Yurii S. Kolomoitsev

Georg-August-Universität Göttingen | GAUG · Institute for Numerical and Applied Mathematics

We show that the Peetre K -functional between the space $$L_p$$ L p with $$0<p<1$$ 0 < p < 1 and the corresponding smooth function space $$W_p^\psi $$ W p ψ generated by the Weyl-type differential operator $$\psi (D)$$ ψ ( D ) , where $$\psi $$ ψ is a homogeneous function of any positive order, is identically zero. The proof of the main results is...

We study approximation properties of multivariate periodic functions from weighted Wiener spaces by sparse grid 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,...

We show that the Peetre $K$-functional between the space $L_p$ with $0<p<1$ and the corresponding smooth function space $W_p^\psi$ generated by the Weyl-type differential operator $\psi(D)$, where $\psi$ 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 Vall\'...

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 repre...

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 repre...

Approximation properties of quasi-projection operators $$Q_j(f,\varphi , \widetilde{\varphi })$$ Q j ( f , φ , φ ~ ) are studied. These operators are associated with a function $$\varphi $$ φ satisfying the Strang–Fix conditions and a tempered distribution $$\widetilde{\varphi }$$ φ ~ such that compatibility conditions with $$\varphi $$ φ hold. Err...

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 measu...

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...

In this paper asymptotic formulas are given for the Lebesgue constants generated by three special approximation processes related to the ℓ1-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 anisotropi...

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...

Multivariate quasi-projection operators Qj(f,φ,φ˜), associated with a function φ and a distribution/function φ˜, are considered. The function φ is supposed to satisfy the Strang-Fix conditions and a compatibility condition with φ˜. Using technique based on the Fourier multipliers, we study approximation properties of such operators for functions f...

We study approximation properties of general multivariate periodic quasi-interpolation operators, which are generated by distributions/functions φ˜j and trigonometric polynomials φj. The class of such operators includes classical interpolation polynomials (φ˜j is the Dirac delta function), Kantorovich-type operators (φ˜j is a characteristic functio...

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 averag...

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: The first based on geometric properties of Banach spaces and the second on Littlewood–Paley and Hörmander-type m...

Approximation properties of the sampling‐type quasi‐projection operators Qj(f,φ,φ˜) for functions f from anisotropic Besov spaces are studied. Error estimates in Lp‐norm are obtained for a large class of tempered distributions φ˜ and a large class of functions φ under the assumptions that φ has enough decay, satisfies the Strang‐Fix conditions, and...

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 anis...

Approximation properties of quasi-projection operators $Q_j(f,\varphi, \widetilde{\varphi})$ are studied. Such an operator is associated with a function $\varphi$ satisfying the Strang-Fix conditions and a tempered distribution $\widetilde{\varphi}$ such that compatibility conditions with $\varphi$ hold. Error estimates in the uniform norm are obta...

Multivariate quasi-projection operators $Q_j(f,\varphi, \widetilde{\varphi})$, associated with a function $\varphi$ and a distribution/function $\widetilde{\varphi}$, are considered. The function $\varphi$ is supposed to satisfy the Strang-Fix conditions and a compatibility condition with $\widetilde{\varphi}$. Using technique based on the Fourier...

Approximation properties of periodic quasi-projection operators with matrix dilations are studied. Such operators are generated by a sequence of functions φj and a sequence of distributions/functions φ˜j. Error estimates for sampling-type quasi-projection operators are obtained under the periodic Strang-Fix conditions for φj and the compatibility c...

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.

In this paper, we discuss various basic properties of moduli of smoothness of functions from Lp(Rd), 0<p≤∞. 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...

We study approximation properties of the general multivariate periodic quasi-interpolation operator $Q_j(f,\varphi_j,\widetilde\varphi_j)$, which is generated by the distribution/function $\widetilde{\varphi}_j$ and some trigonometric polynomial $\varphi_j$. The class of such operators includes classical interpolation polynomials ($\widetilde\varph...

Approximation properties of periodic quasi-projection operators with matrix dilations are studied. Such operators are generated by a sequence of functions $\varphi_j$ and a sequence of distributions/functions $\widetilde{\varphi}_j$. Error estimates for sampling-type quasi-projection operators are obtained under the periodic Strang-Fix conditions f...

Approximation properties of multivariate quasi-projection operators are studied. 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 Lp-spaces for these operators is investigated. The results allow u...

Approximation properties of the sampling-type quasi-projection operators $Q_j(f,\varphi, \widetilde{\varphi})$ for functions $f$ from anisotropic Besov spaces are studied. Error estimates in $L_p$-norm are obtained for a large class of tempered distributions $\widetilde{\varphi}$ and a large class of functions $\varphi$ under the assumptions that $...

In the paper, we study inequalities for the best trigonometric approximations and fractional moduli of smoothness involving the Weyl and Liouville-Grünwald derivatives in Lp, 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 th...

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...

We provide a comprehensive study of interrelations between different measures of smoothness of functions on various domains and smoothness properties of approximation processes. Two general approaches to this problem have been developed: the first based on geometric properties of Banach spaces and the second on Littlewood-Paley and H\"{o}rmander ty...

In the paper, we study inequalities for the best trigonometric approximations and fractional moduli of smoothness involving the Weyl and Liouville-Gr\"unwald 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...

Differential and falsified sampling expansions $\sum_{k\in \mathbb{Z}^d}c_k\phi(M^jx+k)$, where $M$ is a matrix dilation, are studied. In the case of differential expansions, $c_k=Lf(M^{-j}\cdot)(-k)$, where $L$ is an appropriate differential operator. For a large class of functions $\phi$, the approximation order of differential expansions was rec...

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...

We obtain estimates of the $L_p$-error of the bivariate polynomial interpolation on the Lissajous-Chebyshev node points for wide classes of functions including non-smooth functions of bounded variation in the sense of Hardy-Krause. The results show that $L_p$-errors of polynomial interpolation on the Lissajous-Chebyshev nodes have almost the same b...

To analyze the absolute condition number of multivariate polynomial interpolation on Lissajous-Chebyshev node points, we derive upper and lower bounds for the respective Lebesgue constant. The proof is based on a relation between the Lebesgue constant for the polynomial interpolation problem and the Lebesgue constant linked to the polyhedral partia...

We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness ω α ( f , t ) q \omega _\alpha (f,t)_q and ω β ( f , t ) p \omega _\beta (f,t)_p for 0 > p > q ≤ ∞ 0>p>q\le \infty . A similar problem for the generalized K K -functionals and their realizations between the couples ( L p , W p ψ ) (L_p, W_...

We consider two types of fractional integral moduli of smoothness, which are widely used in the theory of functions and approximation theory. In particular, we obtain new equivalences between these moduli of smoothness and the classical moduli of smoothness. It turns out that for the fractional integral moduli of smoothness some pathological effect...

In this paper we obtain new sufficient conditions for representation of a function as an absolutely convergent Fourier integral. Unlike those known earlier, these conditions are given in terms of belonging to weighted spaces. Adding weights allows one to extend the range of application of such results to Fourier multipliers with unbounded derivativ...

Approximation properties of multivariate Kantorovich-Kotelnikov type operators generated by different band-limited functions are studied. In particular, a wide class of functions with discontinuous Fourier transform is considered. The -rate of convergence for these operators is given in terms of the classical moduli of smoothness. Several examples...

In the paper, new estimates of the Lebesgue constant $$ \mathcal{L}(W)=\frac1{(2\pi)^d}\int_{\mathbb{T}^d}\bigg|\sum_{{k}\in W\cap \mathbb{Z}^d} e^{i({k},\,{x})}\bigg| {\rm d}{ x} $$ for convex polyhedra $W\subset\mathbb{R}^d$ are obtained. The main result states that if $W$ is a convex polyhedron such that $[0,m_1]\times\dots\times [0,m_d]\subset...

Several new inequalities for moduli of smoothness and errors of the best approximation of a function and its derivatives in the spaces $L_p$, $0<p<1$, are obtained. For example, it is shown that for any $0<p<1$ and $k,\,r\in \mathbb{N}$ one has $ \omega_{r+k}(f,\d)_p\leq C({p,k,r})\d^{r+\frac{1}{p}-1}\(\int_0^\d\frac{\omega_{k}(f^{(r)},t)_p^p}{t^{2...

We study approximation of functions by algebraic polynomials in the H\"older 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 preci...

We establish necessary and sufficient conditions for the validity of Bernstein-type inequalities for the fractional derivatives of trigonometric polynomials of several variables in spaces with integral metrics. The problem of sharpness of these inequalities is investigated.

The main purpose of the paper is to study sharp estimates of approximation of periodic functions in the H\"older spaces $H_p^{r,\alpha}$ for all $0<p\le\infty$ and $0<\alpha\le r$. 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 pre...

We get new sufficient conditions for Fourier multipliers in Hardy spaces , , and , . Being of a multiplicative character, these conditions are stated in terms of the joint behaviour of `norms' of functions in and Besov spaces .

We obtain new sufficient conditions for Fourier multipliers in the Hardy spaces $ {H_p}({{\mathbb{R}}^n}) $ , 0 < p < 2: These conditions are presented in terms of the joint behavior of a function and its derivatives. The results of the paper generalize the corresponding Miyachi theorems.

Sufficient conditions for the representation of functions as the Fourier integral in ℝd of a function belonging to the space L 1 ∩ L p , where 0 < p < 2 are obtained. The sharpness of these conditions is shown.

Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of Lp integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular ca...

We study the approximation of functions by linear polynomial means of their Fourier series with a function-multiplier φ that is equal to 1 not only at zero, in contrast with classical methods of summability. The exact order of convergence to zero of the sequence $$ \mathop{\max}\limits_{{x\in \left[ {-\pi, \pi } \right]}}\left| {f(x)-\sum\limits_{...

A test for the convergence of the generalized spherical and Bochner-Riesz means in the Hardy spaces , , is obtained, where is the unit polydisc. Precise orders of the approximation of functions by the generalized Bochner-Riesz means in terms of the -functional and special moduli of smoothness are found. Bibliography: 31 titles.

We prove the equivalence of special moduli of smoothness and K-functionals of fractional order in the space H p , p > 0. As applications, we obtain an analog of the Hardy–Littlewood theorem and the sharp estimates of the rate of approximation of functions by generalized Bochner–Riesz means.

Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained in the paper. The results are given in terms of $L^p$ integrability of the function and its partial derivatives, each with the corresponding $p$. These $p$ are subject to certain relations known earlier o...

Some sharp results related to the convergence of means and families of operators generated by the generalized Bochner-Riesz kernels are obtained. The exact order of approximation of functions by these methods via $K$-functional (or its realization in the case of the space $L_p$, $0<p<1$) is derived.

Let B be a set of integers with certain arithmetic properties. We obtain estimates of the best approximation of functions in the space L p , 0 p $$ \{e^{ikx}\}_{k\in \mathbb{Z}\backslash B} $$. Bibliography: 13 titles.

We prove a theorem on the relationship between the modulus of smoothness and the best approximation in L p , 0 < p < 1, and theorems on the extension of functions with preservation of the modulus of smoothness in L p , 0 < p < 1. We also give a complete description of multipliers of periodic functions in the spaces L p , 0 < p < 1.

We obtain a necessary and sufficient condition for the completeness of the trigonometric system with gaps for the classes ϕ(L).