Abstract
We introduce in this paper a generalization of the pointwise Hölder spaces. We give alternative definitions of these spaces, look at their relationship with the wavelets, and introduce a notion of generalized Hölder exponent.
1. Introduction
A real-valued function defined on belongs to the uniform Hölder space () if there exist a constant and a polynomial of degree less than such thatfor all and small enough. The supremum of all these values is called the global Hölder exponent of . One can naturally define the pointwise version of these spaces: a continuous function belongs to the pointwise Hölder space if and only if there exist and a polynomial of degree less than such thatfor all . Of course, the supremum of all these values is called the Hölder exponent of at . If is differentiable at , the Hölder exponent of at is at least 1. The fact that the constant appearing in (1) is uniform for implies that the uniform Hölder exponent is not necessarily the infimum of the pointwise Hölder exponents. A classical example is the function , which is at the origin, everywhere else, while its global Hölder exponent is .
In [1, 2], the properties of generalized uniform Hölder spaces have been investigated. The idea underlying the definition is to replace the exponent of the usual spaces [3, 4] with a sequence satisfying some conditions. The so-obtained spaces generalize the spaces ; the spaces are actually the spaces , but they present specific properties (induced by -norms) when compared to the more general spaces studied in [5–10], for example. Indeed, it is shown in [1, 2] that most of the usual properties holding for the spaces can be transposed to the spaces .
Here, we introduce the pointwise version of these spaces: the spaces , with . As in [1, 2], the idea is again to replace the sequence appearing in (2) with a positive sequence such that and are bounded (for any ); the number stands for the maximal degree of the polynomial (this degree cannot be induced by a sequence ). By doing so, one tries to get a better characterization of the regularity of the studied function ; a usual choice is to replace with (see, e.g., [11–13]). For example, it is well known that, for the Brownian motion , there exist such thatfor any almost surely (see, e.g., [14, 15]). More generally, the behavior of the oscillations of as tends to 0 can reveal specific local behaviors such as approximate similarities [11]. Generalizations of the pointwise Hölder spaces have already been proposed, but, to our knowledge, the definition we give here is the most general version and leads to the sharpest results (in particular, the regularity spaces introduced in [13] are a special case of the spaces ; see Definition 2 and Remark 4). These spaces could also be useful in the study of singularities of PDEs where the function is smooth near except on small sets of points (such a situation could be met in the case of the Navier-Stokes equations in three dimensions [16]).
As a first application, we introduce here the sufficient tools to develop a multifractal formalism based on the wavelet characterization (Theorem 16), in the same spirit as in [12]. The theory presented here also contributes to enlightening the reasons why the customary proofs for the classical spaces work.
This paper is organized as follows. We first give the definitions leading to generalized pointwise Hölder spaces and prove that, under some general conditions, the polynomials appearing in the definition are independent of the scale, as is the case with the usual Hölder spaces. Next, we give some alternative definitions of the spaces , mimicking the different possible definitions of . One of the nicest properties of the Hölder spaces is their relationship with the wavelet theory given in [12]; we show here that this result still holds in the general case. Finally, we give some conditions under which one gets embedded generalized pointwise Hölder spaces and define a generalized pointwise Hölder exponent.
Throughout this paper, denotes the open unit ball of centered at the origin; moreover, we set . The floor function is denoted by and designates the set of polynomials of degree at most . We use the letter for a generic positive constant whose value may be different at each occurrence.
2. Pointwise Generalized Hölder Spaces
To present the generalized pointwise Hölder spaces, we first need to recall some notions concerning the admissible sequences. After having introduced the definitions, we point out a major difference between the usual spaces and the generalized ones: the polynomial arising in the definition depends on the scale. It is then natural to look under which conditions this constraint can be dropped.
2.1. Definition
The generalization of the Hölder spaces we propose here is based on the notion of admissible sequence [10].
Definition 1. A sequence of real positive numbers is called admissible if there exists a positive constant such thatfor any .
If is such a sequence, we setand we define the lower and upper Boyd indices as follows:Since is a subadditive sequence, such limits always exist [17]. The following relations about such sequences are well known (see, e.g., [1]). If is an admissible sequence, let ; there exists a positive constant such that for any , . Let be an admissible sequence:(i)if , then there exists a positive constant such that, for any ,(ii)if satisfies , then there exists a positive constant such that, for any ,In this paper, will always stand for an admissible sequence and for a natural number, possibly zero.
Starting from the definitions of the pointwise Hölder spaces (with ) and the generalized uniform Hölder spaces introduced in [1], we are naturally led to the following definition.
Definition 2. Let ; a continuous function belongs to if there exist such thatfor any .
We trivially have the following alternative definition for .
Definition 3. A continuous function belongs to if there exist such that, for any , there exists a polynomial for which
Remark 4. In [13], a generalization of the pointwise Hölder spaces is introduced by replacing the admissible sequence appearing in Definition 2 by a modulus of smoothness (more precisely, the definition is based on (34)). As shown in [1], given a modulus of smoothness , defines an admissible sequence. The converse is not necessarily true: only the decreasing admissible sequences converging to 0 give rise to a modulus of continuity [1].
Sometimes, we will also need to impose a slightly stronger condition than continuity to the function.
Definition 5. A function is uniformly Hölder if there exists such that .
2.2. Independence of the Polynomial from the Scale
It is important to remark that the polynomial occurring in inequality (11) is a function of the scale . However, for the classical Hölder spaces, such polynomial is independent of . Here, we look under which conditions the independence still holds in the generalized case, that is, under which conditions for any . In this section, will designate a continuous function of (although the continuity hypothesis is very often dropped when dealing with norms, we keep it here to ensure the equivalence with Definition 3).
We will need the following Markov inequality (see, e.g., [18]): Let , , and be a bounded convex set with nonempty interior; one hasfor any , where denotes the partial derivative of following the th variable and where depends only on and . If , we thus havefor any and any , where is a constant (and does not depend on , , or ).
Lemma 6. If with , the sequence of polynomials occurring in (11) satisfiesfor any multi-index such that and .
In particular, is a Cauchy sequence for any multi-index such that .
Proof. Using the Markov inequality, we get for any such that . Therefore, if satisfies , one gets which is the desired result.
Lemma 7. If with and is a sequence of polynomials satisfying inequality (11), then, for any multi-index such that , the limitis independent of the chosen sequence .
Proof. If is another sequence of polynomials satisfying inequality (11), one getsOne has, using the Markov inequality, as , which ends the proof.
For such functions, we can introduce the notion of Peano derivative (see, e.g., [13, 19] for more information).
Definition 8. Under the hypothesis of Lemma 7, the th Peano derivative of at is .
We can now obtain the result concerning the independence of the polynomials.
Theorem 9. If , then if and only if there exist and a unique polynomial such thatfor any sufficiently large.
Proof. Let be a sequence of polynomials for which inequality (11) is satisfied and setOne hasSince Lemma 6 impliesfor any sufficiently large, we haveThis inequality can be used to obtainwhich shows the existence of .
If two polynomials satisfy inequality (20), thenbut if ,for any sufficiently large, so that does not tend to zero.
The polynomial in inequality (20) is the Taylor expansion of , where the derivative is replaced with the Peano derivative.
The spaces are a generalization of the usual Hölder spaces, defined by (2).
Remark 10. Let ; the sequence is an admissible sequence with , . Therefore, if is not a natural number, we haveIt is easy to check that the polynomial satisfying (2) is unique if and only if . If , one rather imposes in order to obtain the uniqueness of the polynomial (one easily verifies that both definitions lead to the same spaces), so that , with . We will use the modified version in the sequel, ensuring the uniqueness of the polynomial.
The following proposition rigorously expresses the idea that the space associated with a sequence that decreases faster than is included in the usual Hölder space .
Corollary 11. If , one has .
Proof. Let , be defined as in Theorem 9, that is,and let us setOne gets since tends to zero.
3. Alternative Definitions of Generalized Hölder Spaces
Since the uniform spaces can be defined via finite differences or convolutions, one can wonder if such characterizations also hold for the pointwise version of these spaces.
3.1. Characterization in Terms of Finite Differences
As usual, will stand for the finite difference of order : given a function defined on and ,for any . We also setwhere is the line segment with end points and . In order to obtain a more general result, we drop the continuity condition of Definition 2 in this section.
Proposition 12. Let ; one has if and only if there exist such thatfor any .
Proof. The theorem of Whitney (see, e.g., [20]) directly implies that if satisfies inequality (34), then : one hasLet us now suppose that and let , . One has, using the Fréchet functional equation, Now, since there exists a polynomial such thatfor any sufficiently large, one getsfor any sufficiently large.
3.2. Characterization in Terms of Convolutions
Let us denote the space of the infinitely differentiable functions with compact support included in a subset of by . In this section, will denote a radial function of such that for any and . Moreover, one sets , for any .
In [1], the following result has been obtained.
Lemma 13. Let ; if satisfiesfor , then, for any multi-index such that , one hasfor .
Using the same ideas as in [1], one gets a similar characterization.
Theorem 14. If , then there exists a function such thatfor any sufficiently large.
Conversely, if , if is uniformly Hölder and satisfies inequality (41) for a function , then for any such that .
Proof. Assume . As in [1] (see also [3]), let us setwhere is large enough (larger than ) and . Using the same arguments as in [1], one getswhich, due to Proposition 12, leads to inequality (41).
Let us show the converse. Let be such that and set, as in [1],for . Since is uniformly Hölder, is uniformly equal to on anduniformly on , for any . For , let , , and be such that , , and . One has where the second term in the right-hand side only appears if .
Using Lemma 13 and the fact that , the mean value theorem allows writing Moreover,Finally, One then has as wanted, in view of Proposition 12.
4. Generalized Pointwise Hölder Spaces and Wavelets
The usual Hölder spaces can “nearly” be characterized in terms of wavelets [12]: for the sufficiency of the condition, the function has to be uniformly Hölder and a logarithmic correction appears. We show here that such a result still holds in the generalized case.
4.1. Definitions
Let us briefly recall some definitions and notations (for more precisions, see, e.g., [21–23]). Under some general assumptions, there exist a real-valued function and real-valued functions defined on , called wavelets, such thatform an orthogonal basis of . Any function can be decomposed as follows:whereLet us remark that we do not choose the normalization for the wavelets, but rather an normalization, which is better fitted to the study of the Hölderian regularity. Hereafter, the wavelets are always supposed to belong to with , and the functions , are assumed to have fast decay (where is a sufficiently large number, i.e., strictly greater than ).
A dyadic cube of scale is a cube of the formwhere . From now on, wavelets and wavelet coefficients will be indexed with dyadic cubes . Since takes values, we can assume that it takes values in ; we will use the following notations: (i),(ii),(iii).
The pointwise Hölderian regularity of a function is closely related to the decay rate of its wavelet leaders.
Definition 15. The wavelet leaders are defined byTwo dyadic cubes and are adjacent if they are at the same scale and if . We denote by the set of the dyadic cubes adjacent to and by the dyadic cube of side length containing ; we then set
4.2. Result
From now on, we will suppose that the wavelets are compactly supported (such wavelets are constructed in [24]) and will stand for a natural number such that the support of is included in , for any .
Theorem 16. If , then there exist and such thatfor any .
Conversely, let be a uniformly Hölder function; if inequality (57) is satisfied for an admissible sequence that tends to zero, then , where is the admissible sequence defined by and is any number satisfying .
Proof. In what follows, is according to (11). If , let be such that . For and , one has which is the desired result.
Now, let us suppose that inequality (57) is satisfied for a function and an admissible sequence tending to 0. Let us setfor . In [2], it has been shown that these functions have the same regularity as the wavelets and that is uniformly equal to . Let us defineand let us choose such that and () impliesLet us also choose such that any ball (, ) is included in a dyadic cube of length . If is such that for any , we finally choose such that . One hasLet us look at the first term of the right-hand side. Let ; using the Taylor expansion, one gets If satisfies , we have, for any , Each coefficient in the last sum is such that . Therefore, if , thenOtherwise, since is uniformly Hölder, . Therefore,for any , which implies For the second term in the right-hand side of (62), let us define as the number such that and decompose the sum as follows:We have Now, for and , one hasIf , the wavelet coefficients in the last sum are such thatand thereforeIn the other case,and thusThese inequalities lead toPutting all these inequalities together, one getsas desired.
The converse part of the previous theorem requires a uniform regularity condition. As shown in [25], a stronger condition than continuity is necessary in the usual case (see also [26], where similar results are obtained (in the usual case) with a Besov regularity assumption). Similarly, the logarithmic correction is best possible in the usual case [25].
5. A Generalized Definition of the Hölder Exponent
The usual Hölder spaces are embedded: implies . A notion of regularity for a function at can thus be given by the so-called Hölder exponent:To do so in the generalized case, one needs some conditions under which .
This generalized exponent naturally leads to the definition of an alternative multifractal formalism, yet similar to the one developed in [12], where, for example, logarithmic corrections can appear.
5.1. A Trivial Illustration
The classical version of Theorem 16 theoretically allows estimating the Hölder regularity at a given point by looking at the behavior of versus the scales [12]. This notion of regularity is given by the Hölder exponent , defined by equality (77). Following the standard wavelet characterization [12], one should haveand a log-log plot can be used to estimate the slope . This method is simpler than directly fitting the curvewith to the function , since in this latter case one has to estimate two parameters ( and ) to retain only one (namely, , which is the estimation of ). However, this approach allows fine-tuning the computation of using Theorem 16. In the case of a Brownian motion, for example, having the law of the iterated logarithm in mind [14, 15], one should rather choosein the definition of . Fitting with different definitions of should help to discern between specific models. In the case proposed here, it could support the detection of the presence of a logarithmic correction, which could be the signature of a Brownian motion.
As an illustration, the wavelet leader of a Brownian motion for some (the middle was arbitrarily chosen) is represented in Figure 1. When trying to fit the curve to using the Levenberg-Marquardt algorithm [27], one gets (see Figure 1). The same computation with the logarithmic correction (using Definition (80)) gives , which is closer to the expected value . Computing the distance, for each point of the signal ( points), between the estimated Hölder exponent and the expected value gives rise to the boxplot represented in Figure 2.
Of course, this application is only a simple illustrative example; more sophisticated work is required to show the effectiveness of this approach and is out of the scope of this work (see [28]). From a numerical point of view, the estimation of is unstable (in particular, the detection of fast and slow points in the Brownian motion seems to be a difficult task, if not insufferable); this is why one rather tries to estimate the size of the set of points sharing the same Hölder exponent. This is the purpose of the multifractal formalism for functions (see, e.g., [12]), which will not be recalled here.
5.2. Preliminary Results
We first need some technical easy results. From now on, if , will stand for the sequence of polynomials of corresponding to the definition. We will write
Lemma 17. Let ; one has
Proof. Using the Markov inequality (13), we get for any such that and sufficiently large. Therefore, we have for any .
Now, let be a multi-index such that ; inequality (82) follows fromwhile inequality (83) can be obtained in the same way, usingvalid for any .
Corollary 18. Let ; one has the following inequalities: (i)if ,(ii)if ,(iii)if ,
Corollary 19. Let ; one has the following inequalities: (i)if ,(ii)if ,(iii)if ,
5.3. Definitions
Before introducing the definition of a generalized Hölder exponent, we must first consider embedded spaces of type . Once the definitions are given, we provide sufficient conditions for generalized Hölder spaces to be embedded.
Definition 20. If, for any , is an admissible sequence, the applicationis called a family of admissible sequences.
Definition 21. Let ; a family of admissible sequences is decreasing for if implies .
Definition 22. Let be a decreasing family of admissible sequences for ; if , then the Hölder exponent of at for the family is given by
The following proposition is a simple corollary of the results obtained in the previous section; it helps to check if a family of admissible sequences is decreasing. If is a family of admissible sequences, we set
Proposition 23. Let be a family of admissible sequences and ; is decreasing for if it satisfies the following two conditions: (1)if , with , then there exist such that for any ,(2)for any , at least one of the following two conditions is satisfied:(a)there exists such that, for any , there exist for which and(i)if , then ,(ii)if , then ,(iii)if , then , for any ,(b)there exists such that, for any , there exist for which and(i)if , then ,(ii)if , then ,(iii)if , then , for any .
This result is similar to the one obtained in [1] (under the hypothesis of Proposition 23, one gets a decreasing family of admissible sequences for the uniform case), but the proof given for these generalized uniform Hölder spaces cannot be adapted for the pointwise case.
Remark 24. Since the family of admissible sequences defined by satisfies the conditions of Proposition 23, this result implies the classical inclusion , for any , satisfying .
Let us consider the following example to grasp the practicality of Proposition 23.
Example 25. Let be a positive function and setOne directly checks that condition (2)(b) is satisfied, so that is a decreasing family of admissible sequences.
Both conditions can be used to obtain decreasing families of admissible sequences; here is an example where the first condition is used.
Example 26. LetIf is defined using identity (98), condition (2)(a) is satisfied so that this sequence is also a decreasing family of admissible sequences.
Data Availability
No data were used to support this study.
Disclosure
Some of the results of this manuscript were presented in the Belgium Congress of Mathematics.
Conflicts of Interest
The authors declare that they have no conflicts of interest.