Relativization of some aspects of the theory of functions of bounded variation

Krishna M. Garg

GDML_Books, (1992), p.

Résumé

We present here relativized versions of some aspects of the theory of functions of bounded variation, viz. relative to a function of bounded variation, without going into relative bounded variation. A few results have been known in this direction for some time when the functions involved are continuous, but due to the complications that arise when the functions are discontinuous, no systematic attempt seems to have been made in this direction in the past.Let B denote the linear space of all real-valued functions of bounded variation defined on a given compact interval I = [a,b]. Given f,g ∈ B, we present here a notion of mutual singularity of f and g, and a notion of absolute continuity (or AC) of f relative to g, which are similar to these notions in the case of signed measures. Further, we present decompositions of these two properties into mutual lower and upper singularities and relative lower and upper absolute continuities.Several characterizations of the above six properties are obtained here in terms of variations of f and g. Further, additivity theorems dealing with the additivity of these properties are obtained, and reduction theorems are obtained which reduce these properties to the discontinuous, AC and continuous singular components of f and g. Also, characterizations of these properties are obtained in terms of derivatives of f and g. These characterizations are based on a refined version of a theorem of de La Vallée Poussin which deals with derivatives of the three variations of f ∈ B in terms of the derivative of f.Next, with the help of the above new notions and results we present relativized versions of some other aspects of the theory of functions of bounded variation. A new notion of normalization f* of f ∈ B and a related normalized version of relative derivative also play significant roles in this development.Firstly, characterizations of all the above six properties are obtained here in terms of normalized relative derivative and the Lebesgue–Stieltjes integral (or LS-integral). Following are some other highlights of the developments:A Radon–Nikodym theorem is obtained for LS-integral where the normalized relative derivative turns out to be the Radon–Nikodym derivative in general. Also, two versions of the fundamental theorem of calculus are obtained for LS-integral, and a theorem dealing with the reconstruction of a function from its relative derivative is obtained.Further, relativized versions of (i) a monotonicity theorem of Lebesgue, (ii) the Lebesgue decomposition theorem, (iii) Lusin’s property (N) and the Banach–Zarecki theorem on AC, (iv) the results on Lebesgue points, and (v) a theorem of Tonelli on arc length are obtained. Also, characterizations of mutual singularity and relative AC in terms of arc length, a general formula for arc length based on relative Lebesgue decomposition, and a solution of an old problem of Denjoy on arc length in higher dimensions are obtained.Next, we consider convergence in B under variation norm relative to which B is known to be a Banach space. Some theorems dealing with the stability of variations and components under norm convergence are obtained here for sequences and series of functions in B.Further, a relativized version of Fubini’s theorem on term-by-term differentiation is obtained, and an extension of Fubini’s (relativized) theorem is obtained which holds in general under a form of convergence which is stronger than norm convergence. Finally, some approximation theorems are obtained which deal with approximation in some closed subspaces of B by certain elementary functions in those subspaces. E.g. the functions in B which are AC relative to some u ∈ B can be approximated in the variation norm by piecewise linear functions relative to u, and also in a sense by polynomials in u.

CONTENTSI. Introduction and preliminaries.....................................5II. Mutual singularities...................................................14III. Relative absolute continuities...................................33IV. Normalized relative derivative..................................50V. Relativization of other classical theorems.................73VI. Convergence in B....................................................95References.................................................................119Index of symbols.........................................................121Index of terms.............................................................123

1991Mathematics Subject Classification:Primary 26-02, 26A42, 26A45, 28-02; Secondary 26A15, 26A24, 26A30, 26A46, 28A75

Détail

Zbl 0762.26007

EUDML-ID : urn:eudml:doc:219292