On localization for double Fourier series.

The localization theorems for Fourier series of functions of a single variable are classical and easy to prove. The situation is different for Fourier series of functions of several variables, even if one restricts consideration to rectangular, in particular square, partial sums. We show that the answer to the problem can be obtained by considering the notion of generalized bounded variation, which we introduced. Given a nondecreasing sequence {lambda(n)} of positive numbers such that Sigma 1/lambda(n) diverges, a function g defined on an interval I of R(1) is said to be of Lambda-bounded variation (LambdaBV) if Sigma|g(a(n)) - g(b(n))|/lambda(n) converges for every sequence of nonoverlapping intervals (a(n), b(n)) [unk]I. If lambda(n) = n, we say that g is of harmonic bounded variation (HBV). The definition suitably modified can be extended to functions of several variables. We show that in the case of two variables the localization principle holds for rectangular partial sums if LambdaBV = HBV, and that if LambdaBV is not contained in HBV, then the localization principle does not hold for LambdaBV even in the case of square partial sums.