Том 71
№ 6

All Issues

Blatt H. P.

Articles: 2
Brief Communications (English)

Sign changes in rational Lw1-approximation

Blatt H. P., Grothmann R., Kovacheva R. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2006. - 58, № 2. - pp. 283–287

Let $f \in L_{1}^{w}[-1, 1]$, let $r_{n, m}(f)$ be a best rational $L_{1}^{w}$-approximation for $f$ with respect to real rational functions of degree at most n in the numerator and of degree at most m in the denominator, let $m = m(n)$, and let $\lim_{n\rightarrow \infty}(n - m(n)) = \infty$. Then we show that the counting measures of certain subsets of sign changes of $f - r_{n,m}(f)$ converge weakly to the equilibrium measure on $[-1, 1]$ as $n\rightarrow \infty$. Moreover, we prove discrepancy estimates between these counting measures and the equilibrium measure.

Article (Ukrainian)

Influence of poles on equioscillation in rational approximation

Blatt H. P.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2006. - 58, № 1. - pp. 3–11

The error curve for the rational best approximation of ƒ ? C[?1, 1] is characterized by the well-known equioscillation property. Contrary to the polynomial case, the distribution of these alternations is not governed by the equilibrium distribution. It is known that these points need not be dense in [?1, 1]. The reason is the influence of the distribution of the poles of rational approximants. In this paper, we generalize the results known so far to situations where the requirements for the degrees of numerators and denominators are less restrictive.