Multilevel Decompositions of Functional Spaces.
Multivariate polynomial inequalities viapluripotential theory and subanalytic geometry methods
We give a state-of-the-art survey of investigations concerning multivariate polynomial inequalities. A satisfactory theory of such inequalities has been developed due to applications of both the Gabrielov-Hironaka-Łojasiewicz subanalytic geometry and pluripotential methods based on the complex Monge-Ampère operator. Such an approach permits one to study various inequalities for polynomials restricted not only to nice (nonpluripolar) compact subsets of ℝⁿ or ℂⁿ but also their versions for pieces...
Nearly Coconvex Approximation
* Part of this work was done while the second author was on a visit at Tel Aviv University in March 2001Let 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 convexity of f except perhaps in some small neighborhoods of the points where f changes its convexity. We obtain...
New error inequalities for the Lagrange interpolating polynomial.
Nonlinear Approximation Theory on Compact Groups.
On a Class of Chebyshev Approximation Problems Which Arise in Connection with a Conjugate Gradient Type Method.
On a polynomial inequality of P. Erdős and T. Grünwald.
On a Tauberian theorem of Keldysh.
On an Extremal Problem concerning Bernstein Operators
* The second author is supported by the Alexander-von-Humboldt Foundation. He is on leave from: Institute of Mathematics, Academia Sinica, Beijing 100080, People’s Republic of China.The best constant problem for Bernstein operators with respect to the second modulus of smoothness is considered. We show that for any 1/2 ≤ a < 1, there is an N(a) ∈ N such that for n ≥ N(a), 1−a≤k, n≤a, sup | Bn (f, k/n) − f(k/n) | ≤ cω2(f, 1/√n), where c is a constant,0 < c < 1.
On Bernstein inequalities for multivariate trigonometric polynomials in ,
Let be the space of all trigonometric polynomials of degree not greater than with complex coefficients. Arestov extended the result of Bernstein and others and proved that for and . We derive the multivariate version of the result of Golitschek and Lorentz for all trigonometric polynomials (with complex coeffcients) in variables of degree at most .
On Bernstein's Inequality and Kahane's Ultraflat Polynomials.
On converse theorems of trigonometric approximation in weighted variable exponent Lebesgue spaces.
On extremal problems related to inverse balayage.
On Jackson type inequality in Orlicz classes.
It is shown that Jackson type inequality fails in the Orlicz classes φ(L) if φ(x) differs essentially from a power function of any order.
On norms of factors of multivariate polynomials on convex bodies.
On some shift invariant integral operators, univariate case
In recent papers the authors studied global smoothness preservation by certain univariate and multivariate linear operators over compact domains. Here the domain is ℝ. A very general positive linear integral type operator is introduced through a convolution-like iteration of another general positive linear operator with a scaling type function. For it sufficient conditions are given for shift invariance, preservation of global smoothness, convergence to the unit with rates, shape preserving and...
On the behavior of algebraic polynomials in regions with piecewise smooth boundary without cusps
We continue studying the estimation of Bernstein-Walsh type for algebraic polynomials in regions with piecewise smooth boundary.
On the best constants in some inequalities for entire functions of exponential type
On the comparison of trigonometric convolution operators with their discrete analogues for Riemann integrable functions.
On the Durrmeyer-type modification of some discrete approximation operators.