Orthogonal polynomials and quadratic extremal problems (Q2781409)

Let \(\{\varepsilon_n\}\) be a sequence of positive numbers. The Hilbert space \({\mathcal H}(\{\varepsilon_n\})\) is the set of all functions \(f(z)= \sum a_n z^n\) analytic in the unit disk \(\Delta\) such that NEWLINE\[NEWLINE\|f\|_{\mathcal H}= \Biggl(\sum^\infty_{n=0} \varepsilon_n|a_n|^2\Biggr)^{{1\over 2}}< \infty.NEWLINE\]NEWLINE As some special cases, \({\mathcal H}(\{\varepsilon_n\})\) is the Hardy \(H^2\) space when \(\varepsilon_n= 1\), the Dirichlet space when \(\varepsilon_n= n+1\), and the Bergman space when \(\varepsilon_n= (n+ 1)^{-1}\). For \(\{\alpha_n\}\) a real sequence and \(f(z)= \sum a_n z^n\in{\mathcal H}(\{\varepsilon_n\})\), define NEWLINE\[NEWLINE\phi(f)= \text{Re }{1\over\pi} \int_\Delta \overline{f(z)} f'(z) dx dy+ \sum^\infty_{n= 0} \alpha_n\varepsilon_n|a_n|^2.NEWLINE\]NEWLINE In the special case where \(\{\alpha_n\}\) is the constant sequence \(\{c\}\), define NEWLINE\[NEWLINE\phi_c(f)= \text{Re }{1\over\pi} \overline{f(z)} f'(z) dx dy+ c\sum^\infty_{n=0} \varepsilon_n|a_n|^2.NEWLINE\]NEWLINE Let \(\ell^2(\{\varepsilon_n\})\) be the space of all complex sequences \({\mathbf a}= \{a_n\}\) such that \(\|{\mathbf a}\|^2= \sum \varepsilon_n|a_n|^2< \infty\). Define a matrix \(A= (d_{i,j})\), where \(d_{i,i}= \alpha_i\), \(d_{i,i-1}= d_{i,i+1}= (2\varepsilon_i)^{-1}\), and \(d_{i,j}= 0\) otherwise, \(0\leq i, j< \infty\), and, for \({\mathbf a}\in\ell^2(\{\varepsilon_n\})\), \(\|{\mathbf a}\|> 0\), define NEWLINE\[NEWLINE\Phi({\mathbf a})= {\langle A{\mathbf a},{\mathbf a}\rangle\over \langle{\mathbf a},{\mathbf a}\rangle}= {\text{Re } \sum^\infty_{n=0} \overline{a_n}a_{n+1}+ \sum^\infty_{n=0} \varepsilon_n \alpha_n|a_n|^2\over \|{\mathbf a}\|^2}.NEWLINE\]NEWLINE It is shown that the functionals \(\phi\) and \(\Phi\) are bounded if and only if both \(\{\alpha_n\}\) is a bounded sequence and \(\liminf_{n\to\infty} \varepsilon_{n-1} \varepsilon_n> 0\), and both \(\phi\) and \(\Phi\) have the same local and global extreme values on the unit spheres of \(F{\mathcal H}(\{\varepsilon_n\})\) and \(\ell^2(\{\varepsilon_n\})\), respectively. Further, \(\phi\) attains an extreme value at \(g(z)= \sum b_n z^n\in{\mathcal H}(\{\varepsilon_n\})\) if and only if \(\Phi\) attains the same extreme value at \(C\{b_n\}\in \ell^2(\{\varepsilon_n\})\), where \(C\) is such that \(C\|\{b_n\}\|= 1\). Extrema are studied by introducing the idea of a critical value and critical point. If \(\lambda\) is a real number such that the recurrence relation NEWLINE\[NEWLINEa_{n+1}= 2\varepsilon_n a_n(\lambda- \alpha_n)- a_{n-1},\quad n\geq 0,\quad a_{-1}= 0,NEWLINE\]NEWLINE is satisfied, then \({\mathbf a}= \{a_n\}\) is called a critical point for \(\Phi\) and \(\Phi({\mathbf a})= \lambda\) is called a critical value of \(\Phi\). It is shown that \({\mathbf a}\) is a critical point for \(\Phi\) corresponding to the critical value \(\lambda\) if \({\mathbf a}\) is an eigenvector of the matrix \(A\) corresponding to the eigenvalue \(\lambda\). Various other results are obtained, including significant connections to orthogonal polynomials. Among the results shown are that \(\phi_c\) attains neither maximum nor minimum values in the unit sphere of \(H^2\) for \(|c|\leq{1\over 2}\). For \(|c|>{1\over 2}\), \(\phi_c\) assumes a single extreme value, which is \(c+ (4c)^{-1}\), and this value is attained for the rational function \(F(z)= C{2c\over 2c- z}\), where \(C\) is chosen so that \(\|F\|_{H^2}= 1\).