Upper estimates of Christoffel function on convex domains

New upper bounds on the pointwise behaviour of Christoffel function on convex domains in ${\mathbb{R}}^d$ are obtained. These estimates are established by explicitly constructing the corresponding "needle"-like algebraic polynomials having small integral norm on the domain, and are stated in terms of few easy-to-measure geometric characteristics of the location of the point of interest in the domain. Sharpness of the results is shown and examples of applications are given

Geometric computation of Christoffel functions on planar convex domains

For arbitrary planar convex domain, we compute the behavior of Christoffel function up to a constant factor using comparison with other simple reference domains. The lower bound is obtained by constructing an appropriate ellipse contained in the domain, while for the upper bound an appropriate parallelepiped containing the domain is constructed. As an application we obtain a new proof of existence of optimal polynomial meshes in planar convex domains

Pointwise behavior of Christoffel function on planar convex domains

We prove a general lower bound on Christoffel function on planar convex domains in terms of a modification of the parallel section function of the domain. For a certain class of planar convex domains, in combination with a recent general upper bound, this allows to compute the pointwise behavior of Christoffel function. We illustrate this approach for the domains $\{(x,y):|x|^\alpha+|y|^\alpha\le1\}$, $1<\alpha<2$, and compute up to a constant factor the required modification of the parallel section function, and, consequently, Christoffel function at an arbitrary interior point of the domain

Convexity, Moduli of Smoothness and a Jackson-Type Inequality

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 $\omega^r(f,t)_B$ 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 derived. As applications of the technique used in this paper, many new theorems are deduced. An $L_p$ space with $1<p<\infty$ satisfies $(*)$ where $s=\max(p,2),$ and many Orlicz spaces are shown to satisfy $(*)$ with appropriate $s.

Convex Multivariate Approximation by Algebras of Continuous Functions

We obtain an analog of Shvedov theorem for convex multivariate approximation by algebras of continuous functions.Comment: 9 page

On Nikol'skii inequalities for domains in RdR^d

Nikol'skii inequalities for various sets of functions, domains and weights will be discussed. Much of the work is dedicated to the class of algebraic polynomials of total degree $n$ on a bounded convex domain $D$. That is, we study $\sigma:= \sigma(D,d)$ for which $$\|P\|_{L_q(D)}\le c n^{\sigma(\frac1p-\frac1q)}\|P\|_{L_p(D)},\quad 0<p\le q\le\infty,$$ where $P$ is a polynomial of total degree $n$. We use geometric properties of the boundary of $D$ to determine $\sigma(D,n)$ with the aid of comparison between domains. Computing the asymptotics of the Christoffel function of various domains is crucial in our investigation. The methods will be illustrated by the numerous examples in which the optimal $\sigma(D,n)$ will be computed explicitly.Comment: accepted in Constructive Approximatio

Christoffel function on planar domains with piecewise smooth boundary

We compute up to a constant factor the Christoffel function on planar domains with boundary consisting of finitely many $C^2$ curves such that each corner point of the boundary has interior angle strictly between $0$ and $\pi$. The resulting formula uses the distances from the point of interest to the curves or certain parts of the curves defining the boundary of the domain

Discrete dd-dimensional moduli of smoothness

We show that on the $d$-dimensional cube $I^d\equiv [0,1]^d$ the discrete moduli of smoothness which use only the values of the function on a diadic mesh are sufficient to determine the moduli of smoothness of that function. As an important special case our result implies for $f\in C(I^d)$ and given integer $r$ that when $0<\alpha<r$, the condition $$\left|\Delta^r_{2^{-n} e_i}f\left(\frac{k_1}{2^n},\dots,\frac{k_d}{2^n}\right)\right|\le M2^{-n\alpha}$$ for integers $1\le i\le d$, $0\le k_i\le 2^n-r$, $0\le k_j\le 2^n$ when $j\ne i$, and $n=1,2,\dots$ is equivalent to $$\Bigl|\Delta^r_{h u}f(x)\Bigr|\le M_1 h^\alpha$$ for $x,u\in\mathbb{R}^d$, $h>0$ and $|u|=1$ such that $x,x+rhu\in I^d$

Constrained Spline Smoothing

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 function in the ${\mathbb L}_p$-(quasi)norm. It is well known that approximating a function by ppf's of minimal defect (splines) avoids introduction of artifacts which may be unrelated to the original function, thus it is always preferable. On the other hand, it is usually easier to construct constrained ppf's with as little requirements on smoothness as possible. Our results allow to obtain shape-preserving splines of minimal defect with equidistant or Chebyshev knots. The validity of the corresponding Jackson-type estimates for shape-preserving spline approximation is summarized, in particular we show, that the ${\mathbb L}_p$-estimates, $p\ge1$, can be immediately derived from the ${\mathbb L}_\infty$-estimates

On Banach-Mazur distance between planar convex bodies

Upper estimates of the diameter and the radius of the family of all planar convex bodies with respect to the Banach-Mazur distance are obtained. Namely, it is shown that the diameter does not exceed $\tfrac{19-\sqrt{73}}4\approx 2.614$, which improves the previously known bound of $3$, and that the radius does not exceed $\frac{117}{70}\approx 1.671$