Kamen G. Ivanov

Strong converse inequalities

Techniques are developed to obtain strong converse inequalities for various linear approximation processes. This will establish equivalence between the approximating rate of a certain linear process and the appropriate PeetreK-functional. Approximation processes that will be treated have to be saturated asK-functionals are saturated. These general methods will lead to new results on the various trigonometric polynomial approximation processes, on holomorphic semigroups, on Bernstein polynomials and on other commonly used approximation processes.

Fast decreasing polynomials

Matching two-sided estimates are given for the minimal degree of polynomialsP satisfyingP(0)=1 and ¦P(x)|≤exp(−ϕ (¦x¦)),x ∈ [−1,1], whereϕ is an arbitrary, in [0, 1], increasing function. Besides these fast decreasing polynomials we also consider bell-shaped polynomials and polynomials approximating well the signum function.

A characterization of weighted Peetre K -functionals

The purpose of this paper is to present a characterization of certain types of generalized weighted Peetre K-functionals by means of a modulus of smoothness. This new modulus is based on the classical one taken on a certain linear transform of the function. A new modulus of smoothness which describes the best algebraic approximation is introduced.

Approximation by polynomials with locally geometric rates

In contrast to the behavior of best uniform polynomial approximants on [0, 1] we show that if f E C[O, 1] there exists a sequence of polynomials {Pn} of respective degree < n which converges uniformly to f on [0, 1] and geometrically fast at each point of [0, 1] where f is analytic. Moreover we describe the best possible rates of convergence at all regular points for such a sequence.

Converse theorems for approximation by bernstein polynomials in Lp[0,1] (1<p<∞)

The class of all continuous functions possessing n−α(1/p<α≤1) order of approximation by Bernstein polynomials inLp[0, 1] is characterized.

Fast memory efficient evaluation of spherical polynomials at scattered points

A method for fast evaluation of band-limited functions (spherical polynomials) at many scattered points on the unit 2-d sphere is presented. The method relies on the superb localization of the father needlet kernels and their compatibility with spherical harmonics. It is fast, local, memory efficient, numerically stable and with guaranteed (prescribed) accuracy. The speed is independent of the band limit and depends logarithmically on the prescribed accuracy. The method can be also applied for approximation on the sphere, verification of spherical polynomials and for fast generation of surfaces in computer-aided geometric design. It naturally extends to higher dimensions.

New estimates of errors of quadrature formulae, formulae of numerical differentiation and interpolation

AbstractВ работе в качестве ха рактеристики функци иf рассматриваются сле дующие ее модули: ωk(f; x; δ)=sup {¦Δhkf(t¦: t, t+kh∈[x-kδ/2, x+kδ/2][a, b], ωk(f; δ)={sup ωk(f; x; δ): х∈[а, Ь].Получены оценки погр ешности квадратурны х формул с помощью модулят. Нап ример, справедливо следующ ее утверждение. Пусть квадратурная формул а точна на отрезке [а, Ь] д ля всех алгебраическ их многочленов степени не вышеk- 1 иRn(f) — погрешностьn-соста вной квадратуры, поро жденнойL(f). Тогда

More quantitative results on walsh equiconvergence: I. Lagrange Case

L(f) = \sum\limits_{i = 0}^m {\sum\limits_{j = 0}^{\alpha _i } {A_i^j f^{(j)} (x_i )} }