I. C. GOHBERG and N. J. KRUPNIK [S]. It was A. V. KOSAK, who pointed out that the theory of the finite section method for these operators can also lie founded upon a local principle. His approach was originated by the local principle of IB SIMONENKO, and it led to a series of remarkable results on multidimensional convolution operators with continuous symbols (cf.[lo],[1 I]). One of the authors of this paper established in [14] a local theory of the finite section method for one-dimensional TOEPLITZ operators, which is based upon the local principle of IC GOHBERG and N. J. KRUPNIK. The advantage of the pinciple developed in [14] is. on the one hand, that the argumentation runs widely parallel to that which leads to the FREDHOLM properties and, on the other hand, the 1) rinciple is especially suitahle for operators with discontinuous symbols. In recent years considerable progress has been made by R. V. DUDU~ AVA in the FREDHOLM theory of multidimensional TOEPLITZ operators with piecewise continuous symbols (cf. 131,[4],[5]) and, of course, the problem of the applicability of the finite section method to such operators it.; currently emerging. The purpose of this paper is to show, how by applying a tensor-algebraical version (cf.[2],[12]) of the theory of [14], comllined with results of [7] and [5], a solution of this problem can he given.

It should I) e noted that the methods developed in the present paper can also he applied to the investigation of FREI) FTOLM properties of multidimensional TOIWLITZ operators with piecewise continuous symbols (cf.[3],[5]), and to estahlish a theory of the finite section method for continuous and discrete con-volutions in the spaces L” and,!”, respectively.