On a uniform approximation of the density function of a string equation using eigenvalues and nodal points and some related inverse nodal problems

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Abstract
Using the eigenvalues and the nodal points of the corresponding eigenfunctions of a string equation with density function rho(x), we construct a sequence of piecewise linear continuous functions which converges to rho-1/2(x) uniformly. We also obtain formulae for derivatives of rho-1/2(x) in terms of these spectral data, and using these formulae we find a necessary and sufficient condition for the convexity of rho-1/2(x) in terms of the nodal points of the eigenfunctions of the string equation.

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