Fréchet variation

One of the numerical characteristics of a function of several variables that can be regarded as a multi-dimensional analogue of the variation of a function of a single variable. Suppose that a real-valued function is given on the -dimensional parallelopipedon

and introduce the notation

Let be an arbitrary partition of by hyperplanes

into -dimensional parallelopipeda, and let take the values in an arbitrary way. The Fréchet variation is defined as follows:

If , then one says that has bounded (finite) Fréchet variation on , and the class of all such functions is denoted by . For , this class was introduced by M. Fréchet [1] in connection with the investigation of the general form of a bilinear continuous functional on the space of functions of the form that are continuous on the square . He proved that every such functional can be represented in the form

where , .

Analogues of many of the classical criteria for the convergence of Fourier series are valid for -periodic functions in the class (, see [2]). For example, if , then the rectangular partial sums of the Fourier series of converge at every point to the number

where the summation is taken over all the possible combinations of the signs . Here, if the function is continuous, the convergence is uniform (an analogue of the Jordan criterion).

References

[1]	M. Fréchet, "Sur les fonctionelles bilinéaires" Trans. Amer. Math. Soc. , 16 : 3 (1915) pp. 215–234
[2]	M. Morse, W. Transue, "The Fréchet variation and the convergence of multiple Fourier series" Proc. Nat. Acad. Sci. USA , 35 : 7 (1949) pp. 395–399