# On 2D Inverse Problems/On inhomogeneous string of Krein

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The following physical model of a vibrating inhomogeneous string (or string w/beads) by Krein provides physical/mechanical interpretation for the study ofStieltjes continued fractions, see [GK]. The model is one-dimensional, but it arises as the restriction ofn-dimensional inverse problems with rotational symmetry.

The string is represented by a non-decreasing positive mass functionm(x)on a possibly infinite interval[0, l]. The right end of the string is fixed. The ratio of the forced oscillation to an applied periodic force @ the left end of the string is the function of frequency, called coefficient ofdynamic complianceof the string, see [KK] and [I2].

The small vertical vibration of the string is described by the following differential equation:

where is the density of the string, possibly including atomic masses. One can express the coefficient in terms of the fundamental solution of the ODE:

where,

The fundamental theorem of Krein and Kac, see [KK] & also [I2], essentially states that an analytic function ) is the coefficient of dynamic compliance of a string if and only if the function

is an analytic automorphism of the right half-plane , that is positive on the real positive ray. The **Herglotz theorem** completely characterizes such functions by the following integral representation:

where,

is **positive measure** of bounded variation on the closed positive ray .

**Exercise(**).**Use the theorem above, change of variables and the Fourier transform to characterize the set of Dirichlet-to-Neumann maps for a disc w/rotationally invariant conductivity.