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

Abstract

AbstractResearchers dealing with real functions $$\ f\left( \cdot \right) \in L^{1}\left( a,b\right) $$ f · ∈ L 1 a , b are often challenged with technical difficulties on trying to prove statements involving the positive $$\ f^{\,+}\left( \cdot \right) $$ f + · and negative $$\ f^{\,-}\left( \cdot \right) $$ f - · parts of these functions. Indeed, the set of points where $$\ f\left( \cdot \right) $$ f · 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 $$\ \left( a,b\right) $$ a , b . To remedy this situation, we regularize these sets through open sets. More precisely, for each zero-average $$\ f\left( \cdot \right) \in L^{\,1}\left( a,b\right) $$ f · ∈ L 1 a , b , we construct, explicitly, a series of functions $$\ \overset{\frown }{f}_{i}\left( \cdot \right) $$ f ⌢ i · having sum $$\ f\left( \cdot \right) $$ f · — a.e. and in $$\ L^{1}\left( a,b\right) $$ L 1 a , b — in such a way that, for each $$\ i\in \left\{ \,0,1,2,\ldots \, \right\} $$ i ∈ 0 , 1 , 2 , … , there exist two disjoint open sets where $$\ \overset{\frown }{f}_{i}\left( \cdot \right) \ge 0$$ f ⌢ i · ≥ 0 a.e. and $$\ \overset{\frown }{f}_{i}\left( \cdot \right) \le 0$$ f ⌢ i · ≤ 0 a.e., respectively, while $$\ \overset{\frown }{f}_{i}\left( \cdot \right) =0$$ f ⌢ i · = 0 a.e. elsewhere. Moreover, its primitive $$\ \int ^{t}f\left( \cdot \right) $$ ∫ t f · becomes the sum of a strongly convergent series of nice AC functions. Applications to calculus of variations & optimal control appear in our next papers.