m-Functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices

We study inverse spectral analysis for finite and semi-infinite Jacobi matricesH. Our results include a new proof of the central result of the inverse theory (that the spectral measure determinesH). We prove an extension of the theorem of Hochstadt (who proved the result in casen = N) thatn eigenvalues of anN × N Jacobi matrixH can replace the firstn matrix elements in determiningH uniquely. We completely solve the inverse problem for (δ _n , (H-z)^-1 δ _n ) in the caseN < ∞. This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-9623121 and DMS-9401491.