Sampling and interpolating sequences in finite dimensional spaces
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 \(n\) and there is a \(\delta>0\) such that \(\lim\inf \mu_n \geq (1+\delta) dS(z)\), while if \(\mathcal{Z}\) is interpolating for \(\{H_n\}\) then \(\#\mathcal{Z}_n\leq (n+1)\) for all \(n\) and there is a \(\delta>0\) such that \(\lim\sup \mu_n \geq (1-\delta) dS(z)\). In the real setting one considers polynomials \(H_{\mathbb{T}}\) of degree at most \(n\) on the unit circle. One says that \(\mathcal{Z}\subset{\mathbb{T}}\) is a set of sampling if there is a constant \(C\) such that \[\frac{1}{Cn}\sum_{j=1}^{m_n} |p(z_{n,j})|^2\leq \int_0^{2\pi } |p(e^{i\theta})|^2\, d\theta\leq \frac{C}{n}\sum_{j=1}^{m_n} |p(z_{n,j})|^2 \] and is a set of interpolation if (\(z=e^{i\theta}\)) for all \(a_0,\dots a_{m_n}\), \[\frac{1}{C}\sum_{j=0}^{m_n} |a_j|^2\leq \int_0^{2\pi} \Bigl| \frac{a_j(1-(\bar{z}_{n,j} z)^{n+1})}{\sqrt{n+1}(1-\bar{z}_{n,j} z)}\Bigr|^2 \, d\theta\leq C\sum_{j=0}^{m_n} |a_j|^2\, .\] The cardinality constraints on \(\#\mathcal{Z}_n\) are the same as in the complex case. The measure constraints are that \(\lim\inf \mu_n \geq d\nu\) (sampling case) and \(\lim\sup \mu_n \leq d\nu\) (interpolating case) where, now, \(\mu_n=\frac{1}{n+1} \sum_{z\in\mathcal{Z}_n}\delta_z\) and \(d\nu\) is normalized Lebesgue measure on the unit circle. The analytical apparatus in both cases is the Kantorovich-Wassertein distance, \[W(\mu,\nu)=\inf_\rho \int\int_{X\times X}\, d(x,y)\, d\rho(x,y) \] where the infimum is taken over all probability measures \(\rho\) on \(X\times X\) with marginals \(\mu\) and \(\nu\). This distance is expressed concretely in relation to reproducing kernels in the complex and real settings, with the view that these examples are illustrations of a general principle that should apply in certain geometric settings. For the entire collection see [Zbl 1396.00017].
scientific article; zbMATH DE number 6979394
scientific article