Around the classical string problem
The problem of reconstructing the density π(x) of a vibrating string given the N first eigenfrequencies of two vibrating configurations admits solutions that minimize certain weighted averages of the density. There exists a simple set of necessary conditions of these weights. In particular, it has been shown that the only weight functions f(x)than can be consistent in all cases with the existence of an extremal density which is made up of a finite number of points masses are polynomials of degree two. In the present paper, it is shown that the weighted averages can be calculated exactly. Explicit formulas are given, which in certain cases depend only on the spectrum of one vibrating configuration.The results strongly suggest applications to the Earth inverse problem. They are also extended to other problems, which suggest applications to non-linear questions. In particular, a new non linear evolution equation is studied, and conservation laws are exhibited.
KeywordsEntire Function Spectral Problem Extremal Solution Extremal Property Linear Evolution Equation
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