real held. Let BV be the space of all E-valued functions on a compact interval [ a, b] ⊂ R which are of bounded variation. When BV is endowed with a norm || f || v = || f( a)|| + v( f), f ∈ BV, where v( f) is a total variation of f on [ a, b], we are concerned with the existence and uniqueness of best approximations by monotone increasing or convex functions of BV.

On uniqueness of bestL1-approximation in sobolev spaces

In this paper we investigate the uniqueness of best approximation of continuous functions on [-a a] in the norm , by elements of Haar and A-spaces. It turns out that the size of the interval is crucial for uniqueness. In most cases we shall determine the exact set of a's for which uniqueness holds. We also show that in the case when r = 1 the approximation by Haar spaces yields uniqueness for every a.

Networks and the best approximation property

Networks can be considered as approximation schemes. Multilayer networks of the perceptron type can approximate arbitrarily well continuous functions (Cybenko 1988, 1989; Funahashi 1989; Stinchcombe and White 1989). We prove that networks derived from regularization theory and including Radial Basis Functions (Poggio and Girosi 1989), have a similar property. From the point of view of approximation theory, however, the property of approximating continuous functions arbitrarily well is not sufficient for characterizing good approximation schemes. More critical is the property ofbest approximation. The main result of this paper is that multilayer perceptron networks, of the type used in backpropagation, do not have the best approximation property. For regularization networks (in particular Radial Basis Function networks) we prove existence and uniqueness of best approximation.

On an inequality of Bynum and Drew

An extension of the Bynum-Drew inequality in the abstract L p space, 1 < p ⩽ 2, is presented. The extended inequality is applied to show the strong uniqueness of best approximations and existence of fixed points for uniformly Lipschitzian mappings and to prove an estimate for the self-Jung constants.

Some remarks on strong uniqueness of best approximation

In the paper we examine properties of strongly unique best approximation in terms of extremal functionals in an abstract normed linear space. Using some new or modified tools (e.g., the shadow of a set, strongly tangent sets, I-sets) we express criteria of strong uniqueness both in linear and nonlinear approximation. Some further remarks are contributed to the discussion on Poreda's problem concerning the behavior of the strong unicity constants and a detailed description of properties of the tangent cone of Dubovitsky-Milyutin is given.

Strong uniqueness of best approximations in an abstract L1 space

We prove that the set of elements in an abstract L 1 space, which have a strongly unique best approximation in a finite dimensional subspace, is dense in the set of elements having a unique best approximation in the subspace.

Uniqueness of best Chebyshev approximations in spline subspaces

This paper deals with the problem of uniqueness of best Chebyshev approximations by subspaces of spline functions on compact subsets T of R . Necessary and sufficient conditions ensuring uniqueness of best approximations are given and a characterization of strongly unique best approximations using best approximations on finite subsets of T is established. Moreover, problems where a best approximation is unique on an interval I but is not a unique best approximation on any finite subset are considered.

Uniqueness and strong uniqueness of best approximations by spline subspaces and other subspaces

A complete characterization is given of those functions in C¦a, b¦ which have a unique best approximation from the subspace of spline functions of degree n with k fixed knots. Also, the relationship between unique and strongly unique best approximations from arbitrary finitedimensional subspaces of C 0( T) is investigated.

Polynomials with positive coefficients: Uniqueness of best approximation

Polynomials with positive coefficients: Uniqueness of best approximation

Best approximation in von Neumann algebras

We investigate here the question of uniqueness of best approximation to operators in von Neumann algebras by elements of certain linear subspaces. Recall that a linear subspace V of a Banach space X is called a Chebyshev subspace if each vector in X has a unique best approximation by vectors in V. Our first main result characterizes the one-dimensional Chebyshev subspaces of a von Neumann algebra. This may be regarded as a generalization of a result of Stampfli [(4), theorem 2, corollary] which states that the scalar multiples of the identity operator form a Chebyshev subspace. Alternatively it may be regarded as a generalization of the commutative situation in which a continuous complex-valued function f on a compact Hausdorff space X spans a Chebyshev subspace of C(X) if and only if f does not vanish on X [(3), p. 215]. Our second main result is that a finite dimensional * subalgebra, of dimension &gt; 1, of an infinite dimensional von Neumann algebra cannot be a Chebyshev subspace. This imposes limits to further generalization of Stampfli's result.

Nonexistence of best alternating approximations on subsets

The existence of best Chebyshev approximations by an alternating family on closed subsets of an interval is considered. In varisolvent approximation, existence on subsets of sufficiently low density is guaranteed if the best approximation on the interval is of maximum degree. The paper studies the case in which the best approximation is not of maximum degree and shows that in many common cases, no such guarantee is possible. Let Y be a compact subset of [a,,/]] and C(Y) be the space of continuous functions on Y. For g E C(Y) define llglly = sup{lg(x)A: x E Y}, lIlgH = 110[al. Let F be an approximating function on [a,,8/] with parameter space P. The Chebyshev problem on Y is, given f E C(Y), to find a parameter A * to minimize lif F(A, .)IIy over A E P. Such a parameter A* is called best and F(A*, ) is called best to f on Y. We will consider only approximating functions for which best approximations on [a,,8]1 are characterized by alternation [10, pp. 17-21], [6], which implies uniqueness of best approximations. Following Cheney, we define the density of Y in [a,,8]1 to be IYI = max{inf{x -yl: y Y} a 0, there is a parameter B E P such that IF(B, a) -y < e and Received by the editors October 2, 1975. AMS (MOS) subject classifications (1970). Primary 41A50. Copyright ? 197'. Aniwrican Mathematical Society