Existence Of Best Approximation Research Articles

Best approximation in b-modular spaces

In this paper, some basic notions and facts in the b-modular space similar to those in the modular spaces as a type of generalization are given. For example, concepts of convergence, best approximate, uniformly convexity etc. And then, two results about relation between semi compactness and approximation are proved which are used to prove a theorem on the existence of best approximation for a semi-compact subset of b-modular space.

Functions Approximation by Spectral Graph Wavelets

Approximating function by using Spectral Graph Wavelets is an interesting direction in approximation theory. We essential to well choosing the space of functions that are approximated by Spectral Graph Wavelets. spaces of functions are fantastic choices to study It is more interesting to take the value . In this paper , new formulas of Spectral Graph Wavelets were constructed and proved to get good rates of approximation. Fundamental properties of Graph Wavelets transform ( GWT) are studied, such as, inversion , scaling Limit and approximation Wavelets. Finally , existence of best approximation can be concluded here for Graph functions in terms of SGWT.

Some Results on Best Approximations without star-shaped

In this paper, we give two theorems in Banach space. The first one is about the existence of a fixed point for generalized affine mapping, and a class of relatively nonexpansive commutative mappings and then we use it in the proof of theorem about existence of best approximation.

Uniformly and locally convex asymmetric spaces

The nonemptyness of the intersections of nested systems of convex bounded closed subsets of uniformly convex asymmetric spaces is studied. The density properties of the points of existence and points of approximative uniqueness are examined for nonempty closed subsets of uniformly convex asymmetric spaces. Problems of the existence and stability of Chebyshev centres are considered; the relationships between $\gamma$-suns, suns and the existence of best approximants are investigated. Sufficient conditions for radial $\delta$-solarity are obtained. Bibliography: 27 titles.

Approximative compactness of linear combinations of characteristic functions

Best approximation by the set of all n-fold linear combinations of a family of characteristic functions of measurable subsets is investigated. Such combinations generalize Heaviside-type neural networks. Existence of best approximation is studied in terms of approximative compactness, which requires convergence of distance-minimizing sequences. We show that for (Ω,μ) a measure space, in Lp(Ω,μ) with 1≤p≤∞ and for all n≥1, compact families of characteristic functions of sets (of finite measure for p<∞) generate approximatively compact n-fold linear spans. Results are illustrated by examples of continuously parametrized sets.

Tree-based tensor formats

The main goal of this paper is to study the topological properties of tensors in tree-based Tucker format. These formats include the Tucker format and the Hierarchical Tucker format. A property of the so-called minimal subspaces is used for obtaining a representation of tensors with either bounded or fixed tree-based rank in the underlying algebraic tensor space. We provide a new characterisation of minimal subspaces which extends the existing characterisations. We also introduce a definition of topological tensor spaces in tree-based format, with the introduction of a norm at each vertex of the tree, and prove the existence of best approximations from sets of tensors with bounded tree-based rank, under some assumptions on the norms weaker than in the existing results.

Best Approximation in Modular Spaces By Type of Nonexpansive Maps

This paper presents results about the existence of best approximations via nonexpansive type maps defined on modular spaces.

Isotone cones in Banach spaces and applications to best approximations of operators without continuity conditions

In this paper, we introduce the concept of isotone cones in Banach spaces. Then, we apply the order monotonic property of the metric projection operator to prove the existence of best approximations for some operators without continuity conditions in partially ordered Banach spaces.

On the Existence of Optimal Unions of Subspaces for Data Modeling and Clustering

Given a set of vectors F={f 1,…,f m } in a Hilbert space $\mathcal {H}$, and given a family $\mathcal {C}$ of closed subspaces of $\mathcal {H}$, the subspace clustering problem consists in finding a union of subspaces in $\mathcal {C}$ that best approximates (is nearest to) the data F. This problem has applications to and connections with many areas of mathematics, computer science and engineering, such as Generalized Principal Component Analysis (GPCA), learning theory, compressed sensing, and sampling with finite rate of innovation. In this paper, we characterize families of subspaces $\mathcal {C}$ for which such a best approximation exists. In finite dimensions the characterization is in terms of the convex hull of an augmented set $\mathcal {C}^{+}$. In infinite dimensions, however, the characterization is in terms of a new but related notion; that of contact half-spaces. As an application, the existence of best approximations from π(G)-invariant families $\mathcal {C}$ of unitary representations of Abelian groups is derived.

Spectral numbers in Floer theories

AbstractThe chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the ‘nondegenerate spectrality’ axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and apply to any Floer-type theory (including Novikov homology) satisfying certain standard formal properties. The key ingredient is a theorem about the existence of best approximations of arbitrary elements of finitely generated free modules over Novikov rings by elements of prescribed submodules with respect to a certain family of non-Archimedean metrics.

On the existence of the best discrete approximation in [formula omitted] norm by reciprocals of real polynomials

For the given data ( w i , x i , y i ) , i = 1 , … , M , we consider the problem of existence of the best discrete approximation in l p norm ( 1 ≤ p < ∞ ) by reciprocals of real polynomials. For this problem, the existence of best approximations is not always guaranteed. In this paper, we give a condition on data which is necessary and sufficient for the existence of the best approximation in l p norm. This condition is theoretical in nature. We apply it to obtain several other existence theorems very useful in practice. Some illustrative examples are also included.

Best Approximation of Gaussian Neural Networks With Nodes Uniformly Spaced

This paper is aimed at exposing the reader to certain aspects in the design of the best approximants with Gaussian radial basis functions (RBFs). The class of functions to which this approach applies consists of those compactly supported in frequency. The approximative properties of uniqueness and existence are restricted to this class. Functions which are smooth enough can be expanded in Gaussian series converging uniformly to the objective function. The uniqueness of these series is demonstrated by the context of the orthonormal basis in a Hilbert space. Furthermore, the best approximation to a given band-limited function from a truncated Gaussian series is analyzed by an energy-based argument. This analysis not only gives a theoretical proof concerned with the existence of best approximations but addresses the problems of architectural selection. Specifically, guidance for selecting the variance and the oversampling parameters is provided for practitioners.

Properties of fixed point set of a multivalued map

Properties of the set of fixed points of some discontinuous multivalued maps in a strictly convex Banach space are studied; in particular, affirmative answers are provided to the questions related to set of fixed points and posed by Ko in 1972 and Xu and Beg in 1998. A result regarding the existence of best approximation is derived.

Discretization and best approximation in Besov spaces

In this paper we prove a theorem about existence of best approximation in a class of spaces involving Besov spaces, via a discretization technique. It is a consequence of this theorem that rational functions and exponencial sums are proximinal subsets of B ∞,q a (π). It is also proved the proximinality of R m n [a, b] in B p,q a (π) for arbitrary p,q and a.

Proximal Mappings

We study differentiability properties and subdifferentiability properties of the Baire approximate and of the Moreau–Yosida approximate of a nonconvex function on a Banach space. These properties are intimately linked with exactness of the infimal convolution defining the approximation. When applied to indicator functions of possibly nonconvex subsets, our results yield existence of best approximations under subdifferentiability assumptions on the distance function and suitable smoothness assumptions on the space.

ON THE THEORY OF SINGULAR EXPANSION IN A TENSOR PRODUCT OF HILBERT SPACES

Approximation of an element in a tensor product of Hilbert spaces by a sum of products of elements in each of these spaces is considered. The existence of best approximation and the convergence of bilinear series are proved. An explicit representation for the best approximation is obtained. Analogous results in the Sobolev spaces are given as corollaries. Bibliography: 20 titles.

Approximation in function modules

We investigate the existence of best approximation of an element α in a function module from a subfunction module whose fibers satisfy the intersection property of balls. Also we investigate the lower semicontinuity of the metric projection associated with such a subfunction module.

Multivalued maps and the existence of best approximants

Some existence of the best approximants in normed linear space is presented. A fixed point theorem for uppersemicontinuous multivalued maps with nonconvex values is used for this purpose. Previous results due to different authors are covered and extended through our method.

Best approximants in modular function spaces

In this paper we consider the existence of best approximants in modular function spaces by elements of sublattices. Modular function spaces are the natural generalization of L p , p > 0, Orlicz, Lorentz, and Köthe spaces. Let ϱ be a pseudomodular, L ϱ the corresponding modular function space, and C a sublattice of L ϱ . Given a function f ∈ L ϱ we consider the minimization problem of finding h ∈ C such that ϱ( f − h) = inf{ f − g: g ∈ C}. Such an h is called a best approximant. Problems of finding best approximants are important in approximation theory and probability theory. In the case where C is L ϱ( B ) for some σ-subalgebra B of the original σ-algebra, finding best approximants is closely related to the problem of nonlinear prediction. Throughout most of the paper we assume only that ϱ is a pseudomodular and except in one section, we do not assume ϱ to be orthogonally additive. This allows, for instance, application to Lorentz type L p spaces. If ϱ is a semimodular or a modular, then L ϱ can be equipped with an F-norm ∥ · ∥ ϱ and one considers the corresponding F-norm minimization problem. This paper gives several existence theorems relating to this problem, a theorem comparing the set of all best ϱ-approximants with the set of all best ∥ · ∥ ϱ-approximants and a uniqueness theorem.

A theorem on best approximations

In this paper a theorem on the existence of best approximation for an approximatively compact subset of a normed space is proved. The result herein contains a recent result of Prolla.