Dany Leviatan

• PhD
• Professor Emeritus at Tel Aviv University

In a recent paper, for a fixed $m\in\mathbb N$, we introduced trigonometric polynomials$$L_n(x):=\frac1{h^m}\underbrace{\int_{-h/2}^{h/2}\dots\int_{-h/2}^{h/2}}_{m\,\text{times}}J_n(x+t_1+\cdots+t_m)\,dt_1\cdots\,dt_m,$$where $J_n$ is a Jackson-type kernel. In the current paper we show that $L_n$ and its first $m-1$ derivatives provide approximatio...

We give here the final results about the validity of Jackson-type estimates in comonotone approximation of 2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\pi $$\end...

We construct trigonometric polynomials that fast decrease towards ±π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm \pi $$\end{document}. We apply them to construct...

We show that certain inequalities involving differences of the Bernstein basis polynomials and values of a function f∈C[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen...

We say that a function f ∈ C[a, b] is q-monotone, q ≥ 2, if f ∈ Cq-2(a, b), i.e., belongs to the space of functions with (q -2)th continuous derivative in (a, b), and f(q-2) is convex in this space. Let f be continuous and 2𝜋-periodic. Assume that it changes its q-monotonicity finitely many times in [-𝜋, 𝜋]. We are interested in estimating the degr...

We show that certain inequalities involving differences of the Bernstein basis polynomials and values of a function f∈C[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{documen...

Let \({{\widetilde{C}}}\) be the space of continuous \(2\pi \)-periodic functions f, endowed with the uniform norm \(\Vert f\Vert :=\max _{x\in {\mathbb {R}}}|f(x)|\), and denote by \(\omega _k(f,t)\), the k-th modulus of smoothness of f. Denote by \({{\widetilde{C}}}^r\), the subspace of r times continuously differentiable functions \(f\in {{\wide...

We call a hybrid polynomial a function \(Q(x)=\alpha x^2+\beta x+T(x)\), where \(\alpha ,\beta \in {\mathbb {R}}\) and T is a trigonometric polynomial. For \(s\ge 1\), let \(Y_s:=\{y_{i}\}_{i\in {\mathbb {Z}}}\), be such that \(y_{i}+2\pi = y_{i+2s}\), \(i\in {\mathbb {Z}}\). We approximate a function \(f(x):=g(x)+\alpha x^2\), where g is a continu...

UDC 517.5 We say that a function f ∈ C [ a , b ] is q -monotone, q ≥ 3 , if f ∈ C q - 2 ( a , b ) and f ( q - 2 ) is convex in ( a , b ) . Let f be continuous and 2 π -periodic, and change its q -monotonicity finitely many times in [ - π , π ] . We are interested in estimating the degree of approximation of f by trigonometric polynomials which are...

We prove that various Durrmeyer-type operators preserve q-monotonicity in [0, 1] or [0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty )$$\end{document} as...

We establish best possible pointwise (up to a constant multiple) estimates for approximation, on a finite interval, by polynomials that satisfy finitely many (Hermite) interpolation conditions, and show that these estimates cannot be improved. In particular, we show that any algebraic polynomial of degree n approximating a function f∈Cr(I), I=[−1,1...

We extend an inequality involving the Bernstein basis polynomials and convex functions on [0, 1]. The inequality was originally conjectured by Raşa about thirty years ago, but was proved only recently. Our extension provides an inequality involving q-monotone functions, q∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \use...

This paper deals with approximation of smooth convex functions f on an interval by convex algebraic polynomials which interpolate f and its derivatives at the endpoints of this interval. We call such estimates “interpolatory”. One important corollary of our main theorem is the following result on approximation of \(f \in \Delta ^{(2)}\), the set of...

We establish best possible pointwise estimates for approximation, on a finite interval, by polynomials which are satisfying finitely many (Hermite) interpolation conditions, and show that these estimates cannot be improved. In particular, we show that {\bf any} algebraic polynomial of degree $n$ approximating a function $f\in C^r(I)$, $I=[-1,1]$, a...

We say that a function $f\in C[a,b]$ is $q$-monotone, $q\ge3$, if $f\in C^{q-2}(a,b)$ and $f^{(q-2)}$ is convex in $(a,b)$. Let $f$ be continuous and $2\pi$-periodic, and change its $q$-monotonicity finitely many times in $[-\pi,\pi]$. We are interested in estimating the degree of approximation of $f$ by trigonometric polynomials which are co-$q$-m...

This paper deals with approximation of smooth convex functions $f$ on an interval by convex algebraic polynomials which interpolate $f$ at the endpoints of this interval. We call such estimates "interpolatory". One important corollary of our main theorem is the following result on approximation of $f\in \Delta^{(2)}$, the set of convex functions, f...

We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as $$\displaystyle{\omega }_{k,r}^\varphi (f^{(r)},t)_{\alpha ,\beta ,p} :=\sup _{0\leq h\leq t}\left \|{\mathcal {W}}_{kh}^{r/2+\alpha ,r/2+\beta }(\cdot )\Delta _{h\varphi (\cdot )}^k (f^{(r)},\cdot )\right \|{ }_{p},$$...

We give an estimate for the general divided differences [x0, . . . ,xm; f], where some points xi are allowed to coalesce (in this case, f is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen the celebrated Whitney and Marchaud inequalities and their generalization to the Hermite interpolation. As one of n...

In this paper, among other things, we show that, given r∈N, there is a constant c=c(r) such that if f∈Cr[−1,1] is convex, then there is a number N=N(f,r), depending on f and r, such that for n≥N, there are convex piecewise polynomials S of order r+2 with knots at the nth Chebyshev partition, satisfying |f(x)−S(x)|≤c(r)(min⁡{1−x²,n⁻¹1−x²})rω2(f(r),n...

We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as \[ \omega_{k,r}^\varphi(f^{(r)},t)_{\alpha,\beta,p} :=\sup_{0\leq h\leq t} \left\| {\mathcal{W}}_{kh}^{r/2+\alpha,r/2+\beta}(\cdot) \Delta_{h\varphi(\cdot)}^k (f^{(r)},\cdot)\right\|_p \] where $\varphi(x) = \sqrt{1-x^...

It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss rece...

We give an estimate of the general divided differences $[x_0,\dots,x_m;f]$, where some of the $x_i$'s are allowed to coalesce (in which case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen Whitney and Marchaud celebrated inequalities in relation to Hermite interpolation. For example, one of the...

Let x 0 , x 1 , . . ., x n ∈ ℝ, be pairwise disjoint, and let θ 0 , θ 1 , . . ., θ n ∈ Set θ := Σ ν=0ⁿ θ ν . For each pair j, p such that 0 ≤ j ≤ n and 0 ≤ p ≤ θ j -1, let y j,p be a complex number. Then there is a unique polynomial, H(x), of degree θ - 1, such that H (p) (x j ) = y j,p , for 0 ≤ p ≤ θ j - 1, 0 ≤ j ≤ n. In particular, there is a un...

In this paper, among other things, we show that, given $r\in N$, there is a constant $c=c(r)$ such that if $f\in C^r[-1,1]$ is convex, then there is a number ${\mathcal N}={\mathcal N}(f,r)$, depending on $f$ and $r$, such that for $n\ge{\mathcal N}$, there are convex piecewise polynomials $S$ of order $r+2$ with knots at the Chebyshev partition, s...

Given a monotone function [Formula presented], r≥1, we obtain pointwise estimates for its monotone approximation by piecewise polynomials involving the second order modulus of smoothness of f(r). These estimates are interpolatory estimates, namely, the piecewise polynomials interpolate the function at the endpoints of the interval. However, they ar...

The inequality on line 8 of page 826 is stated in the wrong direction.

We obtain matching direct and inverse theorems for the degree of weighted Lp-approximation by polynomials with the Jacobi weights (1−x)α(1+x)β. Combined, the estimates yield a constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials. In addition, we prove Whitney type in...

We correct a couple of misprints in the above mentioned article.

Given a nondecreasing function $f$ on $[-1,1]$, we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at $\pm 1$. We establish pointwise estimates of the approximation error by such polynomials that yield interpolation at the endpoints (i.e., the estimates become zero at $\pm 1$). We call such est...

Given a nondecreasing function $f$ on $[-1,1]$, we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at $\pm 1$. We establish pointwise estimates of the approximation error by such polynomials that yield interpolation at the endpoints (i.e., the estimates become zero at $\pm 1$). We call such est...

We obtain matching direct and inverse theorems for the degree of weighted $L_p$-approximation by polynomials with the Jacobi weights $(1-x)^\alpha (1+x)^\beta$. Combined, the estimates yield a constructive characterization of various smoothness classes of functions via the degree of their approximation by algebraic polynomials.

The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights $(1-x)^\alpha(1+x)^\beta$ 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...

The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights $(1-x)^\alpha(1+x)^\beta$ 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...

Given a monotone function f∈Cr[−1,1], r≥1, we obtain pointwise estimates for its monotone approximation by piecewise polynomials involving the second order modulus of smoothness of f(r). These estimates are interpolatory estimates, namely, the piecewise polynomials interpolate the function at the endpoints of the interval. However, they are valid o...

Let f∈C[−1,1] and denote by En(f) its degree of approximation by algebraic polynomials of degree <n. Assume that f changes its monotonicity, respectively, its convexity finitely many times, say s≥2 times, in (−1,1) and we know that for q=1 or q=2 and some 1<α≤2, such that qα≠4, we have En(f)≤n−qα,n≥s+q+1. The purpose of this paper is to prove that...

For each $q\in{\mathbb{N}}_0$, we construct positive linear polynomial approximation operators $M_n$ that simultaneously preserve $k$-monotonicity for all $0\leq k\leq q$ and yield the estimate \[ |f(x)-M_n(f, x)| \leq c \omega_2^{\varphi^\lambda} \left(f, n^{-1} \varphi^{1-\lambda/2}(x) \left(\varphi(x) + 1/n \right)^{-\lambda/2} \right) , \] for...

For each $q\in{\mathbb{N}}_0$, we construct positive linear polynomial approximation operators $M_n$ that simultaneously preserve $k$-monotonicity for all $0\leq k\leq q$ and yield the estimate \[ |f(x)-M_n(f, x)| \leq c \omega_2^{\varphi^\lambda} \left(f, n^{-1} \varphi^{1-\lambda/2}(x) \left(\varphi(x) + 1/n \right)^{-\lambda/2} \right) , \] for...

In this paper, we prove that for $\ell \,=\,1$ or 2 the rate of best $\ell $ - monotone polynomial approximation in the ${{L}_{p}}$ norm $\left( 1\,\le \,p\,\le \,\infty \right)$ weighted by the Jacobi weight ${{w}_{\alpha ,\,\beta }}\left( x \right)\,:=\,{{\left( 1\,+\,x \right)}^{\alpha }}{{\left( 1\,-\,x \right)}^{\beta }}$ with $\alpha ,\,\beta...

Let (Formula Presented), be nonlinear and nondecreasing. We wish to estimate the degree of approximation of f by trigonometric polynomials that are nondecreasing in [-ω,ω]. We obtain estimates involving the second modulus of smoothness of f and show that one in general cannot have estimates with the third modulus of smoothness.

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 $\varphi^rf^{(r)}$ is in $L_p[-1,1]$, where $\varphi(x)=(1-x^2)^{1/2}$. These moduli are equivalent to certain weighted DT moduli, but our defi...

In this paper, we discuss various properties of the new modulus of smoothness \[ \omega^\varphi_{k,r}(f^{(r)},t)_p := \sup_{0 < h\leq t}\|\mathcal W^r_{kh}(\cdot) \Delta_{h\varphi(\cdot)}^k (f^{(r)},\cdot)\|_{L_p[-1,1]}, \] where $\mathcal W_\delta(x) = \bigl((1-x-\delta\varphi(x)/2) (1+x-\delta\varphi(x)/2)\bigr)^{1/2}. $ Related moduli with more...

Let E"n(f) denote the degree of approximation of f@?C[-1,1], by algebraic polynomials of degree 0 and N>=2, n^@aE"n(f)@?1,n>=N. Suppose that f changes its monotonicity s>=1 times in [-1,1]. We are interested in what may be said about its degree of approximation by polynomials of degree =N^*, for some N^*. Clearly, N^*, if it exists at all (we prove...

Let f ∈ C[-1, 1] change its convexity in the interval. We are interested in estimating the degree of approximation of f by polynomials which are coconvex with it, namely, polynomials that change their convexity exactly at the points where f does. We obtain Nikolskii-type pointwise estimates for such constrained approximation.

We estimate the degree of comonotone polynomial approximation of continuous functions f, on [−1,1], that change monotonicity s≥1 times in the interval, when the degree of unconstrained polynomial approximation En(f)≤n^−α, n≥1. We ask whether the degree of comonotone approximation is necessarily ≤c(α,s)n^−α, n≥1, and if not, what can be said. It tur...

We prove that for a 3-monotone function F@?C[-1,1], one can achieve the pointwise estimates |F(x)-@J(x)|@?c@w"3(F,@r"n(x)),x@?[-1,1], where @r"n(x)@?1n^2+1-x^2n and c is an absolute constant, both with @J, a 3-monotone quadratic spline on the nth Chebyshev partition, and with @J, a 3-monotone polynomial of degree @?n. The basis for the construction...

We analyze the degree of shape preserving weighted polynomial approximation for exponential weights on the whole real line. In particular, we establish a Jackson type estimate.

In his famous paper [15] Ibragim Ibishievich Ibragimov has given asymptotic values of the best uniform approximation of functions of the form (a − x) s ln m (a − x), (a ≥ 1). These results have led to the development of a series of new directions in approximation theory, including the following ones, to which we devote this paper. • Constructive ch...

We consider the problem of reconstructing an unknown function $f$ on a domain $X$ from samples of $f$ at $n$ randomly chosen points with respect to a given measure $\rho_X$. Given a sequence of linear spaces $(V_m)_{m>0}$ with ${\rm dim}(V_m)=m\leq n$, we study the least squares approximations from the spaces $V_m$. It is well known that such appro...

We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a finite interval. In this article, "shape" refers to (finitely many changes of) monotonicity, convexity, or q-monotonicity of a function (for definition, see Section 4). It is rather well known that it is possible t...

In Part I of the paper, we have proved that, for every α > 0 and a continuous function f, which is either convex (s = 0) or changes convexity at a finite collection Y s = {y i } s i=1 of points y i ∈ (-1, 1), sup{ na En(2)( f,Ys ):n \geqslant N* } \leqslant c( a, s )sup{ na En(f):n \geqslant 1 }, \sup \left\{ {{n^\alpha }E_n^{(2)}\left( {f,{Y_...

For each r ∈ N we prove the Nikolskii type pointwise estimate for coconvex approximation of functions f ∈ Wr, the subspace of all functions f ∈ C[-1, 1], possessing an absolutely continuous (r - 1)st derivative on (-1, 1) and satisfying f(r) ∈ L∞[-1, 1], that change their convexity once on [-1, 1].

Given a monotone or convex function on a finite interval we construct splines of arbitrarily high order having maximum smoothness which are "nearly monotone" or "nearly convex" and provide the rate of L p -approximation which can be estimated in terms of the third or fourth (classical or Ditzian–Totik) moduli of smoothness (for uniformly spaced or...

Let Wpr (Bd) be the usual Sobolev class of functions on the unit ball Bd in Rd, and Wp{ring operator}, r (Bd) be the subclass of all radial functions in Wpr (Bd). We show that for the classes Wp{ring operator}, r (Bd) and Wpr (Bd), the orders of best approximation by polynomials in Lq (Bd) coincide. We also obtain exact orders of best approximation...

Let ℂ[−1,1] be the space of continuous functions on [−,1], and denote by Δ2 the set of convex functions f ∈ ℂ[−,1]. Also, let E n (f) and E n (2) (f) denote the degrees of best unconstrained and convex approximation of f ∈ Δ2 by algebraic polynomials of degree n, respectively. Clearly, En (f) ≦ E n (2) (f), and Lorentz and Zeller proved that the...

We obtain estimates on the order of best approximation by polynomials and ridge functions in the spaces LqLq of classes of s-monotone radial functions which belong to the space LpLp, 1⩽q⩽p≤∞1⩽q⩽p≤∞.

Several results on constrained spline smoothing are obtained. In particular, we establish a general result, showing how one can constructively smooth any monotone or convex piecewise polynomial function (ppf) (or any $q$-monotone ppf, $q\geq 3$, with one additional degree of smoothness) to be of minimal defect while keeping it close to the original...

Let I be a finite interval, s ∈ ℕ0, and r,ν,n ∈ ℕ. Given a set M, of functions defined on I, denote by Ds + \Delta ^{s}_{ + } M the subset of all functions y ∈ M such that the s-difference Dst y()\Delta ^{s}_{\tau } y(\cdot) is nonnegative on I, ∀τ > 0. Further, denote by WrpW^{r}_{p} the Sobolev class of functions x on I with the seminorm ||x(r)|...

We study nonlinear m-term approximation with regard to a redundant dictionary D\mathcal {D} in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f∈H and any dictionary D\mathcal {D} an expansion into a series f=åj=1¥cj(f)jj(f),jj(f) Î D,j=1,2,¼,f=\sum_{j=1}^{\inft...

Let I be a finite interval, r,n∈N, s∈N0 and 1⩽p⩽∞. Given a set M, of functions defined on I, denote by Δ+sM the subset of all functions y∈M such that the s-difference Δτsy(·) is nonnegative on I, ∀τ>0. Further, denote by Wpr the Sobolev class of functions x on I with the seminorm ∥x(r)∥Lp⩽1. We obtain the exact orders of the Kolmogorov and the line...

The paper deals with approximation of a continuous function, on a finite interval, which changes convexity finitely many times, by algebraic polynomials which are coconvex with it. We give final answers to open questions concerning the validity of Jackson type estimates involving the weighted Ditzian-Totik moduli of smoothness.

It is shown that the rate of L p -approximation of a non-decreasing function in L p , 0 < p < ∞, by "nearly non-decreasing" splines can be esti-mated in terms of the third classical modulus of smoothness (for uniformly spaced knots) and third Ditzian-Totik modulus (for Chebyshev knots), and that estimates in terms of higher moduli are impossible. I...

Estimating the degree of approximation in the uniform norm, of a convex function on a finite interval, by convex algebraic polynomials, has received wide attention over the last twenty years. However, while much progress has been made especially in recent years by, among others, the authors of this article, separately and jointly, there have been l...

We study nonlinear m-term approximation with regard to a redundant dictionary DD in a Banach space. It is known that in the case of Hilbert space H the pure greedy algorithm (or, more generally, the weak greedy algorithm) provides for each f∈Hf∈H and any dictionary DD an expansion into a seriesf=∑j=1∞cj(f)ϕj(f),ϕj(f)∈D,j=1,2,…with the Parseval prop...

We consider 3-monotone approximation by piecewise polynomials with pre-scribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L1 approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone a...

Since the Sobolev set Wpr, 0 < p < 1, in general is not contained in Lq, 0 < q ≤ ∞, we limit ourselves to the set Wpr∩L∞, 0 < p < 1. We prove that the Kolmogorov n-width of the latter set in L q, 0 < q < 1 is asymptotically 1, that is, the set cannot be approximated by n-dimensional linear manifolds in the Lq-norm. We then describe a related set, t...

The Binary Space Partition (BSP) technique is a simple and efficient method to adaptively partition an initial given domain to match the geometry of a given input function. As such the BSP technique has been widely used by practitioners, but up until now no rigorous mathematical justification to it has been offered. Here we attempt to put the techn...

In recent years there have been various attempts at the representations of {\mbox multivariate} signals such as images, which outperform wavelets. As is well known, wavelets are not optimal in that they do not take full advantage of the geometrical regularities and singularities of the images. Thus these approaches have been based on tracing curves...

Let I be a finite interval and r,sN. Given a set M, of functions defined on I, denote by + s M the subset of all functions yM such that the s-difference s y() is nonnegative on I, >0. Further, denote by + s W p r , the class of functions x on I with the seminorm x (r)L p 1, such that s x0, >0. Let M n (h k ):={ i=1 n c i h k...

We prove the following Whitney estimate. Given 0 < p \le \infty, r \in N, and d \ge 1, there exists a constant C(d,r,p), depending only on the three parameters, such that for every bounded convex domain \subset Rd, and each function f \in Lp(), Er-1(f,)p \le C(d,r,p)r(f, diam())p, where Er-1(f,)p is the degree of approximation by polynomials of t...

It is well known that it is possible to enhance the approximation properties of a kernel operator by increasing its support size. There is an obvious tradeoff between higher approximation order of a kernel and the complexity of algorithms that employ it. A question is then asked: how do we compare the efficiency of kernels with comparable support s...

The Bramble-Hilbert lemma is a fundamental result on multivariate polynomial ap- proximation. It is frequently applied in the analysis of Finite Element Methods (FEM) used for numerical solutions of PDEs. However, this classical estimate depends on the geometry of the domain and may 'blow-up' for simple examples such as a sequence of triangles of e...

LetI be a finite interval andr ∈ ℕ. Denote by △ + s L q the subset of all functionsy ∈L q such that thes-difference △ T s y(·) is nonnegative onI, ∀τ>0. Further, denote by △ + s W p r the class of functionsx onI with the seminorm ‖x (r) ‖L p ≤1, such that △ T s x≥0, τ > 0, τ>0. Fors=3,…,r+1, we obtain two-sided estimates of the sha...

We survey some recent results in the theory of multivariate piecewise polynomial approximation. In the univariate case this method is equivalent to Wavelet approximation, but in the multivariate case this is no longer true, since this form of approximation is more adaptive to the geometry of the singularities of the function to be approximated. The...

Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials, and by piecewise polynomials, which are coconvex with it, namely, polynomials and piecewise polynomials that change their convexity exactly at the points where f does. We obtain Jackson type es...

Let I be a finite interval, r∈ N and ρ(t)= dist {t, I} , t∈ I . Denote by Δ s + L q the subset of all functions y∈ L q such that the s -difference Δ s τ y(t) is nonnegative on I , $$\forall$$ τ>0 . Further, denote by $$\Delta^s_+W^r_{p,\alpha}$$ , 0≤α<∞ , the classes of functions x on I with the seminorm ||x (r) ρ α ||_ L p ≤ 1 , such that Δ s...

In the univariate case there are certain equivalences between the nonlinear approximation methods that use piecewise polynomials and those that use rational functions. It is known that for certain parameters the respective approximation spaces are identical and may be described as Besov spaces. The characterization of the approximation spaces of th...

Let f 2 C(¡1;1) change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials which are coconvex with it, namely, polynomials that change their convexity exactly at the points where f does. We discuss some Jackson type estimates where the constants involved depend on the l...

Let I be a finite interval, rε N and p(t)=dist{t,∂I}, tεI. Denote by W rp,α, 0<α<∞, the class of functions x on I with the seminorm ∥x (r)p α∥Lp≤1. We obtain two-sided estimates of the Kolmogorov widths d n(Wrp,α)Lq and of the linear widths d n(Wrp,α)Lqlin

Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials, and by piecewise polynomials, which are nearly coconvex with it, namely, polynomials and piecewise polynomials that preserve the convex-ity of f except perhaps in some small neighborhoods of th...

Let X be a real linear space of vectors x with normx‖x‖x ,W ⊂ X,W ≠ ∅, and V ⊂X,V ≠∅.Let L n be a subspace in X of dimension dim L n ≤ n,n≥ 0.and M n =M n (x 0) :=x 0 + L n be a shift of the subspace L n by an arbitrary vector x 0∈ X.

The motivation for this work is a recently constructed family of generators of shift invariant spaces with certain optimal approximation properties, but which are not refinable in the classical sense. We try to see whether, once the classical refinability requirement is removed, it is still possible to construct meaningful wavelet decompositions of...

We obtain Jackson-type estimates for the simultaneous approximation of a 3monotone twice continuously dierentiable function x by means of quadratic splines with equidistant knots. x1.

We obtain Jackson-type estimates for the simultaneous approximation of a 3monotonetwice continuously dierentiable function x by means of quadratic splines withequidistant knots.x1.

We discuss whether or not it is possible to have interpolatory pointwise estimates in the approximation of a function f∈ C[0,1] , by polynomials. For the sake of completeness, as well as in order to strengthen some existing results, we discuss briefly the situation in unconstrained approximation. Then we deal with positive and monotone constraints...

. We are going to survey recent developments and achievements in shape preserving approximation by polynomials. We wish to approximate a function f defined on a finite interval, say [Gamma1; 1], while preserving certain intrinsic "shape" properties. To be specific we demand that the approximation process preserve properties of f , like its sign in...

. The main achievement of this article is that we show what was to us a surprising conclusion, namely, twice continuously differentiable functions in (0,1) (with some regular behavior at the endpoints) which change monotonicity at least once in the interval are approximable better by comonotone polynomials, than such functions which are merely mono...

. When we approximate a continuous function f which changes its monotonicity finitely many, say s times, in [Gamma1; 1], we wish sometimes that the approximating polynomials follow these changes in monotonicity. However, it is well known that this requirement restricts very much the degree of approximation that the polynomials can achieve, namely,...

Letfbe a continuous function on [−1, 1], which changes its monotonicity finitely many times in the interval, saystimes. In the first part of this paper we have discussed the validity of Jackson type estimates for the approximation offby algebraic polynomials that are comonotone with it. We have proved the validity of a Jackson type estimate involvi...

We show that the degree of copositive polynomial approximation to a function f 2 C[0; 1] which changes sign finitely many times in [0; 1], can be estimatedby C! 2 (f; 1=n) where C is a constant which depends only on the number of sign changes and on the minimal distance between them. The estimate holds for all n sufficiently large, where again how...

. We survey the Jackson and Jackson type estimates for comonotone polynomial approximation of continuous and r-times continuously differentiable functions which change their monotonicity finitely many times in a finite interval, say [Gamma1; 1], with special attention to the constants involved in the estimates. We describe four possibilities rangin...

Let f be a continuous function on [Gamma1; 1], which changes its monotonicity finitely many times in the interval, say s times. In the first part of this paper we have discussed the validity of Jackson type estimates for the approximation of f by algebraic polynomials that are comonotone with it. We have proved the validity of a Jackson type estima...

In this paper we study the existence and characterization of spaces which are images of minimal-norm projections that are required to interpolate at given functionals and satisfy additional shape-preserving requirements. We will call such spaces optimal interpolating spaces preserving shape. This investigation leads to concrete solutions in classic...