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.
We continue studying the estimation of Bernstein-Walsh type for algebraic polynomials in regions with piecewise smooth boundary.
By the Oka-Weil theorem, each holomorphic function f in a neighbourhood of a compact polynomially convex set can be approximated uniformly on K by complex polynomials. The famous Bernstein-Walsh-Siciak theorem specifies the Oka-Weil result: it states that the distance (in the supremum norm on K) of f to the space of complex polynomials of degree at most n tends to zero not slower than the sequence M(f)ρ(f)ⁿ for some M(f) > 0 and ρ(f) ∈ (0,1). The aim of this note is to deduce the uniform version,...
For , precise conditions on the parameters are given under which the particular superposition operator is a bounded map in the Besov space . The proofs rely on linear spline approximation theory.
Crack propagation in anisotropic materials is a persistent problem. A general concept to predict crack growth is the energy principle: A crack can only grow, if energy is released. We study the change of potential energy caused by a propagating crack in a fully three-dimensional solid consisting of an anisotropic material. Based on methods of asymptotic analysis (method of matched asymptotic expansions) we give a formula for the decrease in potential energy if a smooth inner crack grows along a...
C. Watari [12] obtained a simple characterization of Lipschitz classes on the dyadic group using the -modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality of the...
Let be the union of infinitely many disjoint closed intervals where , , , Let be a nonnegative function and a sequence of distinct complex numbers. In this paper, a theorem on the completeness of the system in is obtained where is the weighted Banach space consists of complex functions continuous on with vanishing at infinity.