Sampling and the closure of the polynomials in a weighted Hardy space (Q2725485)

Let \(D\) denote the unit disk in the complex plane, let \(\varphi (z)={1+z \over 1-z}\), for \(c>0\) let \(g_c(z)=\cos {\pi\varphi (z)\over 3c}\), and let \(dm\) denote Lebesgue measure on the unit circle. Let \(\mu_{c,d}= \nu_c+ \sigma_d\), where \(\nu_c=|g_c|^{-1}dm\) and \(d\sigma_d= \sum^\infty_{n=1} \alpha_n \delta_{z_n}\), where \(\{z_n\}\) is the set of zeros of \(g_d\) in \(D\), \(\alpha_n= (|z_n||g_c'(z_n) |)^{-1}\), and \(\delta_{z_n}\) is a point measure on \(z_n\). Further, let \(\mu=\nu_1 +\lambda\), where \(\lambda\) is Lebesgue measure on the line segment \([0,1]\). Let \(P^t(d\mu)\) denote the closure of the polynomials on \(L^t(d\mu)\), \(t\geq 1\). The Paley-Wiener space \(E^2_\pi\) is defined by \(E^2_\pi= \{f:f\) is an entire function of exponential type \(\pi\) where \(\int_R|f(z)|^2 dx<\infty\). Let NEWLINE\[NEWLINE{\mathcal M}=\left\{ \bigl(1+ \varphi (z)\bigr)F \left({\varphi (z)\over 6}\right): F\in E^2_\pi,\;F\left(-{\varphi (z_n)\over 6}\right)= 0,\;n=1,2,3,\dots \right\}NEWLINE\]NEWLINE where \(\{z_n\}\) are the zeros of \(g_2(z)\). Finally, let \(g_2^*(z)= \cos({\pi\over 6}(\varphi (z)+{3 \over 2}))\). The authors present a number of results showing that the zero set of the function \(g_2\) plays an important role in the space \(P^2(d\mu)\). Here is a sample of these results: NEWLINE\[NEWLINEP^2(d\mu) =g_2\cdot H^2(D) \oplus{\mathcal M}\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEg_2\cdot H^2(D)+ g_2^*\cdot H^2(D) \text{ is dense in } P^2(d\mu);\tag{2}NEWLINE\]NEWLINE and, (3) There exists a constant \(M>1\) such that, for each polynomial \(p\), NEWLINE\[NEWLINE{1 \over M}\|p\|_{L^2} (d\mu_{1,2})\leq \|p\|_{L^2} (d\mu)\leq M\|p\|_{L^2} (d\mu_{1,2}).NEWLINE\]