Reparametrization invariant norms

This paper explores the concept of reparametrization invariant norm (RPI-norm), that is any norm invariant under composition with diffeomorphisms. We prove the existence of an infinite family of RPI-norms, called standard RPI-norms, for which we exhibit both an integral and a discrete characterization. Our main result states that, for every one-time differentiable piecewise monotone function with compact support, its standard RPI-norms allow us to compute the value of any other RPI-norm of the same function. This is proved by using the standard RPI-norms in order to reconstruct the function up to reparametrization and an arbitrarily small error with respect to the total variation norm.


Introduction
In recent papers the natural pseudo-distance σ between manifolds endowed with regular real functions has been studied as a tool for comparing the shape of manifolds (cf. [3,4,5]). Each shape is represented by pairs (M, ϕ), where M is a connected manifold and ϕ is a real function defined on it (both M and ϕ are supposed to be sufficiently regular). In this approach, the main idea is to compare two diffeomorphic manifolds by measuring the global change of the real functions they are endowed with, when the manifolds are deformed into each other: σ((M, ϕ), (N , ψ)) = inf h sup p∈M |ϕ(p) − ψ • h(p)|, where h varies among all the diffeomorphisms between M and N . We observe that σ is a Fréchet-like distance (cf., e. g., [6]). Moreover, this line of research is strongly related to the extensive study currently carried out about parametrization-independent shape comparison in Pattern Recognition (cf, e. g., [8]).
The definition of natural pseudo-distance between the pairs (M, ϕ1), (M, ϕ2) can be reformulated as the value inf h∈D F (ϕ1 − ϕ2 • h), where D denotes the set of all diffeomorphisms from M to M and F is the norm that takes each (sufficiently regular) functionφ : M → R to the number maxP ∈M |φ(P )|. In order that inf h∈D F (ϕ1 − ϕ2 • h) is a pseudo-distance, the key property of the functional F is that F is a norm and F (φ • h) = F (φ) for everyφ : M → R and every h ∈ D. In other words, the point is that F is a reparametrization invariant norm. Choosing a different reparametrization invariant norm would allow us to obtain a different pseudo-distance. We conclude this introduction by recalling that the invariance under reparametrization appears to be relevant in several fields of research. Just two examples are Statistics (cf., e. g., the Kolmogorov-Smirnov Test) and the Theory of Interpolation Spaces (with reference, e. g., to the K-Method).

The point and the main ideas in this paper
This paper studies the reparametrization invariant norms that can be defined on a suitable set of regular functions ϕ from R to R. The norms max |ϕ|, max ϕ − min ϕ, the total variation Vϕ of ϕ, and the function p max |ϕ| 2 + V 2 ϕ are simple examples of reparametrization invariant norms, assuming that the derivative of ϕ has compact support and that ϕ vanishes at −∞. Actually, there exist infinitely many examples of such norms, since each linear combination with positive coefficients of reparametrization invariant norms is obviously a reparametrization invariant norm. However, in the set of all these norms, we have succeeded in detecting a particular subset of norms, which we call standard reparametrization invariant (RPI-) norms, such that 1. if the C 1 -function ϕ has compact support and is piecewise monotone, then the knowledge of all the standard RPI-norms of ϕ allows us to reconstruct ϕ up to reparametrization, with an arbitrarily small error ε with respect to the total variation norm; 2. as a consequence of previous property, any other RPI-norm of such a function ϕ is completely determined by the values taken on ϕ by the standard RPI-norms. Therefore, we have focused our research on these norms.
The main idea of this paper originates from the following classical definition of the total variation Vϕ for a regular function ϕ : R → R (see, e.g., [1]): here Ψ is the set of all (sufficiently regular) functions ψ from R to R with |ψ| ≤ 1. We observe that if we substitute Ψ with any subsetΨ of Ψ that is closed with respect to reparametrization, then other reparametrization invariant norms can be obtained (though, in this case, the two suprema in the previous formula may be different). The closure with respect to reparametrizations means that ifψ ∈Ψ then ψ • h ∈Ψ for every orientation-preserving diffeomorphism h : R → R.
In order to apply our idea, in first place we choose a functional space. Many different choices are possible. As a trade-off between generality and simplicity we have chosen the space AS 1 (R) of all almost sigmoidal C 1 functions. Roughly speaking, this space could be defined as the space of all C 1 functions ψ : R → R that "behave as a sigmoid outside a sufficiently large compact" (see Section 2, definition 2.1).
This choice is not very restrictive, since AS 1 (R) contains all the C 1 functions with compact support.
By defining [ψ] as the set containing ψ ∈ AS 1 (R) and all its reparametrizations ψ • h, and by setting e obtain a reparametrizazion invariant norm on AS 1 (R). The norms obtained in this way are precisely the standard reparametrizazion invariant norms, verifying the properties described in previous statements 1 and 2 (Theorem 5.8). The reason for using ϕ(−t) instead of ϕ(t) inside the integral is that this choice allows us to obtain the equality ϕ [ψ] = ψ [ϕ] , thanks to the fact that our functions belong to AS 1 (R).
Incidentally, this motivated also the choice of AS 1 (R) as the functional space to use.
We could proceed analogously for a general setΨ closed with respect to reparametrization in place of [ψ] and obtain a reparametrization invariant norm ϕ Ψ , but the caseΨ = [ψ] is the most interesting one, since ϕ Ψ can be easily expressed as a supremum of standard reparametrization invariant norms.
In order to get the main results of this paper some technicalities will be necessary. In particular, a key role will be played by the Bounding Lemma 2.9, asserting that, after normalization, every reparametrization invariant norm ϕ is upper bounded by the total variation Vϕ and lower bounded by the value limt→+∞ |ϕ(t)|. A stronger Bounding Lemma will be proved for C 1 c (R). It asserts that, after normalization, every reparametrization invariant norm ϕ of a function ϕ having compact support is upper bounded by half the total variation Vϕ and lower bounded by the value max |ϕ(t)|. The proof of these key results will require some computations and a preliminary study of the general properties of reparametrization invariant norms, that will be carried out in Section 2. In particular, we shall examine the role played by two particular functions, called S and Λ. Moreover, in the same section we shall prove the stability of RPI-norms with respect to small perturbations in C 1 and the interesting fact that no inner product can induce a RPI-norm.
The definition of standard reparametrization invariant norm will be introduced in Section 3, together with some examples and some basic properties. However, in order to proceed further, we shall have to represent standard reparametrizazion norms in a simpler way. We know that an alternative definition exists for the total variation, saying that Vϕ equals the value sup n sup τ 0 ≤...≤τ i ≤...≤τn P n−1 i=0 |ϕ(τi+1) − ϕ(τi)|. In Section 4 some computations will be necessary to make available a similar representation also for standard reparametrization invariant norms (Theorem 4.16). This new kind of representation will be used to prove the fundamental results in this paper, i.e. the possibility of reconstructing piecewise monotone functions with compact support up to reparametrizations by means of the standard reparametrization invariant norms and the dependence of reparametrization invariant norms on standard reparametrization invariant norms (Section 5). Section 6 will conclude this paper by illustrating some open problems.
2 Re-parameterization invariant norms: definition and general properties

Some notations and basic definitions
In this paper the symbols C 1 (R) and C 1 c (R) will represent the set of all one time continuously differentiable functions from R to R and the set of all functions in C 1 (R) that have compact support, respectively. The symbol D 1 + (R) will represent the set of all orientation-preserving C 1 -diffeomorphisms from R to R. We shall say that a function f is increasing . A function will be called monotone if it is either decreasing or increasing, and strictly monotone if it is either strictly increasing or strictly decreasing.
A number will be said to be positive when it is strictly greater than zero. The set of positive natural numbers will be denoted by N + .
First of all let us introduce the functional space we shall work in.
Definition 2.1. Let us consider the set of all C 1 -functions ϕ : R → R for which two real values a, b exist such that: • ϕ(t) = 0 for every t ∈ (−∞, a]; We shall denote this set by the symbol AS 1 (R) and call each function in AS 1 (R) an almost sigmoidal function of class C 1 .
Examples of almost sigmoidal functions are shown in Figure 1.
Obviously, C 1 c (R) ⊂ AS 1 (R) and every function in AS 1 (R) has bounded variation. We shall use the symbol 0 to denote the almost sigmoidal function that vanishes everywhere.
The ideas described in this paper can be extended to more general spaces, but we choose this setting in order to simplify our proofs from the technical viewpoint.
For every ψ ∈ AS 1 (R) we shall denote by V + ψ (t) (resp. V − ψ (t)) the positive (resp. negative) variation of ψ, and by V ψ (t) the variation of ψ: Figure 2: We are interested in studying the norms that take both the functions ϕ 1 and ϕ 2 to the same value, since ϕ 2 is obtained by composing ϕ 1 with an orientation-preserving C 1 -diffeomorphism of R.
Since ψ ∈ C 1 (R), the functions V + ψ , V − ψ , and V ψ are C 1 . Moreover we shall denote by V + ψ , V − ψ , V ψ the total positive variation, the total negative variation and the total variation of ψ, respectively: We recall that V + ψ (t) and V − ψ (t) are non-negative increasing functions whose difference is exactly ψ.
Definition 2.2. For any two functions ϕ1, ϕ2 : R → R, we say that ϕ2 is obtained from ϕ1 by a reparametrization (of class C 1 ) if an orientation-preserving diffeomorphism h ∈ D 1 + (R) exists such that ϕ2 = ϕ1 • h. The diffeomorphism h will be called a reparametrization. We denote by ∼ the equivalence relation defined by setting ϕ2 ∼ ϕ1 if and only if ϕ2 is obtained from ϕ1 by a reparametrization. The equivalence class of ϕ1 in AS 1 (R) will be denoted by [ϕ1].
In this paper we study the norms that take equivalent functions to the same value (see Figure 2).

Reparametrization invariant norms
Now we give the main definition in this paper. Definition 2.3. Let us consider the real vector space AS 1 (R). We say that a norm · : AS 1 (R) → R is invariant under reparametrization (or a reparametrization invariant norm) if it is constant over each equivalence class of AS 1 (R)/ ∼.
In the following the reparametrization invariant norms will be often called RPI-norms.
The norms max |ϕ|, max ϕ − min ϕ and the total variation Vϕ of ϕ are simple examples of RPI-norms.
It is quite easy to see that infinitely many RPI-norms exist. Indeed, it is trivial to prove that each linear combination with positive coefficients of a finite number of RPI-norms is still a RPI-norm.
Another simple method to obtain a RPI-norm is to consider the sup of a set of RPI-norms, under the assumption that such a sup is finite at each point.
Remark 2.4. Let · be a RPI-norm on AS 1 (R). If ϕ has compact support, also the composition ϕ • h of ϕ with an orientation-reversing C 1 -diffeomorphism h belongs to AS 1 (R). Hence it makes sense to ask if ϕ equals ϕ • h or not. In other words, the question is whether RPI-norms, that are invariant under orientation-preserving reparametrizations by definition, are invariant also under orientation-reversing reparametrizations.
In general the answer is negative. As a counterexample, consider the RPI-norm In order to proceed, we need to introduce two useful almost sigmoidal functions, represented in Figure   3. Definition 2.5. We shall denote by S the almost sigmoidal C 1 -function from R to R defined by setting We define Λ : R → R by setting Λ(t) = S(t + 1) − S(t − 1).
In the following subsection we shall show that, in some sense, each RPI-norm is controlled by the norm of the function S.

The Bounding Lemma
The Bounding Lemma states that, after normalization, every RPI-norm of ϕ is bounded from above by the total variation of ϕ and from below by limt→+∞ |ϕ(t)|.
This result will be proved as a consequence of the fact that the increasing functions of AS 1 (R) can be approximated arbitrarily well by functions equivalent to multiples of the function S (Prop. 2.6). From this, it follows that the RPI-norms of monotone functions are multiples of the norm of S (Prop. 2.7).
Proof. Let a and b be two real numbers such that a < b, ϕ equals 0 in the interval (−∞, a] and it is constant in the interval [b, +∞). Let us define e S(t) = S and ϕε = ϕ + ε S · e S. We point out that e S = S , since e S is obtained by reparametrizing S. Hence ϕε − ϕ = ε S · e S = ε. Moreover ϕε is clearly an increasing function belonging to AS 1 (R) and max ϕε = max ϕ + ε S . Therefore, we have only to prove that a reparametrization h ∈ D 1 + (R) exists, such that ϕε(t) = max ϕε · S(h(t)) for every t ∈ R. In order to show this, we set where fε : [0, max ϕε] → [−1, 1] denotes the inverse function of the restriction of max ϕε · S to the interval On the one hand, we observe that if t ≤ a then the equality ϕε(t) = ε S · e S(t) holds, and hence for a − ε < t ≤ a the equality (1) becomes ε S · e S(t) = max ϕε · S(h(t)).
If t is also close enough to a−ε we have from (2) that −1 ≤ h(t) ≤ 0 and in this case e S(t) = 2 , because of the definitions of e S and S. Then, by a direct computation we obtain from (2) that if a − ε < t ≤ a and t is close enough to a − ε the equality h(t) = c · (t − a + ε) − 1 holds.
On the other hand, if t ≥ b the equality ϕε(t) = max ϕ + ε S · e S(t) holds, and hence for b ≤ t < b + ε the equality (1) becomes If t is also close enough to b+ε we have from (3) that 0 ≤ h(t) ≤ 1 and in this case e S(t) = 1−2 , once more because of the definitions of e S and S. Then, by a direct computation (recalling that max ϕε = max ϕ + ε S ) we obtain from (3) that if b ≤ t < b + ε and t is close enough to b + ε the equality h(t) = c · (t − b − ε) + 1 holds.
It follows that h is differentiable at both the points a − ε and b + ε, and that at both of them the derivative of h takes the positive value c.
Furthermore, we observe that the restriction of h to the open interval (a−ε, b+ε) has positive derivative, since both the derivative of ϕε is positive in this interval (due to the addend ε S · e S) and the derivative of fε is positive in the open interval (0, max ϕε). Also, h has obviously derivative equal to the positive value c outside the interval [a − ε, b + ε], because of its definition, and at points a and b, since it is C 1 . In conclusion, we have shown that h is an orientation-preserving C 1 -diffeomorphism.
Therefore ϕε is equivalent to the function max ϕε · S and our statement is proved.
Now we can prove the following simple but crucial result, underlining the importance of the function S.
Proof. Set ϕ = |ψ|. By applying the previous Proposition 2.6 and the triangular inequality, we obtain that By passing to the limit for ε tending to 0, we get the equality max ϕ · S − ϕ = 0 and our statement is proved.
Remark 2.8. We have to justify our line of proof of Proposition 2.7, since the passage through Proposition 2.6 could appear a little cumbersome. The point is that the function ϕ max ϕ·S may not tend towards a positive finite constant for t → a + or for t → b − . Furthermore, it may happen that the derivative of ϕ vanishes in the open interval (a, b). In these cases we cannot change directly ϕ into max ϕ · S by a reparametrization, i.e. by composing ϕ with an (orientation-preserving) C 1 -diffeomorphism. Hence we have to change ϕ into an approximation ϕε that does not present the previous problems. Now we are ready to prove the Bounding Lemma. It gives a lower bound and an upper bound for each RPI-norm, involving the norm of S. Lemma 2.9 (Bounding Lemma). Let · : AS 1 (R) → R be a reparametrization invariant norm. Then, for every ϕ ∈ AS 1 (R) the following inequalities hold: Since V + ϕ , V − ϕ are increasing, Proposition 2.7 implies that V + ϕ = V + ϕ · S and V − ϕ = V − ϕ · S , and hence our statement is proved by recalling that Remark 2.10. The inequalities in the Bounding Lemma are sharp, as we can easily see by setting ϕ = S.
Remark 2.12. We observe that the inequality proved in Corollary 2.11 is sharp, since Λ can equal 2 · S . This happens, e.g., when we consider as RPI-norm the total variation. Moreover, it is interesting to note that no positive constant c exists such that the inequality c · S ≤ Λ holds for every RPI-norm · . To see this, it is sufficient to consider the RPI-norm ϕ k = max |ϕ| + k · limt→+∞ |ϕ(t)|, for k ≥ 0.
Since c · S k = c · (1 + k) and Λ k = 1, if k is large enough the inequality c · S k ≤ Λ k does not hold.
Also in this sense, the lower bound in the Bounding Lemma cannot be improved. Incidentally, we observe that the function limt→+∞ |ϕ(t)| defines a seminorm on AS 1 (R) that is reparametrization invariant.

Derivatives and RPI-norms
The main consequence of the Bounding Lemma is that the closeness of two almost sigmoidal functions with respect to the total variation norm implies their closeness with respect to any other RPI-norm. From this we obtain the next proposition, showing that if the derivatives of two functions ϕ, ψ ∈ AS 1 (R) are everywhere close to each other, then ϕ and ψ are close to each other with respect to any other reparametrization invariant norm.
Proposition 2.13. Let · be a reparametrization invariant norm on AS 1 (R). Assume that ϕ, ψ ∈ AS 1 (R) and that the compact support of their derivative is contained in the interval [a, b] with a = b. If By applying the right inequality in the Bounding Lemma 2.9 we obtain

A stronger Bounding Lemma for functions in C 1 c (R)
The inequalities in the Bounding Lemma can be improved if ϕ belongs to C 1 c (R) ⊆ AS 1 (R). In this case ϕ stays somewhere between the norms max |ϕ| and 1 2 Vϕ, up to the multiplicative constant Λ . This section is devoted to prove these stronger bounds.
In order to prove these results we need the following technical lemma,where Vχ and Vχ−χ denotes the total variation of χ andχ − χ, respectively, on [α, β].
Therefore, taking η2 (and hence η1) small enough, by continuity we obtain that It follows that if we chooseε, η1 and η2 small enough, then the inequality Vχ−χ ≤ ε holds.
We defineĥ : (α, β) → (α, β) by settinĝ Now,ĥ is an orientation-preserving C 1 -diffeomorphism becauseχ and the functionχ has been defined to be quadratic asŜ. Therefore, if we extendĥ to the closed interval Now we can prove the following result for Λ, analogous to Proposition 2.7 proved for S.
Proof. Let us assume that ϕ = 0, otherwise the claim is trivial. Consider the smallest interval [a, b] containing the compact support of dϕ dt . Possibly by taking −ϕ instead of ϕ, we can also assume that ϕ is increasing in [a,t] and decreasing in [t, b] so that ϕ ≥ 0. Furthermore, up to a reparametrization, we can assume that a = −2,t = 0 and b = 2.
Let χ1 denote the restriction of ϕ to the interval [−2, 0] and χ2 denote the restriction of ϕ to the interval [0, 2]. Let us apply Lemma 2.14 for some ε > 0 in order to obtain two functionsχ1 andχ2 and the diffeomorphismsh1 andh2 such that Recall also that h1 is the identity in a neighbourhood of −2 and 0, andh2 is the identity in a neighbourhood of 0 and 2.

Consider the function ϕε
We have that ϕε is a function in C 1 c (R), with Vϕ ε −ϕ ≤ ε. So, by applying the Bounding Lemma 2.9, we deduce that ϕε − ϕ ≤ ε · S .
Let us consider the orientation-preserving passing to the limit for ε tending to 0, we get the equality max ϕ · Λ − ϕ = 0 and our statement is proved.
The following result will be useful in the proof of the Reconstruction Theorem 5.8. We omit its proof being quite similar to the ones used for Lemma 2.14 and Proposition 2.15.
Now we are ready to prove the stronger version of the Bounding Lemma for functions with compact support. It gives a lower bound and an upper bound for each RPI-norm, involving the norm of Λ.
Lemma 2.17 (Bounding Lemma for C 1 c (R)). Let · : AS 1 (R) → R be a reparametrization invariant norm. Then, for every ϕ ∈ C 1 c (R) the following inequalities hold: Proof. First of all we prove the left inequality. We take a point tmax where |ϕ| takes its maximum value and consider the function We can easily verify that ψ is continuous also at tmax, because of the two addends appearing in its definition. Then we observe that bothφ and ψ belong to C 1 c (R). In particular, the regularity of ψ follows from the fact that tmax is a critical point for ϕ. Moreover, by computing their derivative we see that ϕ and ψ are increasing in (−∞, tmax] and decreasing in [tmax, +∞). Furthermore maxφ =φ(tmax) = sign(ϕ(tmax)) · max |ϕ| + max ψ.
Since ϕ =φ − ψ, by applying Proposition 2.15 witht = tmax we get As for the proof of the other inequality, we begin by considering an interval [a, b] with a = b, such that the compact support of dϕ dt is contained in [a, b]. Let us define the function Let us now assume that dϕ dt (t) = 0. In this case we set Because of the choice oft, ϕ1 and ϕ2 are continuous also att. Moreover, we observe that both ϕ1 and ϕ2 are C 1 c (R) functions (here we are using the hypothesis dϕ dt (t) = 0). Furthermore they are increasing in (−∞,t ] and decreasing in [t, +∞).
By applying Proposition 2.15 we get Therefore the inequality ϕ ≤ 1 2 Vϕ · Λ is proved, in the case when dϕ dt (t) = 0. Otherwise, if dϕ dt (t) = 0, we observe that for every ε > 0, ϕ can be locally modified neart into a function ϕε ∈ C 1 c (R) such that The change we are using is represented in Figure 5.
Because of what we have just proved in the case dϕ dt (t) = 0, it follows that ϕε ≤ 1 2 Vϕ ε · Λ and hence ϕε Then, the right inequality is proved for any ϕ by passing to the limit for ε tending to 0.
t Figure 5: The change from ϕ (thin) to ϕ ε (thick) in order to get dϕε dt (t) = 0 in the proof of the Bounding Lemma for C 1 c (R).
Remark 2.18. The double inequality that we have just proved shows that, if we confine ourselves to consider functions in C 1 c (R), half the total variation and max |ϕ| are the two extremal cases of RPI-norms. All other RPI-norms are somewhere between them, after normalization with respect to Λ. We also observe that the two new inequalities are sharp, as we can immediately verify by setting ϕ = max |ϕ| and ϕ = Vϕ. Remark 2.19. While the lower bound in the Bounding Lemma 2.9 vanishes for all functions in C 1 c (R), the lower bound in Lemma 2.17 never vanishes for non-zero functions in C 1 c (R). This difference makes the study of functions with compact support easier than the study of general almost sigmoidal functions.

Can a reparametrization invariant norm be induced by an inner product?
We consider the question whether a reparametrization invariant norm can be associated with some inner product. The next result shows that the answer to this question is negative.
Proposition 2.20. No inner product on AS 1 (R) can induce a reparametrization invariant norm.
Proof. Assume that an inner product ·, · exists on AS 1 (R), inducing a reparametrization invariant norm.
The associated norm · satisfies the parallelogram identity: Let us take an almost sigmoidal function ϕ with compact support, and set ϕ1 = V + ϕ and ϕ2 = V − ϕ .
Then, ϕ1 + ϕ2 = V ϕ , ϕ1 − ϕ2 = ϕ. By applying (5) and Proposition 2.7 about the norm of monotone almost sigmoidal functions we get Since for every ϕ with compact support we have that V + ϕ − V − ϕ = 0, every such a function should have a vanishing norm. This contradicts the definition of norm.
However, we remark that there exist degenerate symmetric bilinear maps Φ inducing reparametrization invariant semi -norms on AS 1 (R). An example is given by

Standard reparametrization invariant norms
In this section we introduce a class of reparametrization invariant norms on AS 1 (R). One well-known norm belonging to this class is the L∞-norm. For the sake of conciseness and clearness in exposition, for every ϕ ∈ AS 1 (R) we shall often use the symbol ϕ * to denote the function ϕ * (t) = ϕ(−t). Obviously, in general ϕ * is not an almost sigmoidal function, since it is obtained by composing ϕ with an orientation-reversing diffeomorphism of R.

The integral definition
Lemma 3.1. Let ϕ, ψ ∈ AS 1 (R). The following statements hold: Proof. i) Integrate by parts and observe that ϕ efines a norm on the vector space AS 1 (R), that is invariant under reparametrization. Moreover, if also Note. In the rest of the paper the equality ϕ [ψ] = ψ [ϕ] will be called exchange property.
Proof. First of all, let us prove that · [ψ] is a norm. Clearly ϕ [ψ] is a non-negative real number.
Indeed, the finiteness of the sup follows from Lemma 3.1ii) and the invariance of the total variation under reparametrizations. Moreover it holds λϕ [ψ] = |λ| · ϕ [ψ] for any λ ∈ R, and the triangle inequality is easily verified. Also, if ϕ ≡ 0, then obviously ϕ [ψ] = 0. Therefore, the only thing we have to prove is that ϕ [ψ] = 0 implies ϕ ≡ 0. We prove this statement by contradiction, by assuming that containing the compact support of dψ dt , so that ψ(t) = 0 for t ≤ α. Since ψ is not constant, a point t1 exists with ψ(t1) = 0. It is easy to see that for every ε > 0 an orientation-preserving C 1 -diffeomorphism . Taking ε small enough and remembering that ϕ * (t) = 0 for t ≥ b, we easily get Now we observe that Figure 6: The functions ϕ, ϕ * and ψ (proof of Theorem 3.2).
dt˛is positive, if we have chosen a small enough ε in the costruction of hε. Hence ϕ [ψ] > 0, against our assumption. So, we have proved that · [ψ] is a norm.

Two examples of standard RPI-norms
A simple instance of standard RPI-norm is given by the L∞-norm, as the following proposition states.
Another simple standard RPI-norm on AS 1 (R) is given by max ϕ − min ϕ, as the following proposition states.

The key idea in using standard RPI-norms
The two examples seen in the previous section show that, in some sense, computing standard RPI-norms is equivalent to computing the absolute value of a linear combination of Dirac deltas, applied to the function ϕ * (t). Indeed, it is easy to verify that in order to get ϕ [S] and ϕ [Λ] we have to compute the values sup t |δt(ϕ * )| and sup t 0 ≤t 1 |δt 0 (ϕ * ) − δt 1 (ϕ * )|, where δt is the usual Dirac delta at point t. The "weights" of the Dirac deltas are determined by the integral R 1 at the points where ϕ takes its maximum value and its minimum value (not necessarily in this order). We shall carefully analyze and generalize this approach in Section 4.

Not every RPI-norm is a standard RPI-norm
It is important to observe that some RPI-norms are not standard RPI-norms.
In order to show this, now we give a useful property of standard RPI-norms.
Proposition 3.5. Let · be a RPI-norm. If it can be obtained as a finite linear combination of standard RPI-norms with positive coefficients, then S ≤ Λ .
As a consequence of this property, we can furnish an example of RP I-norm that cannot be represented as a linear combination with positive coefficients of standard RPI-norms.
Corollary 3.6. The RPI-norm ϕ = max |ϕ| + limt→+∞ |ϕ(t)| cannot be represented as a finite linear combination with positive coefficients of standard RPI-norms. In particular, it is not a standard RPI-norm.
Remark 3.7. It could be interesting to know if the norm max |ϕ|+limt→+∞ |ϕ(t)| can be represented either as a sup or as an inf of a suitable set of standard RPI-norms.
Another example of RPI-norm that cannot be expressed as a finite linear combination with positive coefficients of standard RPI-norms is the total variation. Proposition 3.8. The total variation cannot be represented as a finite linear combination with positive coefficients of standard RPI-norms. In particular, it is not a standard RPI-norm.
Proof. If the total variation could be represented as a linear combination with positive coefficients of standard RPI-norms, the equality Vϕ = P k i=1 ai ϕ [ψ i ] would hold for every ϕ ∈ AS 1 (R), when a suitable set {a1, . . . , a k } of positive coefficients is chosen.
By Theorem 3.2 we would have that Vϕ ≤ P k i=1 aiV ψ i · max |ϕ|. This inequality contradicts the fact that we can easily find a functionφ ∈ AS 1 (R) such that maxφ = 0 and the ratio Vφ maxφ is arbitrarily large. Nevertheless, the total variation can be seen as the sup of a suitable set of standard RPI-norms, as shown in the following Section 3.3.1.

The total variation is a sup of standard RPI-norms
We have seen in Proposition 3.8 that the total variation is not a standard RPI-norm. We now show that it is the sup of a family of standard RPI-norms. Proof. Let us prove the first equality. By applying Theorem 3.2 we obtain that ϕ [Ln] ≤ max |Ln| · Vϕ = Vϕ. We just have to show that for every ε > 0 an n exists such that Vϕ − ϕ [Ln ] ≤ ε. This is trivially true if ϕ = 0 so let us assume ϕ = 0. Let [a, b] be a closed interval containing the support of dϕ * dt . Let us recall that Vϕ = Vϕ * = sup k sup a=t 0 <t 1 <...<t k =b P k−1 i=0 |ϕ * (ti+1) − ϕ * (ti)|. Hence, there exist n ≥ 1, and a partition a =τ0 <τ1 < .
The second equality follows from the first one, by observing that the sequence ( ϕ [Ln] ) is increasing.
Remark 3.10. In plain words, our proof of Proposition 3.9 is based on recognizing that P n−1 i=0 |ϕ * (τi+1) − ϕ * (τi)| can be seen as the value taken by the absolute value of the linear functional δτ 0 + P n−1 i=1 (−1) i 2 · δτ i computed at ϕ * , where δt is the usual Dirac delta at point t. The reparametrization hη allows us to and at the otherτi's a signed variation of Ln • hη approximately equal to 1 and ±2, respectively. This last idea will be developed in next Section 4 by using general weights (not just 1 and ±2) for the Dirac deltas, and its generalization will lead to the Representation Theorem 4.16. This result assures that every standard RPI-norm can be seen as the absolute value of a suitable linear combination of Dirac deltas, maximized with respect to the movements that preserve the deltas' position order. We could obtain Proposition 3.9 as a consequence of the Representation Theorem, but we have preferred to anticipate this result for the sake of clarity of exposition. Moreover, this choice allows us to illustrate the ideas that we are going to develop.

Discrete representation of standard RPI-norms
In this section we show how to compute the standard RPI-norms in a simpler, discrete way.
This is a general property of standard RPI-norms, that they can be seen as sup of the absolute value of linear combinations of Dirac deltas. The basic idea underlying this fact is that the sup of is obtained by considering a sequence of reparametrizations more and more concentrating the variation of ψ at suitable points. Passing from the integral definition of standard RPI-norm to linear combinations of Dirac deltas, the sup with respect to orientation-preserving Figure 9: The functions used in the example described in Section 4.
reparametrizations is substituted by a sup with respect to movements that shift the Dirac deltas' centers without changing their ordering (Theorem 4.9). We shall see that the best choice is to place these Dirac deltas' centers at critical points of ϕ (Representation Theorem 4.16).
By way of exemplification, let us consider the norm ϕ [ψ] where ψ(t) and ϕ * (t) are the functions illustrated in Figure 9 convenient to take reparametrizations hη that transform smaller and smaller neighbourhoods of suitable points t0 ≤ t1 ≤ t2 ≤ t3 ≤ t4 ordinately to the intervals (a0 + η, b0 − η), (a1 + η, b1 − η), . . . , (a4 + η, b4 − η), with a smaller and smaller η > 0 (or to (a0 + η, b1 − η) if, say, t0 = t1, and so on). By passing to the limit one obtains that In other words, Now, one easily sees that, in order to get the greatest value, the ti's must be critical points of ϕ * . In particular, in this case the sup is attained when t0 = t1 = τ0, t2 = τ1, t3 = τ2, t4 = τ3, so that We point out that the considered linear combinations can involve infinitely many terms. This is the main difficulty to manage in this section, and will require some computations.
The key result obtained in this section (Representation Theorem 4.16) will be fundamental in the next section, where we shall use it to prove that all the standard RPI-norms of a piecewise monotone C 1 -function ϕ with compact support are sufficient to reconstruct ϕ, up to reparametrization and an arbitrarily small error with respect to the total variation norm.
The first step to get these results is defining a bilinear function F that will be useful in the sequel.

The functional F
All along the remainder of Section 4 we shall assume that two functions ϕ, ψ ∈ AS 1 (R) are given, with Here we set ψ| b i a i = ψ(bi) − ψ(ai) (see example in Figure 10).
Definition 4.1. The set {ti}i∈I is said to be a set of basepoints for the pair (ϕ, ψ).
On the set J (ψ) we shall consider the order induced by the ai's. In other words we shall set Ji Jj if and only if ai ≤ aj . This order will not need to coincide with the order induced by the index i. The symbol σi will denote the sign of dψ dt in the interval Ji, i.e. σi = sign ψ| b i a i .

Definition 4.2.
We define the bilinear functional F : where {ti}i∈I is a set of basepoints for (ϕ, ψ).
In other words, F (ϕ, ψ) = P i∈I ψ| b i a i · δt i (ϕ * ), where δt i is the usual Dirac delta at point ti. Note that the definition of F (ϕ, ψ) does not depend on the particular choice of the set of basepoints for the pair (ϕ, ψ). The idea underlying the definition of basepoint is to maximize each addend ψ| b i a i · ϕ * (t) in the definition of F , when t varies in [ai, bi].

Remark 4.3. It is easy to verify that {ti}i∈I is a set of basepoints for (ϕ, ψ) if and only if for every
Therefore, we get this equivalent definition for F : Moreover, we observe that each set of basepoints for (ϕ, ψ) is contained in the compact support of dψ dt .

Some useful properties of the functional F
Let us consider the set H ψ of all orientation-preserving C 1 -diffeomorphism of the real line that take each interval Ji ∈ J (ψ) to itself. The following lemma shows the key property of the functional F .
Consider the countable set J (ψ) of all maximal open intervals of R where dψ dt does not vanish. Obviously, J (ψ) = J (ψ) and sign dψ dt = sign dψ dt . Furthermore,ψ| b i a i = ψ| b i a i for every index i ∈ I, and {ti}i∈I is a set of basepoints for (ϕ,ψ), i.e.
where σi denotes the sign taken by both dψ dt and dψ dt on the open interval Ji = (ai, bi). Therefore By substituting ψ with −ψ in the previous inequality (observe that J (−ψ) = J (ψ) and where {ti}i∈I is a set of basepoints for (ϕ, −ψ).
where σi denotes the sign taken by dψ ds and dψ ds on the interval Ji. Let us choose an ε > 0. In order to avoid the problem of some ti's possibly belonging to the boundary of Ji, we define a new set {t ′ i }i∈I : for each interval Ji = (ai, bi) ∈ J (ψ) we choose a t ′ i ∈ (ai, bi) such that |ϕ * (ti) − ϕ * (t ′ i )| < ε 2 i . For each positive integer k ≤ |I| let us choose a positive real number η such that η < min for every i ≤ k − 1. Then we can consider an orientation-preserving diffeomorphism h (η,k) ∈ H ψ such that ii) the restriction of h (η,k) to the set R − S k−1 i=0 Ji is the identity.
Recall also that, because h (η,k) ∈ H ψ , for every index i the map h (η,k) takes the interval Ji onto itself.
Since ϕ is continuous, we can also assume to choose η so small that the inequality |ϕ * (t ′ i )−ϕ * (h −1 (η,k) (t))| < ε 2 i holds for any t ∈ (ai + η 2 i , bi − η 2 i ) and any index i ≤ k − 1. Therefore, for any index i ≤ k − 1, by setting t = h (η,k) (s) and recalling that the sign of dψ dt is constant in (ai, bi) (we shall use this fact in several following passages), It follows that, for every positive integer k ≤ |I| and every ε > 0, a small enough positive η = η(k, ε) ≤ ε exists such that, denoting by A k the set S k−1 i=0 (ai, bi), we havę By definition of h (η,k) , the function ψ • h (η,k) equals ψ on R − A k and hence, when I = N, and analogously when k = |I|, Z Therefore, recalling that η ≤ ε, the following inequality holds for every large enough k (if |I| = ∞), and here and in the sequel, when I = N, we set |I| − 1 = ∞).
Since F (ϕ, ψ) = P |I|−1 i=0 ψ| b i a i · ϕ * (ti) and it is finite, if k is large enough (in case |I| = ∞), or k = |I| (in case |I| < ∞) we get˛P Inequality (9) proves that for everyε > 0 an orientation-preserving diffeomorphism h + ∈ H ψ exists such By substituting ψ with −ψ and observing that H −ψ = H ψ we get for everyε > 0 an orientation- Therefore the inequality holds. This implies our claim.
The next result, proved by applying the previous lemma, motivates the introduction of the functional F .
Proof. Lemma 4.4 implies that

Dirac deltas
The next theorem simplifies the computation of the standard RPI-norms, bypassing the concept of basepoints for the pair (ϕ, ψ). First we define a new set T (ϕ, ψ) based on the natural ordering , previously introduced on the set J (ψ). We recall that Ji Jj if and only if ai ≤ aj. This order does not need to coincide with the order induced by the index i. We also recall that the set I indexing J (ψ) is assumed to be either the finite set {0, 1 . . . , n − 1} or the set N. Ji 2 implies τi 1 ≤ τi 2 , for every i1, i2 ∈ I, and τi = b if i ∈ I. The sequences in T (ϕ, ψ) will be said to be compatible with (ϕ, ψ). The terms τi = b with i ∈ I will be called dummy terms of the sequence (τi) ∈ T (ϕ, ψ).
Remark 4.7. When I = N, an ordered set of basepoints is a compatible sequence itself. When I is finite, starting from a set of basepoints, we can obtain a compatible sequence by adding infinitely many dummy terms τi = b to our finite sequence. As a matter of fact, the dummy terms will not be used in our computations.
Remark 4.8. For every (τi) ∈ T (ϕ, ψ), the series Proof. Ifψ ∈ [ψ], an orientation-preserving diffeomorphism h ∈ D 1 + (R) exists such that ψ =ψ • h, and J Because of the definition of F , we obtain that Let`t + i´b e a compatible sequence for (ϕ,ψ) obtained from the set of basepoints {t + i }, and analogously, lett − i´b e a compatible sequence for (ϕ, −ψ) obtained from the set of basepoints {t − i } (recall Remark 4.7). It is easy to see that`t + i´a nd`t − i´a re compatible sequences also for (ϕ, ψ). Therefore, for anyψ ∈ [ψ] the inequality On the other hand, because of the continuity of ϕ, for every (τi) ∈ T (ϕ, ψ), every positive integer k ≤ |I| and every ε > 0, an orientation-preserving diffeomorphism h k,ε ∈ D 1 + (R) exists, such that the distance between the number ϕ * (τi) and each value in the set ϕ * (h k,ε (Ji)) is not greater than ε, for i ≤ k − 1 (it is sufficient to choose a diffeomorphism taking each Ji into an interval contained in a small neighborhood of τi). We point out that here we are using the hypothesis that (τi) is a sequence compatible with (ϕ, ψ).
Let us considerψ k,ε = ψ • h −1 k,ε , and choose a set {t + i } of basepoints for the pair (ϕ,ψ k,ε ) and a set {t − i } of basepoints for the pair (ϕ, −ψ k,ε ). For each index i ≤ k − 1, we define the open interval (α ′ i , β ′ i ) by setting (α ′ i , β ′ i ) = h k,ε (Ji). As before, we observe that J (ψ k,ε ) = {h k,ε (Ji)}i∈I and thatt + i ,t − i belong to the closure of h k,ε (Ji) for every index Furthermore,˛| Since ψ is a function of bounded variation, if k is large enough we get (Here and in the following, k large enough means k = |I| if |I| is finite, and in this case every empty summation is assumed to take the value 0.) Analogously, if k is large enough we get Therefore for every ε > 0 we can find a large enough index k such that By recalling the definition of F we obtain These two last inequalities and the arbitrariness of ε imply that From Prop. 4.5 the inequality ollows. Hence our statement is proved.

Optimal sequences in T (ϕ, ψ)
The previous Theorem 4.9 raises an interesting issue: is the sup equaling ϕ [ψ] actually a max? In Prop.
4.12 we shall give an affirmative answer to this question.
In the next pages, each sequence obtained by the method described in the proof of the previous lemma will be said to be "obtained by a diagonalization process". Proof. On the basis of Theorem 4.9, for each n ∈ N we can take a sequence T n = (t n i ) ∈ T (ϕ, ψ) in such a way that ϕ [ψ] = limn→∞˛P i∈I ψ| b i a i · ϕ * (t n i )˛. By Lemma 4.11, the sequence of sequences (T n ) admits a subsequence (T nr ) that pointwise converges to a sequenceT = (τi) compatible with (ϕ, ψ). If we denote each sequence T nr by (t nr i ) (varying i), the following equalities hold: where the second equality follows from the fact that ψ has bounded variation.
However, a stronger result holds, stating that there exist optimal sequences for (ϕ, ψ) containing only critical points for ϕ * .
Lemma 4.13. From each sequence (T n ) of sequences in O(ϕ, ψ) it is possible to extract a subsequence that pointwise converges to a sequenceT ∈ O(ϕ, ψ).
Now we can prove the following result, improving Proposition 4.12.
Proof. Proposition 4.12 shows that the set O(ϕ, ψ) of all optimal sequences in T (ϕ, ψ) is not empty. Let Kϕ * be the set of all critical points of ϕ * . If T = (τi) ∈ O(ϕ, ψ) we define the weight w(T ) = where γi = min{τi − x|x ∈ Kϕ * , x ≤ τi} (in other words γi is the distance between τi and the first critical point of ϕ * on its left). This positive terms series converges since it is smaller than For any n ∈ N we can take a sequence T n = (τ n i ) ∈ O(ϕ, ψ) in such a way that limn→∞ w(T n ) = inf T ∈O(ϕ,ψ) w(T ). By Lemma 4.13, we can extract from (T n ) a subsequence pointwise converging to an optimal sequence b T = (τi).
Let us set γ n i = min{τ n i − x|x ∈ Kϕ * , x ≤ τ n i } for every n, i ∈ N, and b γi = min{τi − x|x ∈ Kϕ * , x ≤τi} for every i ∈ N. Since the set Kϕ * is closed, we can easily prove that, for every i ∈ N, either b γi = limn→∞ γ n i or b γi = 0 although limn→∞ γ n i = 0. By recalling once again that ψ has bounded variation, it follows that Then an index j ∈ I exists such thatτj ∈ Kϕ * and, since Kϕ * is closed, we can find an η > 0 for which the closure of the open interval U = (τj − η,τj + η) does not meet Kϕ * . We want to show that we can move all points of b T in U leftwards, and get an optimal sequence with a weight that is strictly less than w( b T ). This will generate our contradiction.
In order to do that, let us consider a C 1 function ρ : R → R such that The last hypothesis guarantees that the function ϕ + = ϕ * + ρ is a (possibly orientation-reversing) diffeomorphism from U onto its image ϕ + (U ). Since ϕ + (U ) = ϕ * (U ), we can consider the function from U to U that takes each point t to the unique point t ′ such that ϕ * (t ′ ) = ϕ + (t) (observe that either both ϕ + and ϕ * are strictly increasing in U or both ϕ + and ϕ * are strictly decreasing in U ). We can extend this function to a function h + : R → R by defining it to equal the identity outside U . It is immediate to verify that h + is an orientation-preserving diffeomorphism, since dϕ * dt and dϕ + dt take the same sign in U . Analogously, the function ϕ − = ϕ * − ρ is a diffeomorphism from U onto its image ϕ − (U ) = ϕ * (U ).
Hence we can consider the function from U to U that takes each point t to the unique point t ′ such that ϕ * (t ′ ) = ϕ − (t). We can extend this function to a function h − : R → R by defining it to equal the identity outside U , and h − is an orientation-preserving diffeomorphism.
Now, let us define two new sequences (τ + i ) and (τ − i ). For every i ∈ N we set τ Since h + is an orientation-preserving diffeomorphism,τi ≤τj if and only if τ + i ≤ τ + j . That means that also (τ + i ) is a sequence compatible with (ϕ, ψ). Analogously, also (τ − i ) results to be a sequence compatible with (ϕ, ψ). Since the sequence (τi) is optimal, if P i∈I ψ| b i a i · ϕ * (τi) ≥ 0 the following statements hold: and hence P i∈I ψ| b i a i · ρ(τi) = 0. On the other hand, if P i∈I ψ| b i a i · ϕ * (τi) < 0 the optimality of (τi) implies the following statements: It follows that also the sequences T + = (τ + i ) and T − = (τ − i ) belong to O(ϕ, ψ). Moreover, since ρ(t) > 0 if t ∈ U , it holds that either h + or h − moves every point in U leftwards (according to whether dϕ * dt is negative or positive in U , respectively), while both of them do not move the points outside U . Therefore Proposition 4.14 allows us to obtain immediately the next useful result, strengthening Theorem 4.9.
We state first a new definition. such thatτi is a critical point of ϕ * for every index i ∈ I. We shall say that these sequences are the optimal critical sequences for (ϕ, ψ).
Remark 4.17. Another way to express Theorem 4.16 is stating that the standard RPI-norm ϕ [ψ] equals the value max (τ i )∈C(ϕ,ψ)˛P i∈I ψ| b i a i · δτ i (ϕ * )˛. In other words, previous Theorem 4.16 makes available an equivalent discrete definition for standard RPI-norms. This definition allows for easier computations.
We conclude this section with a remark.
In other words, this means that any finite linear combination of Dirac deltas corresponds to a standard RPI-norm. This is a partial converse of Remark 4.17, stating that any standard RPI-norm corresponds to a (not necessarily finite) linear combination of Dirac deltas. It might be interesting to know under which hypotheses the statement seen in Remark 4.18 is true for series of Dirac deltas.

Relationship between RPI-norms and standard RPI-norms
A piecewise monotone almost sigmoidal function is an almost sigmoidal function that is monotone in each connected component of the complement of a finite set. In this section we shall prove a key result in this paper, showing that all the RPI-norms of piecewise monotone C 1 c -functions are determined by standard RPI-norms (Theorem 5.8).
Before dealing with the technical details of our proofs, it may be useful to sketch the underlying ideas.
The basic question to be answered could be formulated in this way: "How can we use the information contained in the standard RPI-norms in order to reconstruct the function ϕ?" In order to make this point clear, let us consider for instance a function ψ that is associated with the linear combination of Dirac deltas Σ3 = δt 0 − δt 1 + δt 2 , where the values t0, t1, t2 are set equal to three suitable critical points of ϕ * , according to the Representation Theorem 4.16. In order to get some more information about ϕ, we have to change ψ (and consequently Σ3). The simplest way to change ψ and Σ3 is to slightly perturb one of the three weights 1, −1, 1 in our linear combination of deltas. E.g., we can consider the linear combination Σ ε 3 = (1 + ε)δt 0 − δt 1 + δt 2 , associated with a suitable function ψε. For ε small enough, the choice of t0, t1, t2 for which |Σ ε 3 (ϕ * )| is maximum allows also |Σ3(ϕ * )| to attain its maximum value. This "invariance of the basepoints t0, t1, t2 with respect to small changes of the weights" and the fact that Σ ε 3 (ϕ * ) and Σ3(ϕ * ) take the same sign will allow us to write the following equalities: It follows that the function ϕ [ψε ] is differentiable with respect to ε and that d ϕ [ψε] dε (0) equals ϕ * (t0) · sign (Σ3(ϕ * )). So, we get that the value v0 taken by ϕ * at the critical point t0 equals We can repeat the above procedure to obtain the values taken by ϕ * at the other two critical points t1 and t2. Hence, so far, we know that ϕ * is a function that takes the values v0, v1 and v2 in this order, when t varies from −∞ to +∞. By taking functions ψ with an increasing number of oscillations (i.e. ψ corresponding to P n−1 i=0 (−1) i δt i ), we obtain more and more information about the values of ϕ * at critical points. Since ϕ is piecewise monotone, if the number of oscillations of ψ is large enough then ϕ * is monotone between two suitable critical points ti and ti+1 of ϕ * . Obviously, we have not enough information to locate the points ti's, but we are able to reconstruct the oscillations of ϕ * · sign (Σ3(ϕ * )), up to reparametrization and an arbitrarily small error with respect to the variation norm. Roughly speaking, these are the ideas we are going to use.
First of all we recall the formal definition of piecewise monotone function.
Definition 5.1. We say that f : R → R is piecewise monotone if a finite set W ⊂ R exists such that f is monotone in each connected component of the complement of W . Each such a set W will be said to be a separating set for f . If W is also minimal with respect to inclusion, it will be said to be a minimal separating set for f . We define l(f ) as the minimum of the cardinalities of the separating sets for f .
Obviously, l(S) = 0 and l(Λ) = 1. Note that if f ∈ C 1 c (R) and f = 0 then l(f ) ≥ 1. It is easy to show that all minimal separating sets for f take the same cardinality l(f ). This follows from the next simple proposition (we omit the immediate proof): Proposition 5.2. Let f : R → R be a piecewise monotone function. Let W = {t0, . . . , tm} and W ′ = {t ′ 0 , . . . , t ′ n } be two separating sets for f , with t0 < t1 < . . . < tm and W ′ minimal. If for some i and j it holds that tj < t We also observe that the concept of piecewise monotone almost sigmoidal C 1 -function is invariant under reparametrization, and that l(ϕ) = l(ϕ * ). Moreover, the points of a minimal separating set for ϕ * are necessarily critical points for ϕ * .
In the rest of this section, when ϕ is a piecewise monotone almost sigmoidal C 1 -function with nonempty compact support (i.e. ϕ = 0), we let [a, b] denote the minimal interval containing the support of ϕ * . Moreover, if˘t0, . . . , t l(ϕ)−1¯i s a minimal separating set for ϕ * , we assume it is increasingly ordered and we define c = min 0≤i≤l(ϕ) |ϕ(ti) − ϕ(ti−1)|, where we set t−1 = a and t l(ϕ) = b. This meaning of the symbols t−1 and t l(ϕ) will be maintained in the following pages.
Before proceeding, we need to introduce a new family of functions.
In plain words, the function S e n is a perturbation of the function Sn. In particular, for e = (0, . . . , 0), the function S e n equals the function Sn. We observe that the functions S e n are piecewise polynomial and belong to AS 1 (R). We also note that ϕ [Sn] ≤ ϕ [S n+1 ] for every n ≥ 1 and every ϕ ∈ AS 1 (R).
The following lemma is a key passage towards the proof of the Reconstruction Theorem 5.8 for piecewise monotone functions in C 1 c (R).
First of all, we can assume that i) the only repeated point in the sequence T is b since consecutive points appear with opposite weights, and that ii) each τi belongs to {t0, t1, . . . , t l(ϕ)−1 } ∪ {a, b}.
Indeed, if statement ii) were false we could easily find a better sequence than T by using the monotonicity of ϕ * outside the set {t0, t1, . . . , t l(ϕ)−1 }, and hence T would not be optimal. Now, let us note that τ l(ϕ)−1 = b, since τ l(ϕ)−1 is the first dummy point of T ∈ T`ϕ, S l(ϕ)−1´( recall Definition 4.6). Therefore, if τ0 = a we obtain v) for each i with 0 ≤ i ≤ l(ϕ) − 2, exactly one index j exists such that 0 ≤ j ≤ l(ϕ) and τi = tj . Furthermore, if i is even then ϕ * (tj) > ϕ * (tj−1), while if i is odd then ϕ * (tj) < ϕ * (tj−1) (here we set Indeed, if i is even and A) holds property v-A) implies that j is even, so that ϕ * is increasing in [tj −1, tj]. If i is even and B) holds property v-B) implies that j is odd, so that ϕ * is increasing in [tj−1, tj ].
If i is odd and A) holds property v-A) implies that j is odd, so that ϕ * is decreasing in [tj −1, tj]. If i is odd and B) holds property v-B) implies that j is even, so that ϕ * is decreasing in [tj −1, tj]. In summary, if i is even then ϕ * is increasing in [tj −1, tj], while if i is odd then ϕ * is decreasing in [tj −1, tj ]. This proves property v).
In order to check vi), let us proceed by contradiction and assume that τ l(ϕ)−3 = b. Note that necessarily τ l(ϕ)−2 = τ l(ϕ)−1 = b. Since the number of points τi that are different from b is at most l(ϕ) − 3, at least three points of the separating set {t0, t1, . . . , t l(ϕ)−1 } do not coincide with any τi. Let us consider the largest index k ≤ l(ϕ) − 1 such that t k does not belong to the set {τi}. Since, by definition, t k+1 ∈ {τi}, v-A) and v-B) imply that t k−1 does not belong to the set {τi}. Indeed, properties v-A) and v-B) guarantee that for every index i the points τi and τi+1 are separated by an even number of points tj (possibly 0).
The first two cases happen when τ l(ϕ)−2 = b, the last case when τ l(ϕ)−2 = b. Note that if τ l(ϕ)−2 = b, exactly l(ϕ) − 1 points in the separating set must belong to the set˘τ0, τ1, . . . , τ l(ϕ)−2¯, and recall once again that for every index i the points τi and τi+1 are separated by an even number of points tj (possibly 0). Hence the only tj's that can miss in the set˘τ0, τ1, . . . , τ l(ϕ)−2¯a re t0 and t l(ϕ)−1 . This observation produces the cases 1 and 2. If τ l(ϕ)−2 = b a gap of two consecutive tj's is possible, implying the case 3.
An immediate consequence of the previous lemma is the following result, allowing us to deduce the value of l(ϕ) from the knowledge of the standard RPI-norms ϕ [Sn] , n ≥ 1, when ϕ is piecewise monotone and belongs to C 1 c (R).
Corollary 5.5. Let ϕ = 0 be a piecewise monotone C 1 -function with compact support. The value l(ϕ) is equal to the smallest integer N such that ϕ [S N ] = ϕ [Sn] for every n ≥ N .
We now show that the standard RPI-norms of ϕ with respect to S e l(ϕ) allow us to obtain the values of ϕ at the points of each minimal separating set.
We are now ready to prove that given a piecewise monotone C 1 -function ϕ with compact support, it is possible to construct a piecewise polynomial almost sigmoidal C 1 -functionφ such thatφ approximates ±ϕ in the total variation norm up to reparametrization, and ϕ = φ for any RPI-norm on AS 1 (R). The key point here is that this construction is based just on the knowledge of the values taken by the standard RPI-norms at ϕ. In other words, the RPI-norms of piecewise monotone functions in C 1 c (R) are determined by the standard RPI-norms.

Open problems and conclusions
In this paper we have studied the main properties of the reparametrization invariant norms on AS 1 (R), focusing the key role of standard RPI-norms. We have proved that these norms allow for the reconstruction of any piecewise monotone C 1 -function with compact support up to reparametrization, so determining the value of any other RPI-norm on the same function.
However, many problems remain open. First of all the theory has been developed just for the space AS 1 (R) and our last results require also to assume that the considered functions are piecewise monotone and have compact support. The extensions to less regular spaces could be desirable.
Another interesting extension could be the passage from spaces of functions defined on R to spaces of functions defined on R n or a closed manifold. We notice that in the case of a closed manifold M the extension could be intertwined with the study of the topology of M, requiring to substitute the role of the We postpone the research on these issues to other papers.