# Around the classical string problem

## Abstract

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.

## Keywords

Entire Function Spectral Problem Extremal Solution Extremal Property Linear Evolution Equation## Preview

Unable to display preview. Download preview PDF.

## References

- (1).KREIN M.G.: On some case of effective determination of the density of an inhomogeneous cord from its spectral function. Dokl. Akad. Nauk SSSR 93, 617Google Scholar
- (2).Barcilon V.: Ideal solution of an inverse normal mode problem with finite spectral data. Geophys. J. Roy. Soc. March 1979Google Scholar
- (3).SABATIER P.C.: On extrémal solutions of Strum Liouville inverse problems in “ Inverse and improlerly posed problems in differential equations ” Anger Ed. Akademie-VerlagBerlin1979Google Scholar
- (4).KAC I. and KREIN M.G.: On the spectral functions of the string. Am. Math. Soc. Transl. II, 103, 19–102 (1974)Google Scholar