English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Given a non-trivial Borel measure $\mu$ on the unit circle $\mathbb T$, thecorresponding reproducing (or Christoffel-Darboux) kernels with one of thevariables fixed at $z=1$ constitute a family of so-called para-orthogonalpolynomials, whose zeros belong to $\mathbb T$. With a proper normalizationthey satisfy a three-term recurrence relation determined by two sequence ofreal coefficients, $\{c_n\}$ and $\{d_n\}$, where $\{d_n\}$ is additionally apositive chain sequence. Coefficients $(c_n,d_n)$ provide a parametrization ofa family of measures related to $\mu$ by addition of a mass point at $z=1$.  In this paper we estimate the location of the extreme zeros (those closest to$z=1$) of the para-orthogonal polynomials from the $(c_n,d_n)$-parametrizationof the measure, and use this information to establish sufficient conditions forthe existence of a gap in the support of $\mu$ at $z=1$. These results areeasily reformulated in order to find gaps in the support of $\mu$ at any other$z\in \mathbb T$.  We provide also some examples showing that the bounds are tight andillustrating their computational applications.

Extreme zeros in a sequence of para-orthogonal polynomials and bounds for the support of the measure