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3. Spectrum of Hilbert Matrices


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      Problem 3.1.

      [Alfonso Montes Rodriguez] The \(\mathscr{l}^2\)-spectrum for the Hilbert matrix \(H_{m n}=\frac{1}{m+n+1}\) is known. Find the spectrum on spaces of the form \(\mathscr{l}_2(\alpha)={\sum |a_n|^2(n+1)^{\alpha}<\infty}\).


        • Add remark

          Problem 3.2.

          [Alfonso Montes Rodriguez] Define \[u_{\xi, \alpha(z)}=e^{-\alpha\Big(\frac{\xi+z}{\xi-z}\Big)},\] for \(\alpha >0\) and \(|\xi|=1\). Given \(\xi_1, \xi_2, \xi_3\) and \(\alpha_1, \alpha_2, \alpha_3\), can we approximate \[\sum_{m=0}^{\infty} c_m^{(1)}u_1^m+c_m^{(2)}u_2^m+c_m^{(3)}u_3^m?\]

              Cite this as: AimPL: Riemann-Hilbert problems, Toeplitz matrices, and applications, available at http://aimpl.org/riemhilberttoeplitz.