We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of $2π$-periodic functions $φ$, such that $\|φ\|_2\le1$, with fixed generated kernels $Ψ_{\barβ}$, which have Fourier series of the form $\sum\limits_{k=1}^\infty ψ(k)\cos(kt-β_kπ/2), $ where $ψ(k)\ge0,$ $\sumψ^2(k)<\infty, β_k\in\mathbb{R},$ in the space $C$. It is shown that for rapidly decrising sequences $ψ(k)$ (in particular, if $\lim\limits_{k\rightarrow\infty}{ψ(k+1)}/{ψ(k)}=0$) obtained estimates are asymptotic equalities. We establish that asymptotic equalities for widths of this classes are realized by trigonometric Fourier sums.
We find asymptotic equalities for the exact upper bounds of approximations by Fourier sums of Weyl-Nagy classes $W^r_{β,p}, 1\le p\le\infty,$ for rapidly growing exponents of smoothness $r$ $(r/n\rightarrow\infty)$ in the uniform metric. We obtain similar estimates for approximations of the classes $W^r_{β,1}$ in metrics of the spaces $L_p, 1\le p\le\infty$.
We establish asymptotic estimates for the least upper bounds of approximations in the uniform metric by Fourier sums of order $n-1$ of classes of $2π$-periodic Weyl--Nagy differentiable functions, $W^r_{β,p}, 1\le p\le \infty, β\in\mathbb{R},$ for high exponents of smoothness $r\ (r-1\ge \sqrt{n})$. We obtain similar estimates in metrics of the spaces $L_p, 1\le p\le\infty,$ for functional classes $W^r_{β,1}$.
In this paper, we give a characterization of Nikol'ski\uı-Besov type classes of functions, given by integral representations of moduli of smoothness, in terms of series over the moduli of smoothness. Also, necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to a such a class are given. In order to prove our results, we make use of certain recent reverse Copson- and Leindler-type inequalities.
We obtain order estimates of approximation of functions from the classes $S^Ω_{p,θ}B (\mathbb{R}^d)$ in the space $L_q(\mathbb{R}^d)$, $1<p<q<\infty$, by entire functions of exponential type with supports of their Fourier transforms in sets generated by the level surfaces of a function $Ω$.
We find two-sides estimates for the best uniform approximations of classes of convolutions of $2π$-periodic functions from unit ball of the space $L_p, 1 \le p <\infty,$ with fixed kernels, modules of Fourier coefficients of which satisfy the condition $\sum\limits_{k=n+1}^\inftyψ(k)<ψ(n).$ In the case of $\sum\limits_{k=n+1}^\inftyψ(k)=o(1)ψ(n)$ the obtained estimates become the asymptotic equalities.
Exact Jackson-type inequalities are obtained in terms of best approximations and averaged values of generalized moduli of smoothness in the spaces ${\mathcal S}^p$. The values of Kolmogorov, Bernstein, linear, and projective widths in the spaces ${\mathcal S}^p$ are found for classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness.
Order estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in uniform metric
This paper considers the Lorentz space with mixed norm of periodic functions of many variables and of the generalized Nikol'skii -- Besov classes. Estimates for the order of approximation of the generalized Nikol'skii -- Besov classes by partial sums of Fourier's series for multiple trigonometric system in Lorentz spaces with mixed norm are obtained.
We obtain order estimates for the best uniform orthogonal trigonometric approximations of $2π$-periodic functions, whose $(ψ,β)$-derivatives belong to unit balls of spaces $L_{p}, \ 1\leq p<\infty$, in case at consequences $ψ(k)$ are that product $ψ(n)n^{\frac{1}{p}}$ can tend to zero slower than any power function and $\sum\limits_{k=1}^{\infty}ψ^{p'}(k)k^{p'-2}<\infty$ when $1<p<\infty$, $\frac{1}{p}+\frac{1}{p'}=1$ and $\sum\limits_{k=1}^{\infty}ψ(k)<\infty$ when $p=1$. We also establish the analogical estimates in $L_{s}$-metric, $1< s\leq \infty$, for classes of the summable $(ψ,β)$-diffe…
Estimates of the best approximations and approximations of Fourier sums of classes of convolutions of periodic functions of not high smoothness in integral metrics
In the Musielak-Orlicz type spaces ${\mathcal S}_{\bf M}$, exact Jackson-type inequalities are obtained in terms of best approximations of functions and the averaged values of their generalized moduli of smoothness. The values of Kolmogorov, Bernstein, linear, and projective widths in ${\mathcal S}_{\bf M}$ are found for classes of periodic functions defined by certain conditions on the averaged values of the generalized moduli of smoothness.
We prove that kernels of analytic functions of kind $H_{h,β}(t)=\sum\limits_{k=1}^{\infty}\frac{1}{\cosh kh}\cos\Big(kt-\frac{βπ}{2}\Big)$, $h>0$, ${β\in\mathbb{R}}$, satisfies Kushpel's condition $C_{y,2n}$ beginning with some number $n_h$ which is explicitly expressed by parameter $h$ of smoothness of the kernel. As a consequence, for all $n\geqslant n_h$ we obtain lower bounds for Kolmogorov widths $d_{2n}$ of functional classes that are representable as convolutions of kernel $H_{h,β}$ with functions $φ\perp1$, which belong to the unit ball in the space $L_{\infty}$, in the space $C$. The …
We introduce new moduli of smoothness for functions $f\in L_p[-1,1]\cap C^{r-1}(-1,1)$, $1\le p\le\infty$, $r\ge1$, that have an $(r-1)$st locally absolutely continuous derivative in $(-1,1)$, and such that $φ^rf^{(r)}$ is in $L_p[-1,1]$, where $φ(x)=(1-x^2)^{1/2}$. These moduli are equivalent to certain weighted DT moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original DT moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in…
On the sets of $2π$-periodic functions $f$, which are defined with a help of $(ψ, β)$-integrals of the functions $φ$ from $L_{1}$, we establish Lebesgue-type inequalities, in which the uniform norms of deviations of Fourier sums are expressed via the best approximations by trigonometric polynomials of the functions $φ$. We prove that obtained estimates are best possible, in the case when the sequences $ψ(k)$ decrease to zero faster than any power function. In some important cases we establish the asymptotic equalities for the exact upper boundaries of uniform approximations by Fourier sums on …
Here we obtain order estimates for widths of weighted Sobolev classes in the weighted Lebesgue space where parameters of the second weight satisfy some limiting conditions.
In this paper we find the orders of decay for Kolmogorov widths of some Besov classes related to $W^1_1$ (the behaviour of the widths for $W^1_1$ remains unknown): $$ d_n(B^1_{1,θ}[0,1],L_q[0,1])\asymp n^{-1/2}\log^{\max(\frac12,1-\frac{1}θ)}n,\quad 2<q<\infty. $$ The proof relies on the lower bound for widths of product of octahedra in a special norm (maximum of two weighted $\ell_q$ norms). This bound generalizes the theorem of B.S.~Kashin on widths of octahedra in $\ell_q^N$.
In this paper we prove that Kolmogorov widths of weighted Sobolev classes with restrictions $f(a)=\dots=f^{(k-1)}(a)=f^{(k)}(b)=\dots=f^{(r-1)}(b)=0$ in a weighted Lebesgue space and spectral numbers of some non-linear differential equation coincide. Here we suppose that the embedding is compact.
In weighted Orlicz type spaces ${\mathcal S}_{_{\scriptstyle \mathbf p,\,μ}}$ 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 in a certain sense the best. 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 …
Let $P(D)$ be the differential operator induced by a polynomial $P$, and let ${U^{[P]}_2}$ be the class of multivariate periodic functions $f$ such that $\|P(D)(f)\|_2\leq 1$. The problem of computing the asymptotic order of the Kolmogorov $n$-width $d_n({U^{[P]}_2},L_2)$ in the general case when ${U^{[P]}_2}$ is compactly embedded into $L_2$ has been open for a long time. In the present paper, we use convex analytical tools to solve it in the case when $P(D)$ is non-degenerate.
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…
