Constructive Decomposition of Any L^{1}left( a,bright) Function as Sum of a Strongly Convergent Series of Integrable Functions Each One Positive or Negative Exactly in Open Sets

Abstract

Researchers dealing with real functions fleft( cdot right) in L^{1}left( a,bright) are often challenged with technical difficulties on trying to prove statements involving the positive f^{,+}left( cdot right) and negative f^{,-}left( cdot right) parts of these functions. Indeed, the set of points where fleft( cdot right) is positive (resp. negative) is just Lebesgue measurable, and in general these two sets may both have positive measure inside each nonempty open subinterval of left( a,bright) . To remedy this situation, we regularize these sets through open sets. More precisely, for each zero-average fleft( cdot right) in L^{,1}left( a,bright) , we construct, explicitly, a series of functions overset{frown }{f}_{i}left( cdot right) having sum fleft( cdot right) — a.e. and in L^{1}left( a,bright) — in such a way that, for each iin left{ ,0,1,2,ldots , right} , there exist two disjoint open sets where overset{frown }{f}_{i}left( cdot right) ge 0 a.e. and overset{frown }{f}_{i}left( cdot right) le 0 a.e., respectively, while overset{frown }{f}_{i}left( cdot right) =0 a.e. elsewhere. Moreover, its primitive int ^{t}fleft( cdot right) becomes the sum of a strongly convergent series of nice AC functions. Applications to calculus of variations & optimal control appear in our next papers.