An orthogonal polynomial analogue of the Landau-Pollak-Slepian time-frequency analysis (Q1935754)

This paper presents a time-frequency analysis for the orthogonal polynomials on the interval \([-1,1]\), which is in a correspondence to the Landau-Pollak-Slepian theory developed in a series of papers that followed [\textit{H. Landau} and \textit{H. Pollak}, Bell Syst. Tech. J. 40, 65--84 (1961; Zbl 0184.08602)]. The analysis is set in the framework of the Hilbert space \(L^2([-1,1],w)\), where \(w\) is a positive weight function with finite moments, and is based on the following two operators: the multiplication operator \(M_xf(x)=xf(x)\) and the projection operator \(P_n^mf(x)=\sum_{l=m}^n\langle f,p_l\rangle_wp_l(x)\), \(0\le m <n<\infty\). Here \(\{p_l\}_{l=0}^\infty\) denotes an appropriate family of orthonormal polynomials, which is known to be an orthonormal basis in \(L^2([-1,1],w)\). If \(p_l(x,m)\), \(m\in \mathbb{N}\), denote the so-called associate polynomials, then the spectral decomposition of the finite dimensional Hermitian operator \(P_n^m\,M_x\,P_n^m\) has an explicit form presented in Theorem 2.1. The analysis is then followed by investigating the localization properties of \(\psi^m_{n,k} \), the eigenfunctions of \(P_n^m\,M_x\,P_n^m\), and answering the question how the decomposition of a bandlimited function \(f\in \Pi_n^m\) (the subspace of \(L^2([-1,1],w)\) spanned by \(p_l\), \(m\le l\le n\)) in terms of \(\psi^m_{n,k}\) can be used to approximate functions that are well-localized at a point or a subinterval of \([-1,1]\). Finally, in the last section of the paper an uncertainty principle related to the operators \(M_x\) and \(P_n^m\) is discussed.