In the pioneering paper [5], Gelfand and Levitan introduced an elegant and explicit method for computing the potential of a Sturm-Liouville Operator from its eigenvalues and certain values of its eigenfunctions. It has been noted in Burridge [3] that the Gelfand-Levitan method is closely related to inversion methods for recovering the potential in a hyperbolic equation with focused initial state, thus giving a unified theory for the methods of Gelfand-Levitan, of Gopinath and Sondhi [8] and of Parijskij [11] and Blagoveshchenskij [1]; (see also Romanov [12, p. 39,]), the latter ones going unnoticed in [3].
In the present note we analyse discrete analogues of these problems. The spacial differential operator of the Gelfand-Levitan theory is replaced by a symmetric tridiagonal matrix. It will turn out that in this setting the Gelfand-Levitan method reduces essentially to a Cholesky decomposition. The hyperbolic equation in the other theories is replaced by a recursion relation involving an arbitrary tridiagonal matrix, the essential step for the inversion being an LU-decomposition. In our approach, (approximate) transmutation operators as originally used in [5] (see Levitan [10] for a systematic study) play a paramount role.
We are aware of the fact that much work has been done on discrete inverse problems. We mention in particular Burridge [3, p. 514-537,], Case and Kack [4], Bruckstein and Kailath [2], Landau [9], Gladwell and Willms [7]. Our approach differs from others in that it gives a unified treatment in terms of transmutation operators.