Arzelà variation

A numerical characteristic of a function of several variables, which may be regarded as the multi-dimensional analogue of the variation of a function in one unknown. Let be a real-valued function given on an -dimensional parallelepipedon , and let be the class of all continuous vector functions , , such that each of the functions is non-decreasing on , and with , , . Then

where is an arbitrary system of points in . This definition for the case was proposed by C. Arzelà [1] (see also [2], p. 543). If , one says that has bounded (finite) Arzelà variation on , and the class of all such functions is denoted by . For a function to belong to the class it is necessary and sufficient that there exists a decomposition , where and are finite non-decreasing functions on . A function is called non-decreasing on if

for (). The class contains the class of functions of bounded Hardy variation on .

References

[1]	C. Arzelà, Rend. Accad. Sci. Bologna , 9 : 2 (1905) pp. 100–107
[2]	H. Hahn, "Theorie der reellen Funktionen" , 1 , Springer (1921)