Inverse Approximation Theorems Research Articles

Direct and inverse approximation theorems for functions defined in Damek–Ricci spaces

UDC 517.5 We introduce the notion of k th modulus of smoothness and establish the direct and inverse theorems in terms of the quantities E s ( f ) and the moduli of smoothness generated by the spherical mean operator defined on the L 2 -space for the Damek–Ricci spaces. These theorems are analogous to the well-known theorems of Jackson and Bernstein. We also consider some problems related to the constructive characteristics of functional classes defined by the majorants of the moduli of smoothness of their elements.

Best Approximation and Inverse Results for Neural Network Operators

In the present paper we considered the problems of studying the best approximation order and inverse approximation theorems for families of neural network (NN) operators. Both the cases of classical and Kantorovich type NN operators have been considered. As a remarkable achievement, we provide a characterization of the well-known Lipschitz classes in terms of the order of approximation of the considered NN operators. The latter result has inspired a conjecture concerning the saturation order of the considered families of approximation operators. Finally, several noteworthy examples have been discussed in detail

Direct, Inverse Theorems of Approximation by Linear Combinations of Weighted Baskakov–Durrmeyer Operators in Orlicz Spaces

Direct, Inverse Theorems of Approximation by Linear Combinations of Weighted Baskakov–Durrmeyer Operators in Orlicz Spaces

Theorems on direct and inverse approximation by algebraic polynomials and piecewise polynomials in the spaces H<sup>m</sup>(a, b) and B<sup>s</sup><sub>2,q</sub>(a, b)

The best estimates for the approximation error of functions, defined on a finite interval, by algebraic polynomials and piecewise polynomial functions are obtained in the case when the errors are measured in the norms of Sobolev and Besov spaces. We indicate the weighted Besov spaces, whose functions satisfy Jackson-type and Bernstein-type inequalities and, as a consequence, direct and inverse approximation theorems. In a number of cases, exact constants are indicated in the estimates.

Inverse approximation theorem using Bernstein’s Durrmeyer operator in terms of 2nd modulus of smoothness

Many inverse theorems have been proven using the ordinary modulus of smoothness but no one proved the invers theorems for approximation by Bernstein’s Durrmeyer operator in terms 2nd modulus of smoothness.

Hölder classes in the 𝐿^{𝑝} norm on a chord-arc curve in ℝ³

The Hölder classes L p α ( L ) L_p^{\alpha } (L) in the L p ( L ) L_p(L) norm on a chord-arc curve L L in R 3 \mathbb {R}^3 are defined and direct and inverse approximation theorems are proved for functions from these classes by functions harmonic in a neighborhood of the curve. The approximation is estimated in the L p ( L ) L^p(L) norm, the direct theorem is proved for a certain subclass of L p α ( L ) L^\alpha _p(L) and the inverse theorem covers the entire Hölder class.

To multidimensional Mellin analysis: Besov spaces, K-functor, approximations, frames

In the setting of the multidimensional Mellin analysis we introduce moduli of continuity and use them to define Besov–Mellin spaces. We prove that Besov–Mellin spaces are the interpolation spaces (in the sense of J.Peetre) between two Sobolev–Mellin spaces. We also introduce Bernstein-Mellin spaces and prove corresponding direct and inverse approximation theorems. In the Hilbert case we discuss Laplace–Mellin operator and define relevant Paley–Wiener–Mellin spaces. Also in the Hilbert case we describe Besov–Mellin spaces in terms of Hilbert frames.

Two-Layer Neural Networks with Values in a Banach Space

We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped with a partial order, i.e. it is a Riesz space. As the nonlinearity we choose the lattice operation of taking the positive part; in case of $\mathbb R^d$-valued neural networks this corresponds to the ReLU activation function. We prove inverse and direct approximation theorems with Monte-Carlo rates, extending existing results for the finite-dimensional case. In the second part of the paper, we consider training such networks using a finite amount of noisy observations from the regularisation theory viewpoint. We discuss regularity conditions known as source conditions and obtain convergence rates in a Bregman distance in the regime when both the noise level goes to zero and the number of samples goes to infinity at appropriate rates.

Direct and inverse approximation theorems in the Besicovitch – Musielak – Orlicz spaces of almost periodic functions

UDC 517.5 In terms of the best approximations of functions and generalized moduli of smoothness, direct and inverse approximation theorems are proved for Besicovitch almost periodic functions whose Fourier exponent sequences have a single limit point in infinity and their Orlicz norms are finite. Special attention is paid to the study of cases when the constants in these theorems are unimprovable.

Bernstein and Bernstein-Like Inequalities for Modulation Spaces

It is shown that the modulation spaces Mwp can be characterized by the approximation behavior of their elements using Local Fourier bases. In analogy to the Local Fourier bases, we show that the modulation spaces can also be characterized by the approximation behavior of their elements using Gabor frames. We derive direct and inverse approximation theorems that describe the best approximation by linear combinations of N terms of a given function using its modulates and translates.

Approximation by trigonometric polynomials in the variable exponent weighted Morrey spaces

In this paper we investigate the best approximation by trigonometric polynomials in the variable exponent weighted Morrey spaces ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$, where $w$ is a weight function in the Muckenhoupt $A_{p(\cdot)}(I_{0})$ class. We get a characterization of $K$-functionals in terms of the modulus of smoothness in the spaces ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$. Finally, we prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces ${\mathcal{\widetilde{M}}}_{p(\cdot),\lambda(\cdot)}(I_{0},w),$ the closure of the set of all trigonometric polynomials in ${\mathcal{M}}_{p(\cdot),\lambda(\cdot)}(I_{0},w)$.

Direct and inverse theorems on the approximation of almost periodic functions in Besicovitch-Stepanets spaces

Direct and inverse approximation theorems are proved in the Besicovitch-Stepanets spaces $B{\mathcal S}^{p}$ of almost periodic functions in terms of the best approximations of functions and their generalized moduli of smoothness.

The Barron Space and the Flow-Induced Function Spaces for Neural Network Models

One of the key issues in the analysis of machine learning models is to identify the appropriate function space and norm for the model. This is the set of functions endowed with a quantity which can control the approximation and estimation errors by a particular machine learning model. In this paper, we address this issue for two representative neural network models: the two-layer networks and the residual neural networks. We define the Barron space and show that it is the right space for two-layer neural network models in the sense that optimal direct and inverse approximation theorems hold for functions in the Barron space. For residual neural network models, we construct the so-called flow-induced function space and prove direct and inverse approximation theorems for this space. In addition, we show that the Rademacher complexity for bounded sets under these norms has the optimal upper bounds.

The Direct and Converse Theorems for Best Approximation of Algebraic Polynomial in Lp, α (X)

The direct and inverse algebraic polynomials approximation theorems in weighted spaces of unbounded functions are proved by using modulus of smoothness. Also, we obtain sharp Jackson ( direct ) inequality of algebraic approximation of unbounded functions in terms modulus of smoothness. In addition, constructive characterization of modulus of smoothness are considered.

Direct and inverse approximation theorems of functions in the Musielak-Orlicz type spaces

In Musilak-Orlicz type spaces ${\mathcal S}_{\bf M}$, direct and inverse approximation theorems are obtained in terms of the best approximations of functions and generalized moduli of smoothness. The question of the exact constants in Jackson-type inequalities is studied.

Direct and inverse approximation theorems in the weighted Orlicz-type spaceswith a variable exponent

In weighted Orlicz-type spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,\mu}}$ with a variable summation exponent, the direct and inverse approximation theorems are proved in terms of best approximations of functions and moduli of smoothness of fractional order. It is shown that the constant obtained in the inverse approximation theorem is the best in a certain sense. Some applications of the results are also proposed. In particular, the constructive characteristics of functional classes defined by such moduli of smoothness are given. Equivalence between moduli of smoothness and certain Peetre $K$-functionals is shown in the spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,\mu}}$.

Approximation by trigonometric polynomials in weighted Morrey spaces

In this paper we investigate the best approximation by trigonometric polynomials in weighted Morrey spaces $\mathcal{M}_{p,\lambda}(I_{0},w)$, where the weight function $w$ is in the Muckenhoupt class $A_{p}(I_{0})$ with $1 < p < \infty$ and $I_{0}=[0, 2\pi]$. We prove the direct and inverse theorems of approximation by trigonometric polynomials in the spaces $\mathcal{\widetilde{M}}_{p,\lambda}(I_{0},w)$ the closure of $C^{\infty}(I_{0})$ in $\mathcal{M}_{p,\lambda}(I_{0},w)$. We give the characterization of $K-$functionals in terms of the modulus of smoothness and obtain the Bernstein type inequality for trigonometric polynomials in the spaces $\mathcal{M}_{p,\lambda}(I_{0},w)$.

Direct and inverse approximation theorems of functions in the Orlicz type spaces 𝓢M

AbstractIn the Orlicz type spaces 𝓢M, we prove direct and inverse approximation theorems in terms of the best approximations of functions and moduli of smoothness of fractional order. We also show the equivalence between moduli of smoothness and PeetreK-functionals in the spaces 𝓢M.

Approximation theorems for multivariate Taylor-Abel-Poisson means

We obtain direct and inverse approximation theorems of functions of several variables by Taylor-Abel-Poisson means in the integral metrics. We also show that norms of multipliers in the spaces $L_{p,Y}(\mathbb T^d)$ are equivalent for all positive integers $d.$

Inverse results of approximation and the saturation order for the sampling Kantorovich series

In the present paper, we study the saturation order for the sampling Kantorovich series in the space of uniformly continuous and bounded functions. In order to achieve the above result, we first need to establish a relation between the sampling Kantorovich operators and the classical generalized sampling series of P.L. Butzer. Further, for the latter operators, we also need to prove a Voronovskaja-type formula. Moreover, inverse theorems of approximation are also obtained, in order to give a characterization for the rate of convergence of the above operators. Finally, several examples of kernel functions for which the above theory can be applied have been presented and discussed in details.