Inverse Approximation Theorems Research Articles

Exact constants in direct and inverse approximation theorems for functions of several variables in the spaces Sp

In the paper, exact constants in direct and inverse approximation theorems for functions of several variables are found in the spaces Sp. The equivalence between moduli of smoothness and some K-functionals is also shown in the spaces Sp.

A Bernstein type inequality for the Askey–Wilson operator

We establish a Riesz-type interpolation formula on the interval [−1,1] for the Askey–Wilson operator. As consequences, sharp Bernstein inequality and Markov inequality are obtained when differentiation is replaced by the Askey–Wilson operator. Moreover, an inverse approximation theorem is proved using a Bernstein type inequality in L2-space. We conclude our paper with an overconvergence result which is applied to characterize all q-differentiable functions of Brown and Ismail.

Convolution and Jackson inequalities in Musielak–Orlicz spaces

In the present work we prove some direct and inverse theorems for approximation by trigonometric polynomials in Musielak-Orlicz spaces. Furthermore, we get a constructive characterization of the Lipschitz classes in these spaces.

Some Inverse and Simultaneous Approximation Theorems in Weighted Variable Exponent Lebesgue Spaces

In this work we prove some inverse and simultaneous approximation theorems in the weighted variable exponent Lebesgue space and discuss the constructive description problems in the generalized Lipschitz subclasses.

Asymptotic estimates of p-variational and L p -modules of continuity of functions of certain classes

We obtain sharp estimates for upper bound of p-variational or Lp-modules of continuity on classes of functions with given majorants of trigonometric best approximations or modules of continuity. We also obtain refined direct and inverse theorems of approximation in p-variational metric for conjugate function.

On approximation of functions by algebraic polynomials in Hölder spaces

We study approximation of functions by algebraic polynomials in the Hölder spaces corresponding to the generalized Jacobi translation and the Ditzian–Totik moduli of smoothness. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria of the precise order of decrease of the best approximation in these spaces. Moreover, we obtain strong converse inequalities for some methods of approximation of functions. As an example, we consider approximation by the Durrmeyer–Bernstein polynomial operators.

Uniform polynomial approximation with A⁎ weights having finitely many zeros

We prove matching direct and inverse theorems for uniform polynomial approximation with A⁎ weights (a subclass of doubling weights suitable for approximation in the L∞ norm) having finitely many zeros and not too “rapidly changing” away from these zeros. This class of weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian–Totik weights. Main part and complete weighted moduli of smoothness are introduced, their properties are investigated, and equivalence type results involving related realization functionals are discussed.

Sharp estimates of approximation of periodic functions in Hölder spaces

The main purpose of the paper is to study sharp estimates of approximation of periodic functions in the Hölder spaces Hpr,α for all 0<p≤∞ and 0<α≤r. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria for the precise order of decrease of the best approximation in these spaces. Moreover, we obtained strong converse inequalities for general methods of approximation of periodic functions in Hpr,α.

Polynomial approximation with doubling weights having finitely many zeros and singularities

We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights w having finitely many zeros and singularities (i.e., points where w becomes infinite) on an interval and not too “rapidly changing” away from these zeros and singularities. This class of doubling weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian–Totik weights. We approximate in the weighted Lp (quasi) norm ‖f‖p,w with 0<p<∞, where ‖f‖p,w:=(∫−11∣f(u)∣pw(u)du)1/p. Equivalence type results involving related realization functionals are also discussed.

Approximation by polynomials with respect to multiplicative systems in weighted L p -spaces

We study best approximations of polynomials with respect to multiplicative systems in the Lp-spaces with Muckenhoupt weights. Using Jackson’s and Bernstein’s inequalities, we obtain the direct and inverse approximation theorems in terms of the K-functional and the inverse theorem of the Timan-Besov type. In the case of a power weight, we give a criterion for the membership of a function in the weighted Lp-space in terms of the Fourier coefficients with respect to multiplicative systems.

Pointwise approximation for a type of Bernstein-Durrmeyer operators

We give the direct and inverse approximation theorems for a new type of Bernstein-Durrmeyer operators with the modulus of smoothness. MSC:41A25, 41A27, 41A36.

Direct and inverse approximation theorems for the [formula omitted]-version of the finite element method in the framework of weighted Besov spaces, part III: Inverse approximation theorems

This is the third of a series devoted to the direct and inverse approximation theorems of the p-version of the finite element method in the framework of the Jacobi-weighted Besov and Sobolev spaces. In this paper we derive the inverse algebraic approximation in terms of the Jacobi-weighted Besov spaces and prove the inverse theorems for the finite element solutions of the p-version in the Chebyshev-weighted Besov spaces based upon the convergence rate measured in the energy norms.

Formula omitted] Bernstein inequalities and inverse theorems for RBF approximation on [formula omitted

Bernstein inequalities and inverse theorems are a recent development in the theory of radial basis function (RBF) approximation. The purpose of this paper is to extend what is known by deriving Lp Bernstein inequalities for RBF networks on Rd. These inequalities involve bounding a Bessel-potential norm of an RBF network by its corresponding Lp norm in terms of the separation radius associated with the network. The Bernstein inequalities will then be used to prove the corresponding inverse theorem.

The modified P-integral and P-derivative and their applications

The paper is concerned with properties of the modified P-integral and P-derivative, which are defined as multipliers with respect to the generalized Walsh-Fourier transform. Criteria for a function to have a representation as the P-integral or P-derivative of an Lp-function are given, and direct and inverse approximation theorems for P-differentiable functions are established. A relation between the approximation properties of a function and the behaviour of P-derivatives of the appropriate approximate identity is obtained. Analogues of Lizorkin and Taibleson's results on embeddings between the domain of definition of the P-derivative and Hölder-Besov classes are established. Some theorems on embeddings into BMO, Lipschitz and Morrey spaces are proved.Bibliography: 40 titles.

Direct and inverse theorems of rational approximation in the Bergman space

For positive numbers and let denote the Bergman space of analytic functions in the half-plane . For let be the best approximation by rational functions of degree at most . Also let and be numbers such that and . Then the main result of the paper claims that the set of functions such that is precisely the Besov space of analytic functions in .Bibliography: 23 titles.

Trigonometric approximation of functions in generalized Lebesgue spaces with variable exponent

We investigate the approximation properties of the trigonometric system in $ L_{2\pi }^{p\left( \cdot \right)} $ . We consider the moduli of smoothness of fractional order and obtain direct and inverse approximation theorems together with a constructive characterization of a Lipschitz-type class.

Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth

Abstract This work deals with basic approximation problems such as direct, inverse and simultaneous theorems of trigonometric approximation of functions of weighted Lebesgue spaces with a variable exponent on weights satisfying a variable Muckenhoupt A p(·) type condition. Several applications of these results help us transfer the approximation results for weighted variable Smirnov spaces of functions defined on sufficiently smooth finite domains of complex plane ℂ.

Direct and Inverse Approximation Theorems for Baskakov Operators with the Jacobi‐Type Weight

We introduce a new norm and a new K‐functional . Using this K‐functional, direct and inverse approximation theorems for the Baskakov operators with the Jacobi‐type weight are obtained in this paper.

Some inverse theorems for approximation of functions by the Fourier-Laguerre sums

In this paper we prove two inverse theorems for approximation of functions of two variables by Fourier-Laguerre sums in the space L2(ℝ+2;xαyβe−x−y).

$L^p$ Bernstein estimates and approximation by spherical basis functions

The purpose of this paper is to establish L p error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates L p Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the L p norm of the function itself. An important step in its proof involves measuring the L p stability of functions in the approximating space in terms of the l p norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the L P norm. Finally, we give a new characterization of Besov spaces on the n-sphere in terms of spaces of SBFs.