K. A. Kopotun, D. Leviatan, I. A. Shevchuk
In this paper, we discuss various properties of the new modulus of smoothness \[ ω^φ_{k,r}(f^{(r)},t)_p := \sup_{0 < h\leq t}\|\mathcal W^r_{kh}(\cdot) Δ_{hφ(\cdot)}^k (f^{(r)},\cdot)\|_{L_p[-1,1]}, \] where $\mathcal W_δ(x) = \bigl((1-x-δφ(x)/2) (1+x-δφ(x)/2)\bigr)^{1/2}. $ Related moduli with more general weights are also considered.
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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.
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.
In this paper we survey recent developments over the last 25 years on the mixed fractional moduli of smoothness of periodic functions from $L_p$, $1<p<\infty$. In particular, the paper includes monotonicity properties, equivalence and realization results, sharp Jackson, Marchaud, and Ul'yanov inequalities, interrelations between the moduli of smoothness, the Fourier coefficients, and "angular" approximation. The sharpness of the results presented is discussed.
A new modulus of smoothness based on the Euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness, including direct and inverse theorem for the best approximation by polynomials and its equivalence to a $K$-functional, defined via partial derivatives in Euler angles. The set of results on the moduli on the sphere serves as a basis for defining new moduli of smoothness and their corresponding $K$-functionals on the unit ball, which are used to characterize the best approximation by polynomials on the ball.
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…
The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights $(1-x)^α(1+x)^β$ for functions in the Jacobi weighted $L_p[-1,1]$, $0<p\le \infty$, spaces. These moduli are used to characterize the smoothness of (the derivatives of) functions in the weighted $L_p$ spaces. If $1\le p\le\infty$, then these moduli are equivalent to certain weighted $K$-functionals (and so they are equivalent to certain weighted Ditzian-Totik moduli of smoothness for these $p$), while for $0<p<1$ they are equivalent to certain "Realization functionals".
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 prove two-sided inequalities between the integral moduli of smoothness of a function on $\mathbb{R}^d/\mathbb{T}^d$ and the weighted tail-type integrals of its Fourier transform/series. Sharpness of obtained results in particular is given by the equivalence results for functions satisfying certain regular conditions. Applications include a quantitative form of the Riemann-Lebesgue lemma as well as several other questions in approximation theory and the theory of function spaces.
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 algebr…
For a Banach space $B$ of functions which satisfies for some $m>0$ $$ \max(\|F+G\|_B,\|F-G\|_B) \ge (\|F\|^s_B + m\|G\|^s_B)^{1/s}, \forall F,G\in B \ (*) $$ a significant improvement for lower estimates of the moduli of smoothness $ω^r(f,t)_B$ is achieved. As a result of these estimates, sharp Jackson inequalities which are superior to the classical Jackson type inequality are derived. Our investigation covers Banach spaces of functions on $R^d$ or $T^d$ for which translations are isometries or on $S^{d-1}$ for which rotations are isometries. Results for $C_0$ semigroups of contractions are d…
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.
In this paper, exact rate of approximation of functions by linear means of Fourier series and Fourier integrals and corresponding $K$-functionals are expressed via special moduli of smoothness. . Introduction is given in $§1$. In $\S2$ functions on the line $\mathbb{R}$ are studied. A typical (well-known) result is as follows: for each $2π$-periodic function in $L_p$ on the period, for any $p\in[1,+\infty]$ ($L_\infty=C$) and $r\in\mathbb{N}$, there is a trigonometric polynomial $τ_{r,n}(f)$ of degree not greater than $n$ such that \big\|f-τ_{r,n}(f)\big\|_p\asympω_r\Big(f;\frac{1}{n}\Big)_p\a…
In the Orlicz type spaces ${\mathcal S}_{M}$, 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 Peetre $K$-functionals in the spaces ${\mathcal S}_{M}$.
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 …
We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as \[ ω_{k,r}^φ(f^{(r)},t)_{α,β,p} :=\sup_{0\leq h\leq t} \left\| {\mathcal{W}}_{kh}^{r/2+α,r/2+β}(\cdot) Δ_{hφ(\cdot)}^k (f^{(r)},\cdot)\right\|_p \] where $φ(x) = \sqrt{1-x^2}$, $Δ_h^k(f,x)$ is the $k$th symmetric difference of $f$ on $[-1,1]$, \[ {\mathcal{W}}_δ^{ξ,ζ} (x):= (1-x-δφ(x)/2)^ξ(1+x-δφ(x)/2)^ζ, \] and $α,β> -1/p$ if $0<p<\infty$, and $α,β\geq 0$ if $p=\infty$. We show, among other things, that for all $m, n\in N$, $0<p\le \infty$, polynomials $P_n$ of d…
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 multiplier theorems. In particular, we obtain new sharp inequalities for measures of smoothness given by the $K$-functionals or moduli of smoothness. As examples of approximation processes we consider best polynomial and spline approximations, Fourier…
We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $Ω\subset \mathbb{R}^d$. This new modulus of smoothness is defined via finite differences along the directions of coordinate axes, and along a number of tangential directions from the boundary. With this modulus, we prove both the direct Jackson inequality and the corresponding inverse for best polynomial approximation in $L_p(Ω)$. The Jackson inequality is established for the full range of $0<p\leq \infty$, while its proof …
The article examines Nikolskii and Besov spaces with norms defined using "$L_p$-averaged" mixed moduli of continuity of functions of appropriate orders, instead of mixed moduli of continuity of known orders for certain mixed derivative functions. The author builds continuous linear mappings of such spaces of functions defined on cube $ I^d, $ to ordinary Nikolskii and Besov spaces of mixed smoothness in $ \mathbb R^d, $ that are function extension operators, thus incurring coincidence of both kinds of spaces on the cube $ I^d. $
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 fracti…
In this paper we investigate the gamma-relative differentiation by the motivation of amending the order of the weighted polynomial approximation on the semiaxis for certain functions. With the help of this we give some definitions of generalized Sobolev spaces, K-functionals and moduli of smoothness. We prove theorems for estimating these things with each other, in the case of first order we prove equivalence. We remark some possible applications and other generalizations too.
Let $ω_φ^k(f,δ)_{w,L_q}$ be the Ditzian-Totik modulus with weight $w$, $M^k$ be the cone of $k$-monotone functions on $(-1,1)$, i.e., those functions whose $k$th divided differences are nonnegative for all selections of $k+1$ distinct points in $(-1,1)$, and denote $E (X, Π_n)_{w,q} := \sup_{f\in X} \inf_{P\inΠ_n}\|w(f-P)\|_{L_q}$, where $Π_n$ is the set of algebraic polynomials of degree at most $n$. Additionally, let $w_{α,β}(x) := (1+x)^α(1-x)^β$ be the classical Jacobi weight, and denote by $S_p^{α,β}$ the class of all functions such that $\| w_{α,β}f\|_{L_p}=1$. In this paper, we determin…
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.
Moduli of smoothness and growth of the Laguerre transform: two-sided estimates
本文研究Laguerre变换的加权系数估计，建立了由Laguerre平移生成的光滑模与系数范数之间的双边界，并在$p=2$时得到等价性，证明了乘子权重的最优性，恢复了Jackson和Bernstein型估计及Lipschitz类的$L^2$刻画。
Discrete $d$-dimensional moduli of smoothness
本文证明了在d维立方体I^d上，仅使用二进制网格上函数值的离散光滑模足以确定函数的光滑模。对于f∈C(I^d)和整数r，当0<α<r时，离散光滑模条件等价于连续光滑模条件。
Reconstruction derivatives from values of functions belonging to Nikolskii-Besov classes of mixed smoothness in domains of a certain kind
研究在特定有界区域上，利用给定点数的函数值重构混合光滑度Nikolskii-Besov类函数导数的最佳精度，给出了上下界估计，这些估计不弱于甚至强于立方体上的结果，并扩展了适用空间类。
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.
Direct and inverse approximation theorems in the Besicovitch-Museilak-Orlicz spaces of almost periodic functions
本文在 Besicovitch-Musielak-Orlicz 空间中，借助最佳逼近和广义光滑模，为一类 Fourier 指数受限的殆周期函数建立正、逆逼近定理，并考察常数的不可改进性。
Existence and smoothness of extremizers for convolution with compactly supported measures
本文证明紧支撑测度卷积算子在非端点情形下极值函数存在且序列预紧，并具有正性与光滑性。
Polynomial approximation on $C^2$-domains
本文通过坐标轴方向和边界切向方向的有限差分定义新的光滑性模量，在连通紧致 C^2-域上建立了 L_p 多项式最佳逼近的 Jackson 不等式与逆不等式，并推广了 Ivanov 平均模量的逆定理。
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