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 {λn} of positive numbers such that Σ 1/λn diverges, a function g defined on an interval I of R1 is said to be of Λ-bounded variation (ΛBV) if Σ|g(an) — g(bn)|/λn converges for every sequence of nonoverlapping intervals (an, bn) [unk]I. If λn = 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 ΛBV = HBV, and that if ΛBV is not contained in HBV, then the localization principle does not hold for ΛBV even in the case of square partial sums.
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机译：单变量傅里叶级数函数的定位定理是经典的，容易证明。对于几个变量的傅立叶级数函数，情况就不同了，即使将考虑的范围限制为矩形，尤其是正方形的部分和。我们表明，可以通过考虑我们引入的广义有界变化的概念来获得问题的答案。给定一个正数的非递减序列{λn}，使得Σ1 /λn发散，则如果Σ，则在R 1 的间隔I上定义的函数g被称为Λ有界变化（ΛBV）。 | g（an）— g（bn）| /λn对于非重叠间隔（an，bn）[unk] I的每个序列收敛。如果λn= n ，我们说 g 具有谐波有界变化（HBV）。适当修改的定义可以扩展到几个变量的功能。我们表明，在两个变量的情况下，如果ΛBV= HBV，则定位原理适用于矩形部分和，如果HBV中不包含ΛBV，则即使对于平方部分和，定位原理也不适用于ΛBV。
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