A model space approach to some classical inequalities for rational functions

Abstract

Abstract. We consider the set R n of rational functions of degree at most n ≥ 1 with no poles on the unit circle T and its subclass R n, r consisting of rational functions without poles in the annulus ξ : r ≤ |ξ| ≤ 1 r . We discuss an approach based on the model space theory which brings some integral representations for functions in R n and their derivatives. Using this approach we obtain L p -analogs of several classical inequalities for rational functions including the inequalities by P. Borwein and T. Erdélyi, the Spijker Lemma and S.M. Nikolskii's inequalities. These inequalities are shown to be asymptotically sharp as n tends to infinity and the poles of the rational functions approach the unit circle T.

Quadrature formulas with variable nodes and Jackson–Nikolskii inequalities for rational functions

“…Due to the variable nodes, this formula will imply (q, p)-inequalities of Jackson-Nikolskii type for rational functions on the circle γ r . These inequalities refine the results obtained earlier in [3] and [10]. What is more, our (∞, 2)-inequalities are sharp.…”

“…where the sup is taking over all rational functions of degree n with poles outside the annulus A δ . What is more, there is an example in [3] showing the asymptotic sharpness of the constant (3.5) when (q, p) = (∞, 2) and n → ∞.…”

“…As for sharpness, it is shown in [3] that there exists a constant c(p, q) > 0 such that for all n 2 and δ ∈ (0, 1) the following inequality holds:…”

“…Quadrature formulas contributed significantly to rational interpolation and approximation theory, in particular, to the study of approximating properties of special classes of rational fractions such as the (sine) cosine Chebyshev-Markov fractions. Moreover, they were actively used in various extremal problems, for example, for obtaining Markov-Bernstein, Szegő and Nikolskii type inequalities for rational functions on circles and the real axis (see the history of this question and the references in [6], [16, Section 11] and [3,).…”

$H^{\infty}$ interpolation and embedding theorems for rational functions

“…We will use the following simple Bernstein-type inequality for rational functions (see, e.g., [3,Theorem 2.3]). Let σ = (λ 1 , .…”