Bases In Spaces Of Functions Research Articles

Criteria in Nuclear Fréchet Spaces and Silva Spaces with Refinement of the Cannon-Whittaker Theory

Along with the theory of bases in function spaces, the existence of a basis is not always guaranteed. The class of power series spaces contains many classical function spaces, and it is of interest to look for a criterion for this class to ensure the existence of bases which can be expressed in an easier form than in the classical case given by Cannon or even by Newns. In this article, a functional analytical method is provided to determine a criterion for basis transforms in nuclear Fréchet spaces ((NF)-spaces), which is indeed a refinement and a generalization of those given in this concern through the theory of Whittaker on polynomial bases. The provided results are supported by illustrative examples. Then, we give the necessary and sufficient conditions for the existence of bases in Silva spaces. Moreover, a nuclearity criterion is given for Silva spaces with bases. Subsequently, we show that the presented results refine and generalize the fundamental theory of Cannon-Whittaker on the effectiveness property in the sense of infinite matrices.

Spline wavelet bases in function spaces with Muckenhoupt weights

Properties of a class of spline wavelet systems of Battle–Lemarie type are described and applied to wavelet characterisation of Besov spaces and Triebel–Lizorkin spaces with Muckenhoupt weights.

Haar functions in weighted Besov and Triebel–Lizorkin spaces

The paper deals with Haar wavelet bases in function spaces of Besov and Triebel–Lizorkin type with local Muckenhoupt weights. We show that Haar wavelets can be used to characterize such function spaces as far as absolute value of smoothness parameter is small enough and weights fulfill some conditions. The result is based on mapping properties of linear operators involving characteristic functions of dyadic cubes in related spaces and on local means characterization of weighted Besov and Triebel–Lizorkin spaces.

Wavelet decomposition techniques and Hardy inequalities for function spaces on cubes

A rather tricky question is the construction of wavelet bases on domains for suitable function spaces (Sobolev, Besov, Triebel–Lizorkin type). In his monograph from 2008, Triebel presented an approach how to construct wavelet (Riesz) bases in function spaces of Besov and Triebel–Lizorkin type on cellular domains, in particular on the cube. However, he had to exclude essential exceptional values of the smoothness parameter s, for instance the theorems do not cover the Sobolev space W21(Q) on the n-dimensional cube Q for n≥2.Triebel also gave an idea how to deal with those exceptional values for the Triebel–Lizorkin function space scale on the n-dimensional cube Q: he suggested to introduce modified function spaces for the critical values, the so-called reinforced spaces which are subsets of the classical Triebel–Lizorkin function spaces on the cube. In this paper we start examining these reinforced spaces and transfer the crucial decomposition theorems necessary for establishing a wavelet basis from the non-critical values to analogous results for the critical cases now decomposing the reinforced function spaces of Triebel–Lizorkin type.

Lattice-subspaces and positive bases in function spaces

Let x1,...,xn be linearly independent, positive elements of the space RΩ of the real valued functions defined on a set Ω and let X be the vector subspace of RΩ generated by the functions xi. We study the problem: Does a finite-dimensional minimal lattice-subspace (or equivalently a finite-dimensional minimal subspace with a positive basis) of RΩ which contains X exist? To this end we define the function β(t)=\(\frac{1}{{z\left( t \right)}}\)(x1(t),x2(t),...,xn(t)), where z(t)=x1(t)+x2(t)+...+xn(t), which we call basic function and takes values in the simplex Δn of \(\mathbb{R}_ + ^n \). We prove that the answer to the problem is positive if and only if the convex hull K of the closure of the range of β is a polytope. Also we prove that X is a lattice-subspace (or equivalently X has positive basis) if and only if, K is an (n-1)-simplex. In both cases, using the vertices of K, we determine a positive basis of the minimal lattice-subspace. In the sequel, we study the case where Ω is a convex set and x1,x2,...,xn are linear functions. This includes the case where xi are positive elements of a Banach lattice, or more general the case where xi are positive elements of an ordered space Y. Based on the linearity of the functions xi we prove some criteria by means of which we study if K is a polytope or not and also we determine the vertices of K. Finally note that finite dimensional lattice-subspaces and therefore also positive bases have applications in economics.

Approximate identification for robust control

This paper discusses results related to approximate identification for robust control. The criteria for modeling and identification are formulated in terms of L ∞ or H ∞ norm. Emphasis is made on the construction of a model set by specifying bases in function spaces L 2, H 2 or in the disc algebra A . These bases include — besides the most widely used trigonometric basis— the recently introduced rational orthogonal bases and some wavelet bases. The construction of identification algorithms considered as bounded operators mapping measured noisy frequency response data to an element in the model space is discussed. Bounds on the operator norm effecting the approximation and the noise errors are given, too.

Approximate Identification for Robust Control

This paper discusses results related to approximate identification for robust control. The criteria for modeling and identification are formulated in terms of ℒ or ℋ norm. Emphasis is made on the construction of a model set by specifying bases in function spaces ℒ, ℋ or in the disc algebra A. These bases include - besides the most widely used trigonometric basis- the recently introduced rational orthogonal bases and some wavelet bases. The construction of identification algorithms considered as bounded operators mapping measured noisy frequency response data to an element in the model space is discussed. Bounds on the operator norm effecting the approximation and the noise errors are given, too.

Explicit Orthonormal Fixed Bases for Spaces of Functions that are Totally Symmetric Under the Rotational Symmetries of a Platonic Solid

Explicit complete orthonormal fixed bases are computed for subspaces of the space of square-integrable functions on the sphere where the subspaces contain functions that are totally symmetric under the rotational symmetries of a Platonic solid. Each function in the fixed basis is a linear combination of spherical harmonics of fixed l. For each symmetry (icosahedral/dodecahedral, octahedral/cubic, tetrahedral), the calculation has three steps: First, a bilinear equation is derived for the coefficients in the linear combination by equating the expansion of a symmetrized δ function in both spherical harmonics and the fixed basis functions for the appropriate subspace. The equation is parameterized by the location (θ0, ϕ0) of the δ function and must be satisfied for all locations. Second, the dependence on the δ-function location is expressed in a Fourier (ϕ0) and a Taylor (θ0) series and thereby a new system of bilinear equations is derived by equating selected coefficients. Third, a recursive solution of the new system is derived and the recursion is solved explicitly with the aid of symbolic computation. The results for the icosahedral case are important for structural studies of small spherical viruses.

ON UNCONDITIONAL CONVERGENCE WITH RESPECT TO SYSTEMS OF PRODUCTS OF BASES

Decisive conditions are established for unconditionality of expansions with respect to bases in spaces of functions of several variables. The properties of the multiple Haar and Franklin systems are studied. Bibliography: 22 titles.

Existence of continuable bases in spaces of functions, analytic in compacta

Existence of continuable bases in spaces of functions, analytic in compacta

Extendable bases in spaces of functions analytic in multicircular domains

Extendable bases in spaces of functions analytic in multicircular domains