Constructive Decomposition of Any L1a,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ L^{1}\left( a,b\right) $$\end{document} Function as Sum of a Strongly Convergent Series of Integrable Functions Each One Positive or Negative Exactly in Open Sets

Researchers dealing with real functions f·∈L1a,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ f\left( \cdot \right) \in L^{1}\left( a,b\right) $$\end{document} are often challenged with technical difficulties on trying to prove statements involving the positive f+·\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ f^{\,+}\left( \cdot \right) $$\end{document} and negative f-·\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ f^{\,-}\left( \cdot \right) $$\end{document} parts of these functions. Indeed, the set of points where f·\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ f\left( \cdot \right) $$\end{document} 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 a,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ \left( a,b\right) $$\end{document}. To remedy this situation, we regularize these sets through open sets. More precisely, for each zero-average f·∈L1a,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ f\left( \cdot \right) \in L^{\,1}\left( a,b\right) $$\end{document}, we construct, explicitly, a series of functions f⌢i·\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ \overset{\frown }{f}_{i}\left( \cdot \right) $$\end{document} having sum f·\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ f\left( \cdot \right) $$\end{document} — a.e. and in L1a,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ L^{1}\left( a,b\right) $$\end{document} — in such a way that, for each i∈0,1,2,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ i\in \left\{ \,0,1,2,\ldots \, \right\} $$\end{document}, there exist two disjoint open sets where f⌢i·≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ \overset{\frown }{f}_{i}\left( \cdot \right) \ge 0$$\end{document} a.e. and f⌢i·≤0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ \overset{\frown }{f}_{i}\left( \cdot \right) \le 0$$\end{document} a.e., respectively, while f⌢i·=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ \overset{\frown }{f}_{i}\left( \cdot \right) =0$$\end{document} a.e. elsewhere. Moreover, its primitive ∫tf·\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ \int ^{t}f\left( \cdot \right) $$\end{document} becomes the sum of a strongly convergent series of nice AC functions. Applications to calculus of variations & optimal control appear in our next papers.


Introduction
The main focus of this paper is on functions  If we disregard what has zero measure, in this simpler situation the positivity and negativity sets  Detailing the convergence of this series, it is such that, besides As we shall see, this result may also be interpreted as follows: given any AC function and defining we explicitly construct a decomposition of F (·) as the sum of a strongly-W 1,1 convergent series of functions F i (·) := f i (·) , each one also satisfying (1.7), such that F i (·) restricted to each maximal nonempty interval Here : ∃ F (t) = 0 } is the critical set of the given function F (·) (which often has nonempty interior, due to constancy intervals of F (·)); and the reason why one has to exclude C F from (1.8), but not from (1.4), is that while our construction always yields 226 Page 4 of 17 C. Carlota and A. Ornelas e.g. when C F has nonempty interior and each function F i (·) has constant > 0 value along each interval of C F . In case Fourier expansions come to the reader's mind as a possibility to get (1.5), let's recall that Fourier series for L 1 functions may even diverge at all points. (Kolmogorov constructed such an example in [5].) On the other hand, Fourier expansions will in general not satisfy the very useful convergence property (1.8). In fact, starting e.g. from a piecewiseaffine cap-function F (·) one reaches a trigonometrical Fourier series with infinitely many nonzero terms; while our cap-series will, by construction, contain just one nonzero term F 0 (·) = F (·), since F (·) is itself already a cap-function.
Thus the usual approach of harmonic analysis -consisting in fixing e.g. an orthonormal basis in a functional space hence express its functions relative to such basis -is completely different from our own. Indeed, defining F 0 (·) := F (·), our strategy begins instead with the explicit construction, along each maximal interval where F 0 (·) > 0, of a very reasonable cap function F 0 (·) which equals F 0 (·) in (at least) part of that interval; and then we do the same to F i+1 (·) := F i − F i (·), for i = 0, 1, . . .. In particular, our series will contain infinitely many nonzero terms only in case F (·) does consist of infinitely many levels of superposed oscillations (over the bounded interval (a, b)). In the special case where F (·) is not only AC but even analytic, our series will thus become just a finite sum of finitely-piecewise-cap functions, namely ∃ k F ∈ N such that , with each open set t ∈ (a, b) : F i (t) > 0 having finitely many connected components.
Such decomposition of AC functions into cap components is here presented as a technical tool allowing research mathematicians to deal, in an easier way, with positive or negative parts of L 1 functions and with monotonicity intervals of AC functions. Ourselves -in past research on nonconvex calculus of variations & optimal control (see e.g. [1,2,7]) -have felt technical difficulties on dealing with sets of positivity and of negativity of scalar velocities, since these may well have positive measure boundaries. But our new decomposition (1.3) and (1.4) allows us to reach better nonconvex results, and even results on the sign of integrals, in new papers to appear.
On the other hand, using as building blocks our cap-decomposition for scalar positive W 1,1 0 ( [a, b] ) functions, one easily reaches more general formulations. Indeed, the vectorial versions of our above decompositions for L 1 and for AC scalar functions, respectively, turn out to be the following: Moreover, each coordinate of each f i (·) satisfies (1.3).

Corollary 2. Given any function
and consider the critical set: Then we explicitly construct, from Moreover, each coordinate of each F i (·) is a countably-piecewise-cap/cup function.
(What we mean here is that each coordinate of each F i (·) is -locally where it is = 0 -either a cap function or a cup function, calling

Cap or Cup Decomposition of AC Functions
To begin with, associating to each function the corresponding open set  ) and a.e., hence almost uniformly, with f i (·) := F i (·) a.e.. More precisely: (1.5) and (1.6) hold true, with | · | replaced by | · | q , and -using the maximal intervals The idea behind the proof of Th. 1 is the following: after extracting from F 0 (·) := F (·) a first cap level F 0 (·) (= its cap components), we subcomponents), and so on and so forth ad infinitum. Thus F 0 (·) (resp. F 1 (·), F 2 (·), . . . ) yields the cap components (resp. subcomponents, subsubcomponents, . . . ) of F 0 (·); meaning, by "subcomponents" (resp. by "subsubcomponents", . . . ), that F 1 (·) (resp. F 2 (·), . . . ) yields the cap . While this strategy is simple to explain, the real challenge has been how to prove, in a constructive way, strong L 1 -convergence, to zero, of derivatives (as in (2.24)). Notice that since our cap decomposition is explicitly constructed, step by step, it wouldn't be so nice (both aesthetically and from the pragmatic viewpoint of practical implementation of the decomposition) to end up such construction with a nonconstructive proof of convergence.
Proof of Theorem 1. The promised decomposition (2.3) -or (2.4) or, in particular, (1.6) -will be constructed as follows: taking any one defines, recursively, where F i (·), the union of the disjoint cap components of F i (·), is defined as follows: in case is nonempty, we set Considering the set of relevant i 's, then clearly, for i ∈ I, obtaining, by (2.7), using the characteristic functions of On the other hand, differentiating (2.5), a.e. and Applying recursively these equalities one obtains, by (2.11), a.e. (2.14) hence, integrating along [a, b] and letting N → ∞, using (2.14) and (2.13) one gets in particular More precisely than in the lhs of (2.16) we have, by (2.13) and (2.12), On the other hand, by (2.9), which shows, by (2.10), that In particular, this implies that e., and hence, passing to the limit in (2.14) and (2.15), one obtains This completes the proof of Theorem 1.
Remark 1. (Alternative proofs of strong L q -convergence, 1 ≤ q ≤ p < ∞, of ( f i (·) ) to zero) In order to prove strong L q (a, b)-convergence of ( f i (·) ), we did prefer to stick to the constructive proof appearing in (2.17)-(2.20), thereby discarding our prior functional-analysis arguments which follow.
Since Therefore, as above ((2.25) to (2.28)), we can prove that which proves that Let q H be the Hölder conjugate exponent of q. By Hölder's inequality, Remark 3. (Alternative proof of strong L 1 -convergence of ( f i (·) ) to zero) To prove that ( f i (·) ) L 1 −→ 0 strongly we can also use Visintin's theorem [9, th. 1]. Indeed, since, due to (2.13), To begin with, notice that the sequence ( F i (·) ) is equibounded: On the other hand, since ( f i (·) ) is equiintegrable ( (2.29) holds true with q = 1 ), ( F i (·) ) is also equicontinuous. Therefore, by Ascoli-Arzelà theorem, there exist a subsequence ( F i k (·) ) and a continuous h : Since, for each t, the sequence i → F i (t) decreases -and any monotone sequence containing a convergent subsequence is itself convergent to the same limit -we have (2.33) On the other hand, since by (2.33) we conclude that (2.34) in particular By (2.10), .
Remark 7. (Decomposition of any vectorial AC function F (·)) Considering now vectorial functions F (·) ∈ W 1, p ( [a, b], R m ), with each coordinate F ν (·) satisfying (2.1), after determining the series ∞ i=0 (2.4)) for the derivative f 1 (·) of the first coordinate F 1 (·) of F (·), one may determine the series ∞ i=0 f 2 i (·) for the derivative f 2 (·) of the second coordinate F 2 (·) of F (·) and so on and so forth up to f m (·). Thus (2.4) will represent not only a scalar equality in L p (a, b) but also a vectorial equality in L p ( (a, b) , R m ), for the vectorial derivative f (·) of Author contributions Both authors have given an equal contribution to this research and to writing and reviewing this paper.
Funding Open access funding provided by FCT -FCCN (b-on). This research was financially supported by National Funds through FCT-Fundação para a Ciência e a Tecnologia in the framework of the research project "UIDB/04674/2020"-CIMA (Centro de Investigação em Matemática e Aplicações).

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