A comparative analysis of Bernstein type estimates for the derivative of multivariate polynomials

关于多元$C^α$域上的Bernstein型与Marcinkiewicz型不等式

On Bernstein- and Marcinkiewicz-type inequalities on multivariate $C^α$-domains

该研究证明了1≤α≤2的一般紧$C^α$域边界上$L^p$空间的新Bernstein型与Markov型不等式，并将其用于建立多项式$L^p$范数离散化的渐近最优Marcinkiewicz型不等式。

Sharp Constants of Approximation Theory. II. Invariance Theorems and Certain Multivariate Inequalities of Different Metrics

We prove invariance theorems for general inequalities of different metrics and apply them to limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities with the polyharmonic operator for algebraic polynomials on the unit sphere and the unit ball in $\R^m$ and the corresponding constants for entire functions of spherical type on $\R^m$. Certain relations in the univariate weighted spaces are discussed as well.

Inequalities for Multivariate Polynomials

We summarize researches - in great deal jointly with my host Y. Sarantopoulos and his PhD. students V. Anagnostopoulos and A. Pappas - started by a Marie Curie fellowship in 2001 and is still continuing. The project was to study multivariate polynomial inequalities. In the course of work we realized the role of the ``generalized Minkowski functional'', to which we devoted a throughout survey. Resulting from this, infinite dimensional extensions of Chebyshev's extremal problems were tackled successfully. Investigating Bernstein-Markov constants for homogeneous polynomials of real normed spaces …

Quantitative estimates in approximation by Bernstein-Durrmeyer-Choquet operators with respect to monotone and submodular set functions

For the qualitative results of pointwise and uniform approximation obtained in \cite{Gal-Opris}, we present general quantitative estimates in terms of the modulus of continuity and in terms of a $K$-functional, for the generalized multivariate Bernstein-Durrmeyer operator $M_{n, Γ_{n, x}}$, written in terms of the Choquet integral with respect to a family of monotone and submodular set functions, $Γ_{n, x}$, on the standard $d$-dimensional simplex. When $Γ_{n, x}$ reduces to two elements, one a Choquet submodular set function and the other one a Borel measure, for suitable modified Bernstein-D…

Approximation of Random Functions by Random Polynomials in the Framework of Choquet's Theory of Integration

Given a submodular capacity space, we prove the uniform convergence in capacity and also the uniform convergence in the Choquet-mean of order $p\ge1$ with a quantitative estimate, of the multivariate Bernstein polynomials associated to a random function. Applications to quantitative estimates concerning the uniform convergence in capacity in the univariate case are given.

Sharp Constants of Approximation Theory. III. Certain Polynomial Inequalities of Different Metrics on Convex Sets

Let $V\subset\R^m$ be a centrally symmetric convex body and let $V^*\subset\R^m$ be its polar. We prove limit relations between the sharp constants in the multivariate Markov-Bernstein-Nikolskii type inequalities for algebraic polynomials on $V^*$ and the corresponding constants for entire functions of exponential type with the spectrum in $V$.

Sharp Constants of Approximation Theory. V. An Asymptotic Equality Related to Polynomials with Given Newton Polyhedra

Let $V\subset\R^m$ be a convex body, symmetric about all coordinate hyperplanes, and let $\PP_{aV},\, a\ge 0$, be a set of all algebraic polynomials whose Newton polyhedra are subsets of $aV$. We prove a limit equality as $a\to \iy$ between the sharp constant in the multivariate Markov-Bernstein-Nikolskii type inequalities for polynomials from $\PP_{aV}$ and the corresponding constant for entire functions of exponential type with the spectrum in $V$.

Erdélyi多元Laguerre多项式的上界

Upper bounds for Erdélyi's multivariate Laguerre polynomials

本文为Arthur Erdélyi引入的多元Laguerre多项式建立两个不等式，推广了经典的Szegö不等式，并在结论部分给出了上界的比较分析。

Polynomial approximation on $C^2$-domains

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 …