Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations
Seymour V. Parter · Transactions of the American Mathematical Society · 1961
Introduction.Because of their many applications, primarily in the theory of probability, there has been a renewed interest in the theory of Toeplitz forms (see [6]).Of particular interest has been the work of Szegö on the distribution of the eigenvalues of finite sections of Toeplitz forms.More recently Kac, Murdoch and Szegö [lO] obtained estimates on the asymptotic behaviour of the extreme eigenvalues of certain of these finite sections.Later, Widom [19] re-obtained these results and extended them, under suitable restrictions.At the same time, recent interest in iterative methods of solving elliptic difference equations has been accompanied by the development of techniques for estimating the extreme eigenvalues of certain "block" matrices.For example, there are techniques using the theory of non-negative matrices (see [17; 18]), techniques using the classical theory of the direct product of two matrices (see [7; ll]) and the techniques (usually ascribed to Frankel [5], cf.[l; 2]) of separation of variables.In a recent work [14] we studied a very special class of block matrices and obtained some partial results.These enabled us to obtain estimates on the rates of convergence of the "two-line" iterative methods of the Laplace and biharmonic difference equations in rectangular domains(2).In the case of Laplace's equation we obtained an exact asymptotic result.However, in the case of the biharmonic equation we obtained only a "one-sided" estimate.The purpose of this report is two-fold.In § §2, 3, and 4 we extend the results of Kac, Murdoch and Szegö, and Widom.We will make very strong use of Widom's results and technique.In §5 we discuss the application of the preceding results to the general problem of the extreme eigenvalues of "block" Toeplitz matrices.These include the matrices of elliptic difference equations Presented to the Society, January 26, 1961; received by the editors October 18, 1960.(') Some of these results were obtained while the author was at the Brookhaven National Laboratory, summer 1959.(*) The "two-line" iterative methods for the Laplace and biharmonic difference equations were studied by R. S. Varga [18] at the same time.His approach is totally different from the one we investigated in [14].His approach to the solution of the iteration equations is more general and probably preferable.Varga also estimated the rate of convergence in the Laplace case using the theory of non-negative matrices.That theory does not apply to the biharmonic case.