arXiv subjects

Yurii Kolomoitsev

Publications and source records attributed to Yurii Kolomoitsev.

At least 19 recordsLinked to original sources

arXiv2026

Approximation by Kantorovich-type operators in general Banach spaces

This paper investigates the approximation properties of linear Kantorovich-type sampling operators in the setting of general Banach lattices $X$ on the torus $\mathbb{T}$ and the real line $\mathbb{R}$. Under a natural assumption on the uniform boundedness of the Steklov averaging operators, we establish direct and inverse approximation estimates, as well as strong converse inequalities. Our framework does not require the underlying spaces to be translation-invariant, thereby covering a wide variety of classes and extending the estimates for Kantorovich-type operators previously known primarily for Lebesgue spaces $L_p$.

math.FA↗

arXiv2026

Approximation by linear sampling operators in Banach spaces

This paper studies approximation properties of linear sampling operators in general Banach lattices $X$. We obtain matching direct and inverse approximation estimates, convergence criteria, equivalence results involving special $K$-functionals and their realizations by sampling operators, as well as strong converse inequalities, which, to the best of our knowledge, have not been previously established for sampling operators even in the classical spaces $L_p$. The results extend several classical theorems previously known mainly in $L_p$ and apply to all functions $f\in X$ for which the corresponding sampling operator is well defined, thereby substantially enlarging the class of functions that can be considered in this framework.

math.FA↗

arXiv2025

Special measures of smoothness for approximation by sampling operators in $L_p(\Bbb{R}^d)$

Traditional measures of smoothness often fail to provide accurate $L_p$-error estimates for approximation by sampling or interpolation operators, especially for functions with low smoothness. To address this issue, we introduce a modified measure of smoothness that incorporates the local behavior of a function at the sampling points through the use of averaged operators. With this new tool, we obtain matching direct and inverse error estimates for a wide class of sampling operators and functions in $L_p$ spaces. Additionally, we derive a criterion for the convergence of sampling operators in $L_p$, identify conditions that ensure the exact rate of approximation, construct realizations of $K$-functionals based on these operators, and study the smoothness properties of sampling operators. We also demonstrate how our results apply to several well-known operators, including the classical Whittaker-Shannon sampling operator, sampling operators generated by $B$-splines, and those based on the Gaussian.

math.NA↗

arXiv2024

Marcinkiewicz-Zygmund inequalities in quasi-Banach function spaces

We obtain Marcinkiewicz--ygmund (MZ) inequalities in various Banach and quasi-Banach spaces under minimal assumptions on the structural properties of these spaces. Our main results show that the Bernstein inequality in a general quasi-Banach function lattice $X$ implies Marcinkiewicz-Zygmund type estimates in $X$. We present a general approach to obtain MZ inequalities not only for polynomials but for other function classes including entire functions of exponential type, splines, exponential sums, etc.

math.CA↗

arXiv2022

On generalized $K$-functionals in $L_p$ for $0<p<1$

We show that the Peetre $K$-functional between the space $L_p$ with $0<p<1$ and the corresponding smooth function space $W_p^ψ$ generated by the Weyl-type differential operator $ψ(D)$, where $ψ$ is a homogeneous function of any positive order, is identically zero. The proof of the main results is based on the properties of the de la Vallée Poussin kernels and the quadrature formulas for trigonometric polynomials and entire functions of exponential type.

math.CA↗

arXiv2022

Spline Representation and Redundancies of One-Dimensional ReLU Neural Network Models

We analyze the structure of a one-dimensional deep ReLU neural network (ReLU DNN) in comparison to the model of continuous piecewise linear (CPL) spline functions with arbitrary knots. In particular, we give a recursive algorithm to transfer the parameter set determining the ReLU DNN into the parameter set of a CPL spline function. Using this representation, we show that after removing the well-known parameter redundancies of the ReLU DNN, which are caused by the positive scaling property, all remaining parameters are independent. Moreover, we show that the ReLU DNN with one, two or three hidden layers can represent CPL spline functions with $K$ arbitrarily prescribed knots (breakpoints), where $K$ is the number of real parameters determining the normalized ReLU DNN (up to the output layer parameters). Our findings are useful to fix a priori conditions on the ReLU DNN to achieve an output with prescribed breakpoints and function values.

math.NA↗

arXiv2022

Sharp $L_p$-error estimates for sampling operators

We study approximation properties of linear sampling operators in the spaces $L_p$ for $1\le p<\infty$. By means of the Steklov averages, we introduce a new measure of smoothness that simultaneously contains information on the smoothness of a function in $L_p$ and discrete information on the behaviour of a function at sampling points. The new measure of smoothness enables us to improve and extend several classical results of approximation theory to the case of linear sampling operators. In particular, we obtain matching direct and inverse approximation inequalities for sampling operators in $L_p$, find the exact order of decay of the corresponding $L_p$-errors for particular classes of functions, and introduce a special $K$-functional and its realization suitable for studying smoothness properties of sampling operators.

math.CA↗

arXiv2021

Sparse grid approximation in weighted Wiener spaces

We study approximation properties of multivariate periodic functions from weighted Wiener spaces by sparse grids methods constructed with the help of quasi-interpolation operators. The class of such operators includes classical interpolation and sampling operators, Kantorovich-type operators, scaling expansions associated with wavelet constructions, and others. We obtain the rate of convergence of the corresponding sparse grids methods in weighted Wiener norms as well as analogues of the Littlewood-Paley-type characterizations in terms of families of quasi-interpolation operators.

math.NA↗

arXiv2021

Approximation by quasi-interpolation operators and Smolyak's algorithm

We study approximation of multivariate periodic functions from Besov and Triebel--Lizorkin spaces of dominating mixed smoothness by the Smolyak algorithm constructed using a special class of quasi-interpolation operators of Kantorovich-type. These operators are defined similar to the classical sampling operators by replacing samples with the average values of a function on small intervals (or more generally with sampled values of a convolution of a given function with an appropriate kernel). In this paper, we estimate the rate of convergence of the corresponding Smolyak algorithm in the $L_q$-norm for functions from the Besov spaces $\mathbf{B}_{p,θ}^s(\mathbb{T}^d)$ and the Triebel--Lizorkin spaces $\mathbf{F}_{p,θ}^s(\mathbb{T}^d)$ for all $s>0$ and admissible $1\le p,θ\le \infty$ as well as provide analogues of the Littlewood--Paley-type characterizations of these spaces in terms of families of quasi-interpolation operators.

math.CA↗

arXiv2021

Approximation properties of periodic multivariate quasi-interpolation operators

We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions $\widetildeφ_j$ and trigonometric polynomials $φ_j$. The class of such operators includes classical interpolation polynomials ($\widetildeφ_j$ is the Dirac delta function), Kantorovich-type operators ($\widetildeφ_j$ is a characteristic function), scaling expansions associated with wavelet constructions, and others. Under different compatibility conditions on $\widetildeφ_j$ and $φ_j$, we obtain upper and lower bound estimates for the $L_p$-error of approximation by quasi-interpolation operators in terms of the best and best one-sided approximation, classical and fractional moduli of smoothness, $K$-functionals, and other terms.

math.CA↗

arXiv2020

Asymptotics of the Lebesgue constants for bivariate approximation processes

In this paper asymptotic formulas are given for the Lebesgue constants generated by three special approximation processes related to the $\ell_1$-partial sums of Fourier series. In particular, we consider the Lagrange interpolation polynomials based on the Lissajous-Chebyshev node points, the partial sums of the Fourier series generated by the anisotropically dilated rhombus, and the corresponding discrete partial sums.

math.CA↗

arXiv2020

Uniform approximation by multivariate quasi-projection operators

Approximation properties of quasi-projection operators $Q_j(f,φ, \widetildeφ)$ are studied. Such an operator is associated with a function $φ$ satisfying the Strang-Fix conditions and a tempered distribution $\widetildeφ$ such that compatibility conditions with $φ$ hold. Error estimates in the uniform norm are obtained for a wide class of quasi-projection operators defined on the space of uniformly continuous functions and on the anisotropic Besov spaces. Under additional assumptions on $φ$ and $\widetildeφ$, two-sided estimates in terms of realizations of the $K$-functional are also obtained.

math.CA↗

arXiv2020

Approximation by multivariate quasi-projection operators and Fourier multipliers

Multivariate quasi-projection operators $Q_j(f,φ, \widetildeφ)$, associated with a function $φ$ and a distribution/function $\widetildeφ$, are considered. The function $φ$ is supposed to satisfy the Strang-Fix conditions and a compatibility condition with $\widetildeφ$. Using technique based on the Fourier multipliers, we studied approximation properties of such operators for functions $f$ from anisotropic Besov spaces and $L_p$ spaces with $1\le p\le \infty$. In particular, upper and lower estimates of the $L_p$-error of approximation in terms of moduli of smoothness and best approximations are obtained.

math.CA↗

arXiv2020

Asymptotics of the Lebesgue constants for a $d$-dimensional simplex

In this paper an asymptotic formula is given for the Lebesgue constants generated by the anisotropically dilated $d$-dimensional simplex. Contrary to many preceding results established only in dimension two, the obtained ones are proved in any dimension. Also, the "rational" and "irrational" parts are both united and separated in one formula.

math.CA↗

arXiv2020

Properties of moduli of smoothness in $L_p(\mathbb{R}^d)$

In this paper, we discuss various basic properties of moduli of smoothness of functions from $L_p(\mathbb{R}^d)$, $0<p\le \infty$. In particular, complete versions of Jackson-, Marchaud-, and Ulyanov-type inequalities are given for the whole range of $p$. Moreover, equivalences between moduli of smoothness and the corresponding $K$-functionals and the realization concept are proved.

math.CA↗

arXiv2020

Quasi-projection operators in the weighted $L_p$ spaces

Approximation properties of multivariate quasi-projection operators are studied in the paper. Wide classes of such operators are considered, including the sampling and the Kantorovich-Kotelnikov type operators generated by different band-limited functions.The rate of convergence in the weighted $L_p$-spaces for these operators is investigated. The results allow to estimate the error for reconstruction of signals (approximated functions) whose decay is not enough to be in $L_p$.

math.CA↗

arXiv2020

Approximation by sampling-type operators in $L_p$-spaces

Approximation properties of the sampling-type quasi-projection operators $Q_j(f,φ, \widetildeφ)$ for functions $f$ from anisotropic Besov spaces are studied. Error estimates in $L_p$-norm are obtained for a large class of tempered distributions $\widetildeφ$ and a large class of functions $φ$ under the assumptions that $φ$ has enough decay, satisfies the Strang-Fix conditions and a compatibility condition with $\widetildeφ$. The estimates are given in terms of moduli of smoothness and best approximations.

math.CA↗

arXiv2018

Inequalities in approximation theory involving fractional smoothness in $L_p$, $0<p<1$

In the paper, we study inequalities for the best trigonometric approximations and fractional moduli of smoothness involving the Weyl and Liouville-Grünwald derivatives in $L_p$, $0<p<1$. We extend known inequalities to the whole range of parameters of smoothness as well as obtain several new inequalities. As an application, the direct and inverse theorems of approximation theory involving the modulus of smoothness $ω_β(f^{(α)},δ)_p$, where $f^{(α)}$ is a fractional derivative of the function $f$, are derived. A description of the class of functions with the optimal rate of decrease of a fractional modulus of smoothness is given.

math.CA↗