Inverse Approximation Theorems Research Articles

Inverse theorems for approximation by harmonic functions

Inverse theorems for approximation by harmonic functions

The approximate identity kernels of product type for the Walsh system

At present there are only a few approximate identity kernels for the Walsh system, for example, the p N -truncated Dirichlet kernel D p N − 1 ( t) = ∑ j = 0 p N − 1 w j ( t) [6]; the Abel-Poisson kernel λ γ ( t) = ∑ k = 0 ∞ γ k w k ( t) [3], and so on. In [6], Zheng has introduced a new kind of approximate identity kernels for the Walsh system—the kernels of product type. In the present paper we discuss the approximation properties of such product type kernels. Estimates of their moments as well as a direct approximation theorem are obtained. Then, to establish an inverse approximation theorem, we need the p-adic derivative of product type kernels and we estimate this derivative in L 1-norm.

Construction of homogeneous spaces of local functions and inverse approximation theorems

The paper's first part is devoted to the construction of systems, finitary inR n, of functions for approximations of the form The concepts of ℓ-approximating and-interpolating systems of functions are introduced, methods of constructing them are examined, and the labor consumption of the matrices of a variational-difference method is discussed when different systems of functions are used. The paper's second part contains inverse approximation theorems for metric spaces and their application in the case of approximations by spaces of local functions in Sobolev norms.

Thep-Version of the Finite Element Method

In the p-version of the finite element method, the triangulation is fixed and the degree p, of the piecewise polynomial approximation, is progressively increased until some desired level of precision is reached. In this paper, we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation. These properties lead to an a priori estimate of the asymptotic rate of convergence of the p-version. The estimate shows that the p-version gives results which are not worse than those obtained by the conventional finite element method (called the h-version, in which h represents the maximum diameter of the elements), when quasi-uniform triangulations are employed and the basis for comparison is the number of degrees of freedom. Furthermore, in the case of a singularity problem, we show (under conditions which are usually satisfied in practice) that the rate of convergence of the p-version is twice that of the h-version with quasi-uniform mesh. Inverse approximation theorems which determine the smoothness of a function based on the rate at which it is approximated by piecewise polynomials over a fixed triangulation are proved for both singular and nonsingular problems. We present numerical examples which illustrate the effectiveness of the p-version for a simple one-dimensional problem and for two problems in two-dimensional elasticity. We also discuss roundoff error and computational costs associated with the p-version. Finally, we describe some important features, such as hierarchic basis functions, which have been utilized in COMET-X, an experimental computer implementation of the p-version.

An elementary approach to inverse approximation theorems

An elementary approach to inverse approximation theorems

Direct and Inverse Theorems for Multidimensional Spline Approximation

Direct and Inverse Theorems for Multidimensional Spline Approximation

INVERSE APPROXIMATION THEOREMS ON REGULAR COMPACTA OF THE COMPLEX PLANE

We consider compacta which are regular in Dirichlet's problem, and establish inverse approximation theorems for them.

Direct and inverse approximation theorems in Banach spaces and Fourier transforms

An author's summary of a thesis submitted for the degree of Doctor of Physico-Mathematical Sciences. The dissertation was defended December 19, 1969, at a session of the scientific council of the Mechanical-Mathematics Faculty of the M. V. Lomonosov Moscow State University. The official opponents were Professors V. K. Dzyadyk, B. M. Levitan, and S. B. Stechkin, all Doctors of physico-mathematical sciences.