On 2D Inverse Problems/Cauchy matrices

Let be an ordered set of n complex numbers. The corresponding Cauchy matrix is the matrix . Principal submatrices of a Cauchy matrix are Cauchy matrices. The determinant of a Cauchy matrix is given by the following formula: It follows, that if 's are distinct positive numbers, then the Cauchy matrix 's positive definite. Exercise (*). Prove that for any positive numbers there is a Stieltjes continued fraction, interpolating the constant unit function at these numbers, . (Hint.) Use the solution of the Pick-Nevanlinna interpolation problem w/the appropriate Cauchy matrix. The latter exercise has the following functional equation corollary for the discrete and continuous Dirichlet-to-Neumann maps. Exercise (**). Prove that for any positive definite matrix M there is a Stieltjes continued fraction, such that . The next chapter is devoted to the applications of the functional equation.