Approximation Of Functions In Spaces Research Articles

Combining Nyström Methods for a Fast Solution of Fredholm Integral Equations of the Second Kind

In this paper, we propose a suitable combination of two different Nyström methods, both using the zeros of the same sequence of Jacobi polynomials, in order to approximate the solution of Fredholm integral equations on [−1,1]. The proposed procedure is cheaper than the Nyström scheme based on using only one of the described methods . Moreover, we can successfully manage functions with possible algebraic singularities at the endpoints and kernels with different pathologies. The error of the method is comparable with that of the best polynomial approximation in suitable spaces of functions, equipped with the weighted uniform norm. The convergence and the stability of the method are proved, and some numerical tests that confirm the theoretical estimates are given.

Spherical Approximation on Unit Sphere

In this paper we introduce a Jackson type theorem for functions in LP spaces on sphere And study on best approximation of functions in spaces defined on unit sphere.
 our central problem is to describe the approximation behavior of functions in spaces for by modulus of smoothness of functions.

Stable and convergent approximation of two-dimensional vector fields on unstructured meshes

A new framework is proposed for analyzing staggered-grid finite difference finite volume methods on unstructured meshes. The new framework employs the concept of external approximation of function spaces, and gauge convergence of numerical schemes through the quantities of vorticity and divergence, instead of individual derivatives of the velocity components. The construction of a stable and convergent external approximation of a simple but relevant vector-valued function space is demonstrated, and the new framework is applied to establish the convergence of the MAC scheme for the incompressible Stokes problem on unstructured meshes.

Topological Dual Systems for Spaces of Vector Measurep-Integrable Functions

We show a Dvoretzky-Rogers type theorem for the adapted version of theq-summing operators to the topology of the convergence of the vector valued integrals on Banach function spaces. In the pursuit of this objective we prove that the mere summability of the identity map does not guarantee that the space has to be finite dimensional, contrary to the classical case. Some local compactness assumptions on the unit balls are required. Our results open the door to new convergence theorems and tools regarding summability of series of integrable functions and approximation in function spaces, since we may find infinite dimensional spaces in which convergence of the integrals, our vector valued version of convergence in the weak topology, is equivalent to the convergence with respect to the norm. Examples and applications are also given.

Nonlinear approximation of function spaces of mixed smoothness

We study the approximative properties of Lq-greedy algorithms with respect to the wellknown system Ud of shifts of Dirichlet kernels on the Nikol′skiĭ-Besov classes \(SB_{p\theta }^r (\mathbb{T}^d )\) and the Lizorkin-Triebel classes \(SF_{p\theta }^r (\mathbb{T}^d )\) of functions of mixed smoothness.

State space model of the direct methanol fuel cell

In this report we propose a state space model of a direct methanol fuel cell. This dynamical model allows analysis and optimisation of the fuel cell in a control-theoretical framework, for instance to improve the overall dynamics of the fuel cell or to reject environmental disturbances. The model is particularly well suited to synthesise controllers, which are necessary to ensure stable, robust and efficient operation of a fuel cell. To yield a state space model of the direct methanol fuel cell, a system of nonlinear partial differential equations is transformed into a system of ordinary initial value problems by Laplace transforms, approximation in function spaces and the method of weighted residuals.

Quantum approximation I. Embeddings of finite-dimensional Lp spaces

We study approximation of embeddings between finite-dimensional L p spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The results show that for certain regions of the parameter domain quantum computation can essentially improve the rate of convergence of classical deterministic or randomized approximation, while there are other regions where the best possible rates coincide for all three settings. These results serve as a crucial building block for analyzing approximation in function spaces in a subsequent paper (Quantum approximation II. Sobolev Embeddings, J. Complexity, submitted).

Positive operators and approximation in function spaces on completely regular spaces

We discuss the approximation properties of nets of positive linear operators acting on function spaces defined on Hausdorff completely regular spaces. A particular attention is devoted to positive operators which are defined in terms of integrals with respect to a given family of Borel measures. We present several applications which, in particular, show the advantages of such a general approach. Among other things, some new Korovkin‐type theorems on function spaces on arbitrary topological spaces are obtained. Finally, a natural extension of the so‐called Bernstein‐Schnabl operators for convex (not necessarily compact) subsets of a locally convex space is presented as well.

Daugavet's Equation and Operators on L 1 (μ)

Generalizing a result of Babenko and Pichugov, it is shown that if T is a weakly compact operator on L1 (It), where 1A is a a-finite nonatomic measure, then III + TIl = 1 + 1ITIh. A characterization of all operators T on L' (p) having this property is also given. In [2] Daugavet proved that if T is a compact operator on C(,i), then I + TiI = 1 + 11ThI, while Babenko and Pichugov [1] subsequently showed the same is true for a compact operator on L'(IL) (see [61 for recent extensions). In general, if X is a Banach space and T E L(X), then T is said to satisfy Daugavet's equation [4] if III TiI = 1 + 1.ITIh. This property of an operator arises naturally in the consideration of problems of best approximation in function spaces where it has been utilized by a number of authors (e.g. [4], and the references cited in [1]). In a recent paper [5] it was shown that Daugavet's equation actually holds for any weakly compact operator on C(,u). The purpose of this note is to give an analogous extension of the theorem of Babenko and Pichugov to weakly compact operators by proving that if , is a a-finite nonatomic measure on a space S, and T is a weakly compact operator on L1(I), then IlI + TII = 1 + IITII (Theorem 1). Related results concerning extensions and generalizations of this theorem, including a characterization of all operators T on L1 (it) for which IIITII = 1+ lT II (Theorem 2) are also given. We begin with a simple result concerning the norming of operators on Ll (,u). As usual we denote by L(X) the space of all bounded linear operators on a Banach space X, by XA the characteristic function of a measurable set A c S, and by IIf! II and IgIk0, the norms of functions f E L1(,V) and g E L??(,i), respectively. The complement of a set A in S is denoted by A'. PROPOSITION 1. Let T E L(L1('i)). Then given any E > 0 there is a measurable set A C S for which 0 ITIh E. PROOF. Since it is a-finite, (Ll(IL))* = L??(,t). Therefore if T E LZ(L'(,it)), then T* E L(LOO (,i)) and IITII = IIT*II = sup11 g 11= =T* g I I Hence given any e > 0 there is a function g E L??(I,) for which higilgo = 1 and IIT*glloo > ITIh e/2. Since IIT*g I I = ess supt I(T*g)(t)I it follows that there is a measurable set A C S for which m(A) > 0 and I(T*g)(t)l > 11TIh E for all t E S. Replacing A (if necessary) by a subset B C A for which 0 < m(B) < E and on which the sign of (T*g)(t) is constant (which may be done since it is nonatomic), and replacing g by (-g) if Received by the editors April 7, 1986. 1980 Mathematics Sject Clssfication (1985 Revision). Primary 47B38, 47A30; Secondary 46E15.

Rational approximation and smoothness of functions

In the paper one obtains sharp results, yielding an explicit description of the classes of functions, defined on a segment, which are determined in terms of the best approximations by rational functions. Moreover, in the case of a segment, the deviation from rational functions is measured in the norm of BMO, while in the case of domains G in the norm of the space For the proof one makes use of results, obtained earlier by the author, on the rational approximation in spaces of functions, analytic in the unit circle, and of E. M. Dyn'kin's results on the invariance of the Besov classes under Faber mappings.

On the approximation of function spaces in the calculus of variations

On the approximation of function spaces in the calculus of variations