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 $L_p[-1,1]$ (no weight), by means of algebraic polynomials. Moreover, we also prove matching inverse theorems thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials.
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.
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.
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.
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.
This is a survey on best polynomial approximation on the unit sphere and the unit ball. The central problem is to describe the approximation behavior of a function by polynomials via smoothness of the function. A major effort is to identify a correct gadget that characterizes smoothness of functions, either a modulus of smoothness or a $K$- functional, the two of which are often equivalent. We will concentrate on characterization of best approximations, given in terms of direct and converse theorems, and report several moduli of smoothness and $K$-functionals, including recent results that giv…
We prove lower bounds for the approximation error of the variation-diminishing Schoenberg operator on the interval $[0,1]$ in terms of classical moduli of smoothness depending on the degree of the spline basis using a functional analysis based framework. Thereby, we characterize the spectrum of the Schoenberg operator and investigate the asymptotic behavior of its iterates. Finally, we prove the equivalence between the approximation error and the classical second order modulus of smoothness as an improved version of an open conjecture from 2002.
In this paper we study direct and inverse approximation inequalities in $L^{p}(\mathbb{R}^{d})$, $1<p<\infty$, with the Dunkl weight. We obtain these estimates in their sharp form substantially improving previous results. We also establish new estimates of the modulus of smoothness of a function $f$ via the fractional powers of the Dunkl Laplacian of approximants of $f$. Moreover, we obtain new Lebesgue type estimates for moduli of smoothness in terms of Dunkl transforms. Needed Pitt-type and Kellogg-type Fourier--Dunkl inequalities are derived.
In Musilak-Orlicz type spaces ${\mathcal S}_{\bf M}$, direct and inverse approximation theorems are obtained in terms of the best approximations of functions and generalized moduli of smoothness. The question of the exact constants in Jackson-type inequalities is studied.
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 several forms of the Bernstein polynomials with integer coefficients possess the property of simultaneous approximation, that is, they approximate not only the function but also its derivatives. We establish direct estimates of the error of that approximation in uniform norm by means of moduli of smoothness. Moreover, we show that the sufficient conditions under which those estimates hold are also necessary.
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 $L_p$-rate of convergence for these operators is given in terms of the classical moduli of smoothness. Several examples of the Kantorovich-Kotelnikov operators generated by the ${\rm sinc}$-function and its linear combinations are provided.
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 $H_p^{r,α}$.
We prove matching direct and inverse theorems for uniform polynomial approximation with $A^*$ weights (a subclass of doubling weights suitable for approximation in the $L_\infty$ norm) having finitely many zeros and not too "rapidly changing" away from these zeros. This class of weights is rather wide and, in particular, includes the classical Jacobi weights, generalized Jacobi weights and generalized Ditzian-Totik weights. Main part and complete weighted moduli of smoothness are introduced, their properties are investigated, and equivalence type results involving related realization functiona…
We prove a weak converse estimate for the simultaneous approximation by several forms of the Bernstein polynomials with integer coefficients. It is stated in terms of moduli of smoothness. In particular, it yields a big $O$-characterization of the rate of that approximation. We also show that the approximation process generated by these Bernstein polynomials with integer coefficients is saturated. We identify its saturation rate and the trivial class.
In this paper, we introduce new modifications of Szasz-Mirakyan operators based on (p,q)-integers. We first give a recurrence relation for the moments of new operators and present explicit formula for the moments and central moments up to order 4. Some approximation properties of new operators are explored: the uniform convergence over bounded and unbounded intervals is established, direct approximation properties of the operators in terms of the moduli of smoothness is obtained and Voronovskaya theorem is presented. For the particular case p=1, the previous results for q-Szasz-Mirakyan operat…
Approximative properties of linear summation methods of Fourier series are considered in the Orlicz type spaces ${\mathcal S}_{M}$. In particular, in terms of approximations by such methods, constructive characteristics are obtained for classes of functions whose smoothness moduli do not exceed a certain majorant.
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.
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 a…
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 a…
