On the total variation for functions of several variables and a multidimensional analog of Helly’s selection principle

Abstract

ABSTRACT. We introduce the new notion of total variation for the Hardy class of functions of several variables and state various properties, similar to those in the one-dimensional case, for functions belonging to this class. In particular, we prove a precise version of Helly's selection principle for this class.

Functions of bounded variation, signed measures, and a general Koksma–Hlawka inequality

“…It is easily seen that f + (0) = f − (0) = 0, and thus the functions f + and f − from (26) and (27) have the properties requested in Theorem 2.…”

“…Furthermore, since the function f 2 in Lemma 3 is completely monotone, the same is true for the function f − in (27). Now set g = −f .…”

“…Subsequently, we introduce completely monotone functions, following [27,Section 3]. We note that a comprehensive survey on HK-variation and its properties can be found in [36].…”

“…It is a generalization of the well-known Jordan decomposition theorem for functions of bounded variation in the one-dimensional case (see for example [51, §12, Section III]). The key ingredient in its proof is a decomposition theorem of Leonov [27,Theorem 3]. The statement of the theorem uses the notion of a completely monotone function, which is defined in Section 2 below.…”

Maps of several variables of finite total variation. I. Mixed differences and the total variation

“…Under some approaches to the multidimensional variation [2,8,36], which involve certain integration procedures, Helly-type theorems are rather concerned with the almost everywhere convergence of extracted subsequences, and no stronger convergence can be expected in this case. On the other hand, for real-valued functions of several variables there are definitions of the notion of variation [29,34], which go back to Vitali [41], Hardy [27] and Krause [1,23], such that a complete analogue of the Helly theorem holds with respect to the pointwise convergence of extracted subsequences. Moreover, it was shown recently [17,34] that certain counterparts of properties (a) and (b) hold for the variation in the sense of Vitali-Hardy-Krause.…”