An introduction to extended Gevrey regularity

Gevrey classes are the most common choice when considering the regularities of smooth functions that are not analytic. However, in various situations, it is important to consider smoothness properties that go beyond Gevrey regularity, for example when initial value problems are ill-posed in Gevrey settings. Extended Gevrey classes provide a convenient framework for studying smooth functions that possess weaker regularity than any Gevrey function. Since the available literature on this topic is scattered, our aim is to provide an overview to extended Gevrey regularity, highlighting its most important features. Additionally, we consider related dual spaces of ultradistributions and review some results on micro-local analysis in the context of extended Gevrey regularity. We conclude the paper with a few selected applications that may motivate further study of the topic.


Introduction
Gevrey type regularity was introduced in the study of fundamental solutions of the heat equation in [1] and subsequently used to describe regularities stronger than smoothness (C ∞ -regularity) and weaker than analyticity.This property turns out to be important in the general theory of linear partial differential equations, such as hypoellipticity, local solvability, and propagation of singularities, cf.[2].In particular, the Cauchy problem for weakly hyperbolic linear partial differential equations (PDEs) can be well-posed in certain Gevrey classes, while at the same time being ill-posed in the class of analytic functions, as shown in [3,2].
Since there is a gap between Gevrey regularity and smoothness, it is important to study classes of smooth functions that do not belong to any Gevrey class.For example, Jézéquel [4] proved that the trace formula for Anosov flows in dynamical systems holds for certain intermediate regularity classes, and Cicognani and Lorenz used a different intermediate regularity when studying the well-posedness of strictly hyperbolic equations in [5].
A systematic study of smoothness that goes beyond any Gevrey regularity was proposed in [6,7].This was accomplished by introducing two-parameter dependent sequences of the form (p τ p σ ) p , where τ > 0, σ > 1.These sequences give rise to classes of ultradifferentiable functions E τ,σ (R d ), which differ from classical Carleman classes C L (R d ) (cf. [8]), are larger than Jézéquel's classes, and which go beyond Komatsu's approach to ultradifferentiable functions as described in, for example, [9].On one hand, these classes, called Pilipović-Teofanov-Tomić classes in [10], serve as a prominent example of the generalized matrix approach to ulradifferentiable functions.On the other hand, they provide asymptotic estimates in terms of the Lambert functions, which have proven to be useful in various contexts, as discussed in [5,11,12].
Different aspects of the so-called extended Gevrey regularity, i.e., the regularity of ultradifferentiable functions from E τ,σ (R d ), have been studied in a dozen papers published in the last decade.Our aim is to offer a self-contained introduction to the subject and illuminate its main features.We provide proofs that, in general, simplify and complement those in the existing literature.Additionally, we present some new results, such as Proposition 3.1, Proposition 3.3, and Theorem 3.1 for the Beurling case, as well as Theorem 3.3.This survey begins with preliminary Section 2, which covers the main properties of defining sequences, the Lambert function, and the associated function to a given sequence.We emphasize the remarkable connection between the associated function and the Lambert W function (see Theorem 2.1), which provides an elegant formulation of decay properties of the (short-time) Fourier transform of f ∈ E τ,σ (R d ), as demonstrated in Proposition 3.4 and Corollary 4.1.In Section 3, we introduce the extended Gevrey classes E τ,σ (R d ) and the corresponding spaces of ultradistributions.We then present their main properties, such as inverse closedness (Theorem 3.1) and the Paley-Wiener type theorem (Theorem 3.3).
In Section 4, we give an application of extended Gevrey regularity in micro-local analysis.More precisely, we introduce wave-front sets, which detect singularities that are "stronger" than classical C ∞ singularities and, at the same time, "weaker" than any Gevrey type singularity.
To provide a flavor of possible applications of extended Gevrey regularity, in Section 5, we briefly outline some results from [5] and [10].More precisely, we present a result from [5] concerning the wellposedness of strictly hyperbolic equations in E 1,2 (R d ), and observations from [10], where the extended Gevrey classes are referred to as Pilipović-Teofanov-Tomić classes and are considered within the extended matrix approach to ultradifferentiable classes.
We end this section by introducing some notation that will be used in the sequel.
1.1.Notation.We use the standard notation: N, N 0 , Z, R, R + , C, denote sets of positive integers, non-negative integers, integers, real numbers, positive real numbers and complex numbers, respectively.The length of a multi-index α We write L p (R d ), 1 ≤ p ≤ ∞, for the Lebesgue spaces, and S(R d ) denotes the Schwartz space of infinitely smooth (C ∞ (R d )) functions which, together with their derivatives, decay at infinity faster than any inverse polynomial.By S ′ (R d ) we denote the dual of S(R d ), the space of tempered distributions, and ), the space of compactly supported infinitely smooth functions.
We use brackets f, g to denote the extension of the inner product f, g = f (t)g(t)dt on L 2 (R d ) to the dual pairing between a test function space A and its dual | for some C > 0 and x in the intersection of domains for f and g.
Translation, modulation, and dilation operators, T , M, and D respectively, when acting on f ∈ L 2 (R d ) are defined by x ∈ R d , a > 0. Then for f, g ∈ L 2 (R d ) the following relations hold: The Fourier transform, convolution, T , M, and D are extended to other spaces of functions and distributions in a natural way.
Let (M p ) be a positive monotone increasing sequence that satisfies (M.1).Then (M p /p!) 1/p , p ∈ N is an almost increasing sequence if there exists C > 0 such that , p ≤ q, and lim This property is related to inverse closedness in C ∞ (R d ), see [13].
If (M p ) and (N p ) satisfy (M.1), then we write M p ⊂ N p if there exist constants A > 0 and B > 0 (independent on p) such that If, instead, for each B > 0 there exists A > 0 such that (1) holds, then we write Assume that (M p ) satisfies (M.1) and (M.3) ′ .Then p! ≺ M p .Let R denote the set of all sequences of positive numbers monotonically increasing to infinity.For a given sequence (M p ) and (r p ) ∈ R we consider It is easy to see that if (M p ) satisfies (M.1) and (M.3) ′ , then (N p ) satisfies (M.1) and (M.3) ′ as well.In addition, one can find (r p ) ∈ R so that (M p p j=1 rj ) satisfies (M.2) if (M p ) does.This follows from the next lemma.
Proof.It is enough to consider the sequence (r p ) given by r1 = r 1 and inductively rj+1 = min r j+1 , j + 1 j rj , j ∈ N.

2.2.
Defining sequences for extended Gevrey regularity.To extend the class of Gevrey type ultradifferentiable functions we consider two-parameter sequences of the form M τ,σ p = p τ p σ , p ∈ N, τ > 0, σ > 1. ¿From Stirling's formula, and the fact that there exists C > 0 (independent of p) such that sp ≤ Cτ p σ , p ∈ N, for any s, σ > 1, and τ > 0, it follows that p! s ≤ C 1 p τ p σ , for a suitable constant C 1 > 0.
2.3.The Lambert function.The Lambert W function is defined as the inverse of ze z , z ∈ C. By W (x), we denote the restriction of its principal branch to [0, ∞).It is used as a convenient tool to describe asymptotic behavior in different contexts.We refer to [15] for a review of some applications of the Lambert W function in pure and applied mathematics, and to the recent monograph [16] for more details and generalizations.It is noteworthy that the Lambert function describes the precise asymptotic behavior of associated function to the sequence (M τ,σ p ).This fact was firstly observed in [17].Some basic properties of the Lambert function W are given below: (W 1) W (0) = 0, W (e) = 1, W (x) is continuous, increasing and concave on [0, ∞), (W 2) W (xe x ) = x and x = W (x)e W (x) , x ≥ 0, (W 3) W can be represented in the form of the absolutely convergent series with suitable constants c km and x 0 , wherefrom the following estimates hold: The equality in (4) holds if and only if x = e.Note that (W 2) implies W (x ln x) = ln x, x > 1.
By using (W 3) we obtain and therefore for any C > 0. We refer to [15,16] for more details about the Lambert W function.
2.4.Associated functions.Let (M p ) be an increasing sequence positive numbers which satisfies (M.1), and M 0 = 1.Then the Carleman associated function to the sequence (M p ) is defined by This function is introduced in the study of quasi-analytic functions, see, e.g.[18].We use the notation from [19].
In Komatsu's treatise of ultradistributions [9], the associated function to (M p ) is instead given by Lemma 2.3.Let (M p ) be an increasing sequence positive numbers which satisfies (M.1), and M 0 = 1, and let the functions µ and T be given by ( 5) and ( 6) respectively.Then Proof.Clearly, which is (7).
When (M p ) is (equivalent to) the Gevrey sequence, M p = p sp , p ∈ N, s > 1, an explicit calculation gives Thus (7) implies that there exist constants k > 0, and C > 0 such that see also [19,Ch IV,2.1].By using (6) we define the associated function to the sequence M τ,σ p = p τ p σ , p ∈ N, τ > 0, σ > 1, as follows: It is a remarkable fact that T τ,σ (h) can be expressed via the Lambert W function.
Proof.The proof follows from [20, Proposition 2] and estimates (30) given in its proof.More precisely, it can be shown that for large enough h > 0, and suitable constants A σ , B σ , A τ,σ , B τ,σ > 0.
We also notice that (W 3) (from subsection 2.3) implies , for h large enough.
2.5.Associated function as a weight function.The approach to ultradifferentiable functions via defining sequences is equivalent to the Braun-Meise-Taylor approach based on weight functions, when the defining sequences satisfy conditions (M.1), (M.2) and (M.3), see [22,21].Since M τ,σ p = p τ p σ , p ∈ N, does not satisfy (M.2), to compare the two approaches in [20], the authors used the technique of weighted matrices, see [23].One of the main results from [20] can be stated as follows.
Recall, a weight function is non-negative, continuous, even and increasing function defined on R + ∪ {0}, ω(0) = 0, if the following conditions hold: where ln + x = max{0, ln x}, x > 0.Moreover, ω(t) = |t| s is a weight function if and only if 0 < s ≤ 1.We refer to [23] for the weighted matrices approach to ultradifferentiable functions.It is introduced in order to treat both Braun-Meise-Taylor and Komatsu methods in a unified way, see also subsection 5.2.

Extended Gevrey regularity
3.1.Extended Gevrey classes and their dual spaces.Recall that the Gevrey space for all x ∈ K and for all α ∈ N d 0 .In a similar fashion we introduce new classes of smooth functions by using defining sequences M τ,σ p = p τ p σ , p ∈ N, τ > 0, σ > 1.
Definition 3.1.Let there be given τ > 0, σ > 1, and let for all x ∈ K and for all α ∈ N d 0 .The extended Gevrey class of Beurling type for all x ∈ K and for all α ∈ N d 0 .The spaces E {τ,σ} (R d ) and E (τ,σ) (R d ) are in a usual way endowed with projective and inductive limit topologies respectively, we refer to [6] for details.In particular, they are nuclear spaces, see [6,Theorem 3.1].
Note that (11), ( 12) and ( 13) imply where ֒→ denotes continuous and dense inclusion.The set of functions We use the abbreviated notation τ, σ for {τ, σ} or (τ, σ) to denote Next we give an equivalent description of extended Gevrey classes by using sequences from R, see subsection 2.2.We note that such descriptions are important when dealing with integral transforms of ultradifferentiable functions and related ultradistributions, cf.[24,25,14].The result follows from a lemma which is a modification of [25,Lemma 3.4], and [24, Lemma 2.2.1].
Put ⌊x⌋ := max{m ∈ N : m ≤ x} (the greatest integer part of x ∈ R + ).Lemma 3.1.Let there be given σ > 1, a sequence of positive numbers (a p ), (r j ) ∈ R, and put Then the following is true.i) There exists h > 0 such that ii) There exists (r j ) ∈ R such that The proof of Lemma 3.1 is given in the Appendix.Note that in ( 12) and ( 13) we could put h ⌊|α| σ ⌋ instead of h |α| σ (this follows from the simple inequality ⌊p σ ⌋ ≤ p σ ≤ 2⌊p σ ⌋, p ∈ N).Proposition 3.1.Let there be given τ > 0, σ > 1, and let and for any (r p ) ∈ R and R p,σ given by ( 14), there exists C K,(rp) > 0 such that |α| , for all x ∈ K, and all α ∈ N d 0 .
ii) φ ∈ E (τ,σ) (R d ) if and only if for every compact set K ⊂⊂ R d there is a sequence (r p ) ∈ R and a constant C K > 0 satisfying for all x ∈ K and for all α ∈ N d 0 , where R p,σ given by (14).Proposition 3.1 follows from Lemma 3.1.We end this subsection by introducing spaces of ulradistributions as dual spaces of E τ,σ (R d ), τ > 0, σ > 1.In subsection 3.5 we will prove a Paley-Wiener type theorem for such ultradistributions.Definition 3.2.Let τ > 0 and σ > 1, and let and (•, •) denotes standard dual pairing.In a similar way Example of a compactly supported function.The nonquasianalyticity condition (M.3) ′ provides the existence of nontrivial compactly supported functions in E τ,σ (R d ) which can be formulated as follows.
Proposition 3.2.Let τ > 0 and σ > 1.For every a > 0 there exists Of course, any compactly supported Gevrey function from G τ (R d ) will suffice.However, the construction in Proposition 3.2, is sharp in the sense that φ a does not belong to any Gevrey class, i.e. φ a ∈ t>1 G t (R d ).We refer to the proof of [8, Lemma 1.3.6.] for more details.
Proof.We give a proof when d = 1, and for d ≥ 2 the proof follows by taking the tensor product.
Since D τ,σ (R) is closed under dilation and multiplication by a constant, it is enough to show the result for a = 1, and set φ 1 = φ. From (2(p + 1)) for any m ∈ N 0 and any given σ > 1, it follows that there exists a sequence of nonnegative integers (N m ) such that Thus the sequence a p , p ∈ N 0 , given by Then we define the sequence of functions (φ p ) by when p ≥ N m .Let there be given n ∈ N 0 and τ > 0. Then we choose m, p ∈ N 0 so that 1/m < τ , and N m + n < p.
By using (18) and the fact that (M.2 for some C = C(q) > 0, we obtain where C depends on τ .
By the Leibniz formula and (M.1) we have and obtain Thus, for any given h > 0 we can choose h < ( h/2) 1/2 σ to get where C > 0 depends on K and h, that is, φψ ∈ E (τ,σ) (R d ).
3.4.Inverse closedness and composition.We need some preparation related to the decompositions that appear when using the generalized Faà di Bruno formula.
It remains to show that for each h > 0 there exists C > 0 such that This can be done by induction with respect to the length of the multiindex α ∈ N d .The proof for |α| = 1 is the same as in d = 1.Now, if (21) holds for |α| < n, the case |α| = n, follows from the induction step and Proposition 3.3.We omit details.
The proof of Theorem 3.2 for (the Roumieu case) can be found in [7].Theorem 3.1 is a consequence of Theorem 3.2, but, as we see, it can be proved independently.

Paley-Wiener theorems. Let
A more general statement than Proposition 3.4 is given in [17, Theorem 3.1].Proposition 3.4.Let σ > 1, and let f ∈ D (σ) (R d ).Then f , the Fourier transform of f , is analytic function, and for every h > 0 there exists a constant where W denotes the Lambert function.
Let f ∈ D (σ) (R d ), and let K denote the support of f .Since f ∈ D τ 2 ,σ (R d ), by Definition 3.1 for every α ∈ N d we get the following estimate: for a suitable constant C 2 > 0. Now, the relation between the sequence (M τ,σ p ) and its associated function T τ,σ given by ( 8) implies that for suitable C 3 > 0.Then, from the left-hand side of (10) we get with C 4 = C 3 e − Bτ,σ .For any given h > 0 we choose τ = (B σ /h) σ−1 , to obtain (22), which proves the claim.

AN INTRODUCTION TO EXTENDED GEVREY REGULARITY
ii) It is sufficient to prove that for every τ > 0 there exists constant C > 0 such that sup Notice that h > 0 large enough in (10) can be replaced by (1 + |ξ|) for all ξ ∈ R d .

4.
Wave-front sets for extended Gevrey regularity 4.1.Wave-front set and singular support.Wave-front sets measure different types of directional singularities.For example, where u ∈ D ′ (R d ), WF is the classical (C ∞ ) wave-front set, WF t is the Gevrey wave-front set, and WF A is analytic wave-front set, we refer to [27,8,2] for precise definitions.
In this section we introduce wave-front sets which detect singularities that are "stronger" then the classical C ∞ singularities and "weaker" than any Gevrey singularity.Moreover, the usual properties (such as pseudo-local property), which hold for wave-front sets quoted in (25), are preserved when considering the new type of singularities.
Here, u ∈ E {τ,σ} (Ω) means that u satisfies the conditions of Definition 3.1, i.e. (12), with R d replaced by its open subset Ω at each occurrence.
The next result is a consequence of Definition 4.1 and 4.2, we refer to [7] for the proof.
4.2.Characterization of wave-front sets via the STFT.For the estimates of the short-time Fourier transform it is convenient to consider the following refinement of the associated function T τ,σ (see (8)).
Lemma 4.1.Let T τ,σ (h, k) be given by (27), and let T τ,σ (k) be given by (8).Then for any given h > 0 and τ 2 > τ > τ 1 > 0 there exists A, B ∈ R such that It is known that the classical wave-front set WF(u) can be described by the means of the short-time Fourier transform, see [28].Related characterization of WF {τ,σ} (u) is given in [29].Here we provide a slightly different statement, and a more detailed proof.
Let there be given f, g ∈ L 2 (R d ).The short-time Fourier transform (STFT) of f with respect to the window g is given by We observe that the definition of V g f makes sense when f and g belong to any pair of dual spaces, extending the inner product in L 2 (R d ) as it is mentioned in section 1.1.
Proof.We follow the idea presented in [28], and give the proof to enlighten the difference between WF(u) and WF {τ,σ} (u).
(⇒) Assume that there is a conic neighborhood Γ of ξ 0 , a compact set K 1 in R d , so that for any φ ∈ D K 1 {τ,σ} (R d ), such that φ(x 0 ) = 0, the estimate (29) holds for some C, h > 0. Without loss of generality, we may assume that K 1 = B r (x 0 ) for some r > 0.
By (29) it follows that the set Let K = B r/2 (x 0 ), and consider the window g ∈ D Kx 0 {τ,σ} (R d ), such that g = 0 on a neighborhood of 0. Then , and φ = 0 on a neighborhood of x 0 .By the equicontinuity of H h it follows that ¿From the definition of STFT it follows that This, together with (32) implies for all x ∈ K, and for some constants C, h > 0, which gives (31).Notice that we actually proved that (31) holds for any g satisfying (30).
By using Lemma 4.1 and Theorem 2.1 we can express the decay estimate (31) in terms of the Lambert function as follows.
As a combination of results from Theorem 3.3 ii) and Theorem 4.1, we can use Corollary 4.1 to characterize local regularity of u ∈ D ′ (R d ).Namely if (33) holds, then x 0 ∈ singsupp {τ,σ} (u), so that u ∈ E {τ,σ} (Ω) in a neighborhood Ω of x 0 (see Definition 4.2).4.3.Propagation of singularities.One of the main properties of wave-front sets is microlocal hypoelipticity.We first recall the notion of the characteristic set of an operator and the main property of its principal symbol.
If P (x, D) = |α|≤m a α (x)D α is a differential operator of order m in R d and a α ∈ C ∞ (R d ), |α| ≤ m, then its characteristic set is given by is the principal symbol of P (x, D).If Char(P (x, D)) = ∅, then the operator P (x, D) is hypoelliptic.
Noe, for the Roumieu wave-front WF {τ,σ} (u) we have the following theorem on the paopagation of singularities.
The proof of Theorem 4.3 uses inverse closedness (Theorem 3.1), Paley-Wiener type estimates (Theorem 3.3), and contains nontrivial modifications of the proof of [8,Theorem 8.6.1].We refer to [7] for a detailed proof.Note that if σ = 1 and τ > 1, we recover the result for propagation of singularities when the coefficients are Gevrey regular functions, and WF 0,σ (f ) = WF 0,σ (u) in (34) reveals the hypoellipticity of (P (x, D).

5.1.
A strictly hyperbolic partial-differential equation.Cicognani and Lorenz in [5] considered the Cauchy problem for strictly hyperbolic m-th order partial-differential equations (PDEs) of the form where where f and g k , k = 1, . . ., m, satisfy certain Sobolev type regularity conditions (cf.(SH3-W) and (SH4-W) in [5]).and studied well-posedness when the coefficients are low-regular in time, and smooth in space.More precisely, it is assumed that the coefficients a m−j,γ satisfy conditions of the form where µ is a modulus of continuity, and (K |β| ) is a defining sequence (also called a weight sequence).
The modulus of continuity µ is used to describe the (low) regularity in time, whereas (K |β| ) describes the regularity in space.
When µ is a weak modulus of continuity, (Log-Log [m] -Lip-continuity), a suitable weight function η which defines the solution space is chosen to be where ε > 0 is arbitrarily small and c m > 0 such that η(s) ≥ 1 for all s > 1.
We refer to [5] for a detailed analysis of (35), and note that the relation between the modulus of continuity µ and the weight function η is given by lim while the condition which links the weight sequence (K p ) to the weight function η is given by inf for some h, C > 0, which is essentially the relation between the Carleman associated function and the Komatsu associated function as given in Lemma 2.3.One of the conclusions in [5] is that the Cauchy problem ( 35) is wellposed if a m−j,γ (t, x) ∈ E {1,2} (R d ) uniformly in x for every fixed t.In other words, the sequence (K |β| ) in (36) is given by K p = p p 2 .[10] that the extended Gevrey classes are prominent example of ultradifferentiable functions defined in the framework of generalized weighted matrices approach.

Generalized definition of ultradifferentiable classes. It is recently demonstrated in
The main idea behind the weighted matrices approach as given in [30] and [23] is to establish a general framework for considering the Braun-Meise-Taylor and Komatsu approach to ultradifferentiable functions in a unified way.To include the extended Gevrey classes which are called PTT-classes in [11] and [10] (after Pilipović-Teofanov-Tomić), the so-called exponential sequences Φ = (Φ p ) p∈N , and the related generalized weighted matrix setting are introduced in [10].One of the main observations in [10] is that the exponential sequences Φ (such as (h p σ ) p∈N , for some h > 0) yield "ultradifferentiable classes beyond geometric growth factors", under mild regularity and growth assumptions on Φ.In such context, PTT-classes constitute a genuine examples of of ultradifferentiable functions defined by weight matrices.
This approach reveals that, apart from stability properties mentioned in Section 3, PTT-classes enjoy almost analytic extension [31], and almost harmonic extension [32].Moreover, PTT-classes are a convenient tool for the study of Borel mappings.More precisely, the asymptotic Borel mapping, which sends a function into its series of asymptotic expansion in a sector, is known to be surjective for arbitrary openings in the framework of ultraholomorphic classes associated with sequences of rapid growth.By using the PTT-classes E {τ,σ} (R d ), given by M τ,σ p = p τ p σ , p ∈ N, τ > 0, 1 < σ < 2, Jiménez-Garrido, Lastra and Sanz, presented a constructive proof of the surjectivity of the Borel map in sectors of the complex plane for the ultraholomorphic class associated with those specific sequences.In fact, the asymptotic behavior of the associated function given in terms of the Lambert function (see Theorem 2.1) plays a prominent role in these investigations.We refer to [11] for more details.
The logarithm of the first term on the right hand side of the inequality can be estimated as follows: By taking exponential we obtain and by replacing the roles of p and q we get (p + q) τ 2 σ−1 q σ ≤ q τ 2 σ−1 q σ e τ 2 σ−1 (p+q) σ , thus (p + q) τ (p+q) σ ≤ p τ 2 σ−1 p σ q τ 2 σ−1 q σ e τ 2 σ (p+q) σ , and (M.Proof.(proof of Lemma 3.1) i) (⇒) Let a p ≤ Ch p σ , p ∈ N 0 , for some C, h > 0, let (r j ) be any sequence in R, and let j 0 ∈ N 0 be such that h r j ≤ 1, for all j ≥ j 0 .Then for large enough p ∈ N 0 and suitable C 1 > 0. This proves (15).(⇐) The opposite part we prove by contradiction.Assume that (15) holds for arbitrary (r j ) ∈ R, and that sup a p h p σ : p ∈ N 0 = ∞ for every h > 0.
Thus, for every n ∈ N and h := n there exists p n ∈ N such that a pn n ⌊p σ n ⌋ > n.If n = 1, then there exists p 1 ∈ N such that a p 1 > 1, and obviously a p 1 r 1 r 2 . . .r ⌊p σ By the construction it follows that (r j ) ∈ R, and for the sequence we obtain sup a p R p,σ : p ∈ N 0 = ∞, which contradicts (15).
We define It is easy to see that (H j ) is a well defined sequence which satisfies (M.1), and that H j /h j tends to infinity as j → ∞, for all h ≥ 1.
Therefore (r j ) ∈ R, where r j = H j H j−1 , j ∈ N. We note that