Inverse Theorems Of Approximation Theory Research Articles

On the constants in abstract inverse theorems of approximation theory

In the classical inverse theorems of constructive function theory, structural characteristics of an approximated function are estimated in terms of its best approximations. Most of the known proofs of the inverse theorems utilize Bernstein’s idea to expand the function in polynomials of its best approximation. In the present paper, Bernstein’s proof is modified by using integrals instead of sums. With this modification, it turns out that desired inequalities are based on identities similar to Frullani integrals. The considerations here are quite general, which allows one to obtain analogs of the inverse theorems for functionals in abstract Banach or even seminormed spaces. Then these abstract results are specified and inverse theorems in concrete spaces of functions are deduced, including weighted spaces, with explicit constants.

Direct and inverse theorems of approximation theory in Lebesgue spaces with Muckenhoupt weights

Direct and inverse theorems of approximation theory in Lebesgue spaces with Muckenhoupt weights

Some Theorems of Approximation Theory in Weighted Smirnov Classes with Variable Exponent

Let $${ G\subset {\mathbb {C}} }$$ be a Jordan domain with rectifiable Dini smooth boundary $$\varGamma $$. In this work, we investigate approximation properties of matrix transforms constructed via Faber series in weighted Smirnov classes with variable exponent. Moreover, direct and inverse theorems of approximation theory in weighted Smirnov classes with variable exponent are proved and some results related to constructive characterization in generalized Lipschitz classes are obtained.

Fractional Smoothness in Lp with Dunkl Weight and Its Applications

We define fractional power of the Dunkl Laplacian, fractional modulus of smoothness and fractional $K$-functional in $L^p$-space with the Dunkl weight. As application, we prove direct and inverse theorems of approximation theory, and some inequalities for entire functions of spherical exponential type in fractional settings.

Positive $$L^p$$ L p -Bounded Dunkl-Type Generalized Translation Operator and Its Applications

We prove that the spherical mean value of the Dunkl-type generalized translation operator $$\tau ^y$$ is a positive $$L^p$$ -bounded generalized translation operator $$T^t$$ . As applications, we prove the Young inequality for a convolution defined by $$T^t$$ , the $$L^p$$ -boundedness of $$\tau ^y$$ on radial functions for $$p>2$$ , the $$L^p$$ -boundedness of the Riesz potential for the Dunkl transform, and direct and inverse theorems of approximation theory in $$L^p$$ -spaces with the Dunkl weight.

Direct and inverse theorems of approximation theory

In this paper, we prove analogues of direct and some inverse theorems, for the Dunkl harmonic analysis, using the function with bounded spectrum and generalized spherical mean operator.

Approximation in weighted generalized grand smirnov classes

Let G be a finite simple connected domain in the complex plane C, bounded by a Carleson curve Γ := ∂G. In this work the direct and inverse theorems of approximation theory by the algebraic polynomials in the weighted generalized grand Smirnov classes εp),θ(G,ω) and , 1 < p < ∞, in the term of the rth, r = 1, 2,..., mean modulus of smoothness are proved. As a corollary the constructive characterizations of the weighted generalized grand Lipschitz classes are obtained.

Approximation problems in the Lebesgue spaces with variable exponent

In the variable exponent Lebesgue space, the r-th modulus of smoothness (r=1,2,...) is defined and in this term, the direct and inverse theorems of approximation theory are proved. Moreover, the constructive characterization problems for the some subclasses are discussed.

Estimates of the best approximations of functions with a spectrum with a given majorant

Assessment of the best approximations of functions through its modulus of smoothness (direct theorems of approximation theory or theorems of Jackson type) and of modulus of smoothness through its best approximation in a given metric (inverse theorems of approximation theory or theorems similar to Bernstein's theorem) by orthogonal systems have been the subject of research of many generations of mathematicians. The relevance of this topic is determined by the numerical analysis, particularly with the development of computer technology. This topic is naturally linked to the further development of the tasks aiming to estimate the closest approximation of a function, approximation of inequalities of Bernstein and Nikolskiy, and of approximation theory [1-8].

Best approximations of periodic functions in generalized lebesgue spaces

In generalized Lebesgue spaces with variable exponent, we determine the orders of the best approximations in the classes of (ψ; β)-differentiable 2π-periodic functions, deduce an analog of the well-known Bernstein inequality for the (ψ; β)-derivative, and apply this inequality to prove the inverse theorems of approximation theory in these classes.

On the connection between the best approximation by algebraic polynomials and the modulus of smoothness of order r

We introduce a family of asymmetric generalized translations. In terms of these operators, we define generalized moduli of smoothness for which the direct and inverse theorems of approximation theory are proved.

Approximation by polynomials and rational functions in weighted rearrangement invariant spaces

Let Γ be a Dini-smooth curve in the complex plane, and let G : = Int Γ . We prove some direct and inverse theorems of approximation theory by algebraic polynomials and rational functions in the weighted rearrangement invariant Smirnov spaces E X ( G , ω ) defined on G.

Refined theorems of approximation theory in the space of p-absolutely continuous functions

In this paper, we prove direct and inverse theorems of approximation theory in the space of p-absolutely continuous functions which generalize Terekhin’s results in the same way as Timan’s results in Lp generalize the classical theorems of approximation theory. The main theorems are refined for functions with quasimonotone Fourier coefficients and, in a number of cases, the resulats are shown to be sharp.

Approximation in weighted Smirnov-Orlicz classes

In this work some direct and inverse theorems of approximation theory in the weighted Smirnov-Orlicz classes, defined in the domains with a Dini-smooth boundary, are proved. In particular, a constructive characterization of the generalized Lipschitz classes $Lip^{*}\alpha (M,\omega )$, $\alpha >0$, is obtained.

Properties of a Family of Operators

In this paper we define a family of new nonsymmetric operators of generalized translation and describe their properties. For each of these operators, we introduce a generalized modulus of smoothness, for which the direct and inverse theorems of approximation theory are given.

Прямая и обратная теоремы теории приближений для одного обобшенного модуля гладкости

An asymmetric operator of generalized translation is introduced in this paper. Using this operator, we define a generalized modulus of smoothness and prove direct and inverse theorems of approximation theory for it.

APPROXIMATION OF SOLUTIONS OF THE EQUATION $ \overline\partial^jf=0$, $ j\geq1$, IN DOMAINS WITH QUASICONFORMAL BOUNDARY

This article establishes direct and inverse theorems of approximation theory (of the same type as theorems of Dzyadyk) that describe the quantitative connection between the smoothness properties of solutions of the equation , , and the rate of their approximation by module polynomials of the form In particular, a constructive characterization is obtained for generalized Holder classes of such functions on domains with quasiconformal boundary.Bibliography: 19 titles.