arXiv subjects

Yurii Kolomoitsev

Publications and source records attributed to Yurii Kolomoitsev.

27 records · Page 2Linked to original sources

arXiv2018

On the growth of Lebesgue constants for convex polyhedra

In the paper, new estimates of the Lebesgue constant $$ \mathcal{L}(W)=\frac1{(2π)^d}\int_{\mathbb{T}^d}\bigg|\sum_{{k}\in W\cap \mathbb{Z}^d} e^{i({k},\,{x})}\bigg| {\rm d}{ x} $$ for convex polyhedra $W\subset\mathbb{R}^d$ are obtained. The main result states that if $W$ is a convex polyhedron such that $[0,m_1]\times\dots\times [0,m_d]\subset W\subset [0,n_1]\times\dots\times [0,n_d]$, then $$ c(d)\prod_{j=1}^d \log(m_j+1)\le \mathcal{L}(W)\le C(d)s\prod_{j=1}^d \log(n_j+1), $$ where $s$ is a size of the triangulation of $W$.

math.CA↗

arXiv2017

On $L_p$-error of bivariate polynomial interpolation on the square

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 behavior as the polynomial interpolation in the case of the tensor product Chebyshev grid.

math.CA↗

arXiv2017

Hardy-Littlewood and Ulyanov inequalities

We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness $ω_α(f,t)_q$ and $ω_β(f,t)_p$ for $0<p<q\le \infty$. A similar problem for the generalized $K$-functionals and their realizations between the couples $(L_p, W_p^ψ)$ and $(L_q, W_q^φ)$ is also solved. The main tool is the new Hardy-Littlewood-Nikol'skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity $$ \sup_{T_n} \frac{\Vert \mathcal{D}(ψ)(T_n)\Vert_q}{\Vert \mathcal{D}(φ)(T_n)\Vert_p},\qquad 0<p<q\le \infty, $$ where the supremum is taken over all nontrivial trigonometric polynomials $T_n$ of degree at most $n$ and $\mathcal{D}(ψ), \mathcal{D}(φ)$ are the Weyl-type differentiation operators. We also prove the Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for the Lipschitz and Besov spaces.

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arXiv2017

Lebesgue constants for polyhedral sets and polynomial interpolation on Lissajous-Chebyshev nodes

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 partial sums of Fourier series. The magnitude of the obtained bounds is determined by a product of logarithms of the side lengths of the considered polyhedral sets and shows the same behavior as the magnitude of the Lebesgue constant for polynomial interpolation on the tensor product Chebyshev grid.

math.NA↗

arXiv2017

On moduli of smoothness and averaged differences of fractional order

We consider two types of fractional integral moduli of smoothness, which are widely used in 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 fractional integral moduli of smoothness some pathological effects arise.

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arXiv2016

Best approximations and moduli of smoothness of functions and their derivatives in $L_p$, $0<p<1$

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 $ ω_{r+k}(f,\d)_p\leq C({p,k,r})\d^{r+\frac{1}{p}-1}$\int_0^\d\frac{ω_{k}(f^{(r)},t)_p^p}{t^{2-p}}{\rm d}t$^\frac{1}{p},$ where the function $f$ is such that $f^{(r-1)}$ is absolutely continuous. Similar inequalities are obtained for the Ditzian-Totik moduli of smoothness and the error of the best approximation of functions by trigonometric and algebraic polynomials and splines. As an application, positive results about simultaneous approximation of a function and its derivatives by the mentioned approximation methods in the spaces $L_p$, $0<p<1$, are derived.

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arXiv2016

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.

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arXiv2015

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 $H_p^{r,α}$ for all $0<p\le\infty$ and $0<α\le r$. 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 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 $H_p^{r,α}$.

math.CA↗

arXiv2011

On approximation methods generated by generalized Bochner-Riesz kernels

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.

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