Hardy variation

One of the numerical characteristics of a function of several variables. Let , be a function on the -dimensional parallelepiped

and let

Next, let be an arbitrary partition of the parallelepiped by hypersurfaces

into -dimensional parallelepipeds and let be the class of all functions for which

Finally, let , , be an integer-valued vector whose coordinates satisfy the inequalities and let be the integer-valued vector of dimension such that its coordinates form a strictly-increasing sequence and consist of all those numbers that are not contained among . Then every point can be written in the form . If the coordinates of a point are fixed to the values , then one writes .

The Hardy variation of on is:

If , then one says that the function has bounded (finite) Hardy variation on the parallelepiped , and the class of all such functions is denoted by . For this class was introduced by G.H. Hardy in [1] (see also [2]) in connection with the study of the convergence of Fourier double series. He proved that the rectangular partial sums of the Fourier double series of a function of class (), of period in each variable, converge at every point to the number

where

For a function to belong to the class it is necessary and sufficient that it can be represented in the form , where and are finite functions on such that , , for all and for all admissible increments . The class is contained in the class of functions having bounded Arzelà variation on .

References

[1]	G.H. Hardy, "On double Fourier series and especially those which represent the double zeta-function with real and incommensurable parameters" Quarterly J. Math. , 37 (1905) pp. 53–79
[2]	H. Hahn, "Theorie der reellen Funktionen" , 1 , Springer (1921)