Riemann-Hilbert problems, Toeplitz matrices, and applications

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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 $\mathscr{l}_2(\alpha)={\sum |a_n|^2(n+1)^{\alpha}<\infty}$ .
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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.