Q 1. Introduction
The singular integral operator S, on the half-line R,, m being the simplest example of a WIENER-HOPF integral operator with piecewise continuous symbol, suggests that there ought to be some reason to consider such operators not only in L2(R+) but also in Lp(R+) (1 < p < 0 0 ) : A'+ is invertible in Lp(R+) if and only if p + 2. So the FREDHOLM theory of WIENER-HOPP integral operators has been worked out alniost completely to date for both p = 2 and p =+ 2: see [20, 15, 14, 23, 7, 8, 111 for the one-dimensional case and in the higher-dimensional case we refer to [27,22] for continuous and to [9, 101 for piecewise continuous symbols.
A very natural procedure for the approxiinate solution of WIENER-HOPF equations is the finite section method. For the discrete analogue of WIENER-HOPF integral operators, the TOEPLITZ operators, the problem of the applicability of the finite section method is solved in all cases of interest for us: in the one-dimensional case see [15, 23, 24, 291 for continuous and [15] ( p = 2), [2S, 3, 261 for piecewise continuous symbols: in the higher-dimensional case we refer to [16, 17, 18, 12, 291 for continuous and to [4] ( p = 2), [2] for piecewise continuous symbols. Much less is known about the applicability of the finite section method to WIENER-HOPF integral operators. An answer to this question was given for general p in the case of continuous symbols (see [15,23,24,29] for one and 117, 19,291 for more dimensions), but for piecewise continuous symbols solutions are known only for p = 2 (even in the one-dimensional case): see [15] and [4]. Only recently (cf.
[l]) the problem of the applicability of the finite section method to one-dimensional WIENER-HOPF integral operators with piecewise continuous symbol was solved for general p by the author (this result is implicitly reproved here). It is the aiiii of this paper to solve the analogous problem for two-dimensional WIENER-HOPF operators.
Note that the methods applied here are closely related with those of [2], and therefore allow not only to develop a theory of the finite section method, but a Fredholm theory for the operators considered here, too. For lack of space we renounce to do this and rely on [9] and [lo] for FREDHOmneSS.
Furthermore, since we shall proceed in analogy to [2], some facts will be outlined only briefly here, and we restrict ourselves to accentuate the peculiarities of the integral case.