Sampling and interpolating sequences in finite dimensional spaces
type: Item
instance of: scholarly article
title: Sampling and interpolating sequences in finite dimensional spaces
zbMATH Open document ID: 1446.94045
DOI: 10.4171/176-1/17
author: Joaquim Ortega-Cerdà
publication date: 2018-11-14T00:00:00Z
review text: Conditions on sampling and interpolation are studied on spaces of polynomials in a complex variable (complex case) and on the unit circle (real case). These settings are fundamentally different: in the real case there are families of points that can be both sampling and interpolating (e.g., the roots of unity) but in the complex case there are no sets of both sampling and interpolation. In the complex case one considers the space \(H_n\) of polynomials of degree at most \(n\) with norm \[\|p\|_n^2=\int_{\mathbb{C}}\frac{|p(z)|^2}{(1+|z|^2)^n}\, d S(z) \] where \(S(z)=dm(z)/(\pi(1+|z|^2)^2\) is the push-forward to the plane of the normalized surface measure on the unit sphere under stereographic projection. One says that a set \(\mathcal{Z}=\{z_{n,j}\}_{n=0,\dots;j=0,\dots, m_n}\) is a set of sampling for \(\{H_n\}_{n=0}^\infty\) if there is a constant \(C\) such that, for each \(n\) and each \(p\in H_n\), \[\frac{1}{Cn}\sum_{j=0}^{m_n} \frac{|p(z_{n,j})|^2}{(1+|z_{n,j}|^2)^n} \leq \int_{\mathbb{C}}\frac{|p(z)|^2}{(1+|z|^2)^n}\, d S(z) \leq \frac{C}{n}\sum_{j=0}^{m_n} \frac{|p(z_{n,j})|^2}{(1+|z_{n,j}|^2)^n}\, . \] One says that \(\mathcal{Z}\) is a set of interpolation for \(\{H_n\}\) if for each sequence \(a_0,\dots, a_{m_n}\) one has \[\frac{1}{Cn}\sum_{j=0}^{m_n} |a_j|^2\leq \int_{\mathbb{C}} \Bigl| \frac{a_j(1+\bar{z}_{n,j} z)^n}{(1+|z_{n,j}|^2)^{n/2}}\Bigr|^2 \frac{dS(z)}{(1+|z|^2)^n}\leq \frac{C}{n}\sum_{j=0}^{m_n} |a_j|^2 \] (equivalently, each sequence of values admits a suitably bounded polynomial with those values at the interpolation points). Fix \(\mathcal{Z}\), assumed to be separated: there is an \(\epsilon>0\) such that \(\inf\{d(z,w), z,w\in\mathcal{Z}_n\}>\epsilon/\sqrt{n}\) where \(d(z,w)=|z-w| ( (1+|z|^2)(1+|w|^2))^{-1/2}\). Let \(\mu_n\) denote the measure \(\mu_n=\frac{1}{n}\sum_{z\in \mathcal{Z}_n} \delta_z\). In the complex case it is proved that if \(\mathcal{Z}\) is a set of sampling for \(\{H_n\}\) then \(\#\mathcal{Z}_n\geq (n+1)\) for all…
Mathematics Subject Classification ID: 94A20
Mathematics Subject Classification ID: 30C10
zbMATH DE Number: 6979394
zbMATH Keywords: sampling
zbMATH Keywords: interpolation
zbMATH Keywords: frame
zbMATH Keywords: Kantorovich-Wasserstein distance
reviewed by: Joseph Lakey
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/176-1/17
OpenAlex ID: W4239144901
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description: scientific article; zbMATH DE number 6979394