Sparse Sets in Time and Frequency Related to Diophantine Problems and Integrable Systems
Reconstruction of signals and their Fourier transforms lead to the theory of prolate functions, developed by Slepian and Pollack. We look at prior contributions by Szegö to these problems. We present a unified framework for solutions of Szegö like problems for signals supported by an arbitrary union of intervals, using the techniques of Garnier isomonodromy equations. New classes of completely integrable equations and Darboux–Backlund transformations that arise from this framework are similar to the problems encountered in transcendental number theory. A particular example for the Hilbert matrix is studied in detail.
KeywordsHilbert matrix Hankel matrices Padé approximations
- 1.Alberto Grünbaum, F.: A remark on Hilbert’s matrix. Linear Algebra Appl. 43, 119–124 (1982)Google Scholar
- 3.Chudnovsky, D.: Riemann monodromy problem, isomonodromy deformation equations and completely integrable system. In: Bifurcation phenomena in mathematical physics and related topics, pp. 385–448. D. Reidel, Dordrecht (1980)Google Scholar
- 4.Chudnovsky, G.: Padé approximation and Riemann monodromy problem. In: Bifurcation phenomena in mathematical physics and related topics, pp. 449–410. D. Reidel, Dordrecht (1980)Google Scholar
- 11.Mahler, K.: Perfect systems. Compos. Math. pp. 95–166 (1968)Google Scholar
- 13.Okamoto, K.: Isomonodromic deformation and Painleve equation, and the Garnier system. J. Fac. Sci. Univ. Tokyo Sec. IA(33), 575–618 (1986)Google Scholar