1 Introduction

The detection of the local roughness of mathematical objects, such as functions, measures, stochastic processes or fractal sets, has gained a relevant place in Mathematics over the past decades. One key goal is to identify the most representative singularities of the object.

Singularities of functions or signals are points at which the function lacks regularity. The identification and characterization of singularities are an important topic in signal processing, since they contain significant information about the phenomena.

There are several types of singularities that can be illustrated in known examples, e.g. the function \(f(x)=|x-x_{0}|^{\alpha }\), \( 0< \alpha < 1\), has a cusp type singularity at \(x_0\), which is non oscillating .

In contrast, the functions \( f(x)=|x-x_{0}|^{\alpha } \sin \left( |x-x_{0}|^{-\beta } \right) \), with \( 0< \alpha < 1\) and \(\beta > 0 \), and \( f(x)=|x-x_{0}| \sin \left( |x-x_{0}|^{-1} \right) +|x-x_{0}|^{3/2} \) have an oscillatory behaviour around \( x_0 \).

Oscillating and non oscillating are a first and clear distinction among singularities. However, the intuitive notion of oscillation is not enough to characterize more complex structures. For example, in the examples above though both are oscillating the first one has a chirp type singularity whereas the second one not.

From a mathematical point of view, it is important to characterize the different singularities. Classical functions such as the Weierstrass function [8], Riemann functions and other examples [15, 17] present cusp or oscillating type singularities at almost every point.

Likewise, the relevance of characterizing singularities is also important in applications, because it is fundamental to accurately describe natural and social phenomena. In particular the detection of oscillatory behaviors is a challenge in many scientific areas. In fact, many signals from natural and social phenomena (EEG, ECG, data from financial markets, among others) usually present cusp type singularities. But it is also possible to detect local oscillating behaviors in signals from physical and natural phenomena [10]. For example, echolocation waves emitted by bats [19] and hydrodynamic turbulence phenomena [4] present these local oscillating patterns. Recently, it has been confirmed that local oscillating structures appear at gravitational waves, recorded by the laser interferometry gravitational-wave observatories LIGO (located in the USA) and Virgo (located in Europe), during the merger of two black holes in 2015 [2] and from a binary neutron star inspiral in 2017 [3].

The complete characterization of a pointwise singularity requires several parameters. One of the most commonly used parameter in signal processing is the pointwise Hölder exponent. However, the information provided by this exponent is insufficient to distinguish oscillating singularities from non-oscillating ones. To complement this information, other parameters have been proposed: the local Hölder exponent [24], the oscillation and chirp exponents [1, 17] and the weak scaling exponent [22]. Recently, a new quantification of local regularity based on p-exponents, for \(p>0\), has been presented in [18].

Under some global regularity assumptions of the function f, the classical regularity exponents, such as the pointwise Hölder exponent, the local Hölder exponent, the weak scaling exponent and the oscillation and chirp exponents can all be extracted from a concave downward curve in \(\mathbb {R}^2\): the 2-microlocal frontier at \(x_0\) (for more details see [22] and [21]). This curve is defined by means of the 2-microlocal spaces \({C}_{x_{0}}^{s,s'}\), with the parameters \(s,s'\in \mathbb {R}\).

2-microlocal spaces were introduced by Bony [7] to examine the propagations of singularities for semi-linear hyperbolic equations. They are defined as functional spaces embedded in the space of tempered distributions \(\mathscr {S}'(\mathbb {R}) \) and their fundamental property is that they are stable under the action of differential and integral operators [7], that is

$$\begin{aligned} f \in {C}_{x_{0}}^{s,s'} \quad \Longleftrightarrow \quad f^{(n)} \in {C}_{x_{0}}^{s-n,s'} \;\;\; \forall n \in \mathbb {N}. \end{aligned}$$

The original definition of \({C}_{x_{0}}^{s,s'}\) is associated to conditions on Littlewood-Paley decompositions of tempered distributions. In [13], S. Jaffard reformulates these conditions by means of the wavelet transform, providing another characterization of the 2-microlocal spaces of J.M Bony.

In this work we will use the wavelet approach to define these spaces. Recall that a function \(\psi \in L^2(\mathbb {R})\) is called a mother wavelet if \(\{\psi _{j,k}= 2^{j/2} \psi (2^{j} x - k)\}_{j,k\in \mathbb {Z}}\) forms an orthonormal basis of \(L^2(\mathbb {R})\). In that case the wavelet coefficients of \(f\in L^2(\mathbb {R})\) are

$$\begin{aligned} c_{j,k} = 2^{j/2}\langle f,\psi _{j,k}\rangle \;\; \text { with } \; \; \psi _{j,k}= 2^{j/2} \psi (2^{j} x - k). \end{aligned}$$
(1)

To extend the wavelet expansion to functions or distributions that belong to the space of tempered distributions we will require that the mother wavelet \(\psi \) belongs to the Schwartz space of functions \( \mathscr {S} (\mathbb {R}) \). In fact it is not necessary to have \(\psi \in \mathscr {S} (\mathbb {R}) \) when only looking at the local behaviour of f. It would be enough to require that \(\psi \) has sufficient vanishing moments and its first derivatives are of fast decay [22]. Moreover, when analysing the local behaviour of f at \( x_0\) it is not necessary that f is even defined at infinity.

Meyer and Jaffard [17, 22] define the local \( {C}_{x_{0}}^ {s, s '} \) spaces, embedded in the space of distributions \( \mathscr {D}'(V) \), where V is an open set containing \(x_0\). They state their wavelet characterization, which is summarized in the following equivalence. We will use it as definition.

Definition 1.1

Let \(\psi \) be a mother wavelet in the Schwartz space of functions \( \mathscr {S} (\mathbb {R}) \). \(f\in {C}_{x_{0}}^{s,s'}\) if and only if there exists \(C > 0\) such that

$$\begin{aligned} |c_{j,k}| \le C 2^{-j s}\;(1 + | k - 2^{j} x_{0} | )^{-s'} \end{aligned}$$
(2)

for all j and \(k \in {\mathbb Z}\) such that \(j\ge 0\) and \(\left| \frac{k}{2^j} - x_0 \right| < 1\)

Remark 1.1

If the mother wavelet \(\psi \) satisfies that \(\psi \) has N vanishing moments, and that its first r derivatives have fast decay, then the previous definition is valid if \( s, s' \) verify:

$$\begin{aligned} r+s+\inf \ \left\{ s',1\right\}>0 \text { and } N>\sup \left\{ s,s+s'\right\} . \end{aligned}$$
(3)

In this work we will consider \(\psi \) to be in the Schwartz class for simplicity, for example the Meyer wavelet which has all its vanishing moments and all its derivatives of fast decay. Consequently, the conditions required in (3) are satisfied for all pairs \( (s, s') \).

It should be noted that for some \((s,s')\) there also exist characterizations of local 2-microlocal spaces in time domain [9, 20, 21, 25]. Further, the notion of 2-microlocal regularity has been extended recently to the stochastic setting [5, 6, 12].

In order to give a geometric description of the singular behaviour of a function, in [22] Y. Meyer defines a convex set \(D(f,x_0)\in \mathbb {R}^2\) which is called the 2-microlocal domain of f at \( x_0 \):

$$\begin{aligned} D(f,x_0)= \{(s,s'): f\in {C}_{x_{0}}^{s,s'}\}. \end{aligned}$$
(4)

Definition 1.2

The boundary of the set \( D (f, x_0) \) is the 2-microlocal frontier of f at \( x_{0} \), which in the \((s,s')\) plane can be defined by the parametrization

$$\begin{aligned} s'\longmapsto \sup \{s : f \in {C}_{x_{0}}^{s,s'} \}. \end{aligned}$$

Denoting by \( \sigma = s + s' \), the 2-microlocal frontier of f at \( x_{0} \) is the concave downward and decreasing curve, in the \((\sigma , s) \) plane:

$$\begin{aligned} S(\sigma )= & {} \sup \{s : f \in {C}_{x_{0}}^{s,\sigma -s} \}. \end{aligned}$$
(5)

Inspired by [11] we use the \((\sigma , s)\) plane instead of the \((s,s')\) or the \((s', \sigma )\) plane, but any of the other planes could have been chosen.

Under global regularity conditions on f, all the classical regularity exponents can be captured from the curve \( {S} (\sigma ) \), obtaining a complete description of the local regularity. For example, f satisfies a uniform regularity condition if f belongs to the Hölder \(C^{\alpha }(\mathbb {R})\) spaces, i.e., f is a bounded function and for \(\alpha \in (0,1)\) there exists C such that for all x, y:

\(\left| f(x)-f(y)\right| <C\left| x-y\right| ^\alpha \)

or, if \(\alpha >1\), \(\alpha \notin \mathbb {N}\), there exists all derivatives of f of order less than \(\alpha \) and C such that for all x, y:

\(\left| f^{(\left[ \alpha \right] )}(x)-f^{(\left[ \alpha \right] )}(y)\right| <C\left| x-y\right| ^{\alpha -\left[ \alpha \right] }.\)

Then, if \(S(0)>0\), the pointwise Hölder exponent at \(x_0\) is S(0); the local Hölder exponent at \(x_0\) is \( \sigma \) such that \( S (\sigma ) = \sigma \); the chirp exponent at \(x_0\) is the opposite of the slope of the asymptote to the left of \( S (\sigma ) \), the oscillation exponent is the opposite of the slope of the tangent line to the left of \( S (\sigma ) \) at (0, S(0)); and the weak scaling exponent at \(x_0\) is \(\displaystyle {\lim _{\sigma \rightarrow -\infty } S(\sigma )}\).

It is therefore relevant to design prototype functions with a predetermined singularity structure. In this sense, in [8, 14, 24], using different methods, several functions with prescribed pointwise Hölder exponent are constructed. Also, in [24], given a non-negative lower semi-continuous function \(\alpha _l (x)\), the authors construct a function which has \(\alpha _l (x)\) as its local Hölder exponent at all \( x\in \mathbb {R}\). Moreover, in [16] both, the pointwise Hölder and the chirp exponents, are prescribed. More precisely, given h(x) and \( \beta _c (x) \) bounded non-negative functions defined on [0, 1] , which are lower limits of continuous functions, the author provides a function f such that h(x) and \( \beta _c (x) \) are its respective pointwise Hölder and chirp exponents at \( x \in [0,1]\setminus E \), for E a set of measure 0.

It is therefore natural to ask whether, given a decreasing and concave downwards curve \( S (\sigma ) \), we can construct a function or distribution f such that the 2-microlocal frontier of f at \( x_0 \) is \( S (\sigma ) \). In [11, 21, 22] the authors address this question and in each work construct, using the wavelet coefficients of f, a unique function or distribution f having the predetermined 2-microlocal frontier at \(x_0\). These distributions or functions are different in each of the three cited articles. Further, in [21], the authors extend their result to find a function having prescribed 2-microlocal frontiers at a countable dense set of points.

In this work we present a generic formula that provides a wide class of functions or distributions that have \(S(\sigma )\) as the 2-microlocal frontier at \(x_0\), where \( S (\sigma ) \) is a decreasing function defined on \(\mathbb {R}\), such that either \( S (\sigma ) \) is concave downwards with \( S'' (\sigma ) <0 \) or \( S (\sigma ) \) is a line, obtaining the constructions built in [11, 22] and [21] as particular cases. For the case that \( S (\sigma ) \) is a line, we are able to completely characterize the functions or distributions such that \( S (\sigma ) \) is the 2-microlocal frontier at \(x_0\).

The functions or distributions with predetermined 2-microlocal frontier at \(x_0\) that we obtain are given through the definition of their wavelet coefficients.

The wavelet coefficients \(c_{j,k}\) of the prototype family of distributions, with prescribed 2-microlocal frontier at \( x_0 \), will satisfy:

$$\begin{aligned} c_{j,k}\le \mathscr {C}_{j,k}\;. \inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS(\sigma )}\left( \frac{1+\left| k-2^j x_0\right| }{\lambda _{j,k}}\right) ^{S (\sigma )-\sigma }\right\} , \end{aligned}$$
(6)

with \(\mathscr {C}_{j,k} \) and \(\lambda _{j,k} \) positive sequences that satisfy some specific conditions (see Theorems 2.1 and 2.2).

Choosing \( \mathscr {C}_{j, k} \) and \(\lambda _{j, k}\) appropriately (see Sect. 2.3), the formulas explicitly built in [11, 22] and [21], satisfy the general formula (6). Therefore the functions proposed in the three articles are members of this prototype family. In particular formula (6) resembles the construction in [11] introducing the sequences \( \mathscr {C}_{j, k} \) and \(\lambda _{j, k}\). The possibility of choosing the coefficients \(c_{j,k}\) from a large set allows to construct functions with some specific properties (see Example 2.1).

Finally, we can extend our result to give a generic formula for a function or distribution to have prescribed 2-microlocal frontiers at a countable dense set.

2 A Generalization of the 2-Microlocal Frontier Prescription at One Point

In this section we state and prove two main results of this manuscript. In Theorem 2.2 we characterize all functions (or distributions) whose 2-microlocal frontier at a given point is a line. Theorem 2.1 yields a wide class of functions (or distributions) whose 2-microlocal frontier at the given point is a given concave downward function \(S(\sigma )\) with \(S''(\sigma )<0\). Each one of the functions constructed in [11, 21, 22] belongs to the family of functions we construct.

We start with some needed preliminary results.

2.1 Preliminary Results

We state two lemmas that will be useful to prove the theorems of this section. Formula (6) is based on the calculation of the infimum of the functions \(a^{S(\sigma )} b^{ S(\sigma )-\sigma },\) defined on \(\mathbb {R}\), with fixed \(a,b>0\).

If S is the line \(S(\sigma )= M\sigma +d\) with \(M\le 0\), it is easy to prove that

$$\begin{aligned} \inf _{\sigma \in \mathbb {R}}\{ a^{S(\sigma )} b^{S(\sigma )-\sigma } \}= \left\{ \begin{array}{lll} 0&{}\quad \text {if} &{} \quad {(ab)}^{M} \ne b\\ \\ (ab)^d &{}\quad \text {if} &{} \quad {(ab)}^{M} = b, \\ \end{array} \right. \end{aligned}$$
(7)

We will focus on the case \(0<a\le 1\), since we are interested in the case \(a=2^{-j}\) with \(j\ge 0\). We then have the following lemma.

Lemma 2.1

(Lemma 4.2.10, [23]) Let \( S (\sigma ) \) be a decreasing function defined on \(\mathbb {R}\), such that either \( S (\sigma ) \) is concave downwards with \( S'' (\sigma ) <0 \) or \( S (\sigma ) \) is a line. Let \(0<a\le 1\) and \(b>0\). Then

  • For \(a=b=1\):

    $$\begin{aligned} \inf _{\sigma \in \mathbb {R}}\{ a^{S(\sigma )} b^{S(\sigma )-\sigma } \}=1.\end{aligned}$$

  • For \(1\le b<\frac{1}{a}\):

    \(\inf _{\sigma \in \mathbb {R}}\{ a^{S(\sigma )} b^{S(\sigma )-\sigma } \}=\left\{ \begin{array}{lll} a^{S(\sigma _1)} b^{S(\sigma _1)-\sigma _1} &{}\quad \text {if}&{}\quad \exists \, \sigma _1 :(ab)^{ S'(\sigma _1)}=b \\ \\ \displaystyle {\lim _{\sigma \rightarrow +\infty } a^{S(\sigma )} b^{S(\sigma )-\sigma }}&{}\quad \text {if}&{}\quad (ab)^{ S'(\sigma )}<b \,\forall \sigma \\ \\ \displaystyle {\lim _{\sigma \rightarrow -\infty } a^{S(\sigma )} b^{S(\sigma )-\sigma }}&{}\quad \text {if}&{}\quad (ab)^{ S'(\sigma )}>b \,\forall \sigma .\\ \end{array} \right. \).

  • For any other a and b \(\inf _{\sigma \in \mathbb {R}}\{ a^{S(\sigma )} b^{S(\sigma )-\sigma } \}=0.\)

Proof

The case when S is a line, is given in (7). If S is not a line, the proof will be obtained by analysing whether the function \(h(\sigma )=a^{S(\sigma )} b^{S(\sigma )-\sigma }\) has an absolute minimum, or is asymptotically bounded below by a horizontal line. The proof uses that \(S''(\sigma ) < 0\) and therefore there exists a unique \(\sigma _1\) such that \((ab)^{ S'(\sigma _1)}=b\). (For more details, see Lemma 4.2.10, [23].) \(\square \)

Remark 2.1

Lemma 2.1 can be extended to the case \(a>1\) if \( S (\sigma ) \) is a decreasing and concave downwards function, with \( S'' (\sigma ) <0 \).

Lemma 2.2

(Lemma 4.3.2, [23]) Let \( S (\sigma ) \) be a decreasing function defined on \(\mathbb {R}\), such that either \( S (\sigma ) \) is concave downwards with \( S'' (\sigma ) <0 \) or \( S (\sigma ) \) is a line. Let \(\varepsilon >0\) and \(s_0=S(\sigma _0)\). Then, there exists \(B<0\) such that

$$\begin{aligned} \frac{S'(\sigma )(\sigma _0-\sigma )+S(\sigma )-s_0+\varepsilon }{S'(\sigma )-1}<B \quad \quad \forall \sigma \in \mathbb {R}. \end{aligned}$$
(8)

Proof

By the hypothesis on \(S(\sigma )\) we have that the tangents to the graph of \(S(\sigma )\) majorize the graph of \(S(\sigma )\). So, if \(S'(\sigma )\) is bounded, the inequality is immediate. If not, using L’Hospital one sees that

$$\begin{aligned} \lim _{\sigma \rightarrow +\infty }\frac{S'(\sigma )(\sigma _0-\sigma )+S(\sigma )-s_0+\varepsilon }{S'(\sigma )-1} = -\infty , \end{aligned}$$

and so we obtain the desired result. \(\square \)

2.2 Main Results

Inspired by the construction in [22] we write \(\mathbb {N}\) as an infinite union of disjoint infinite sets of integers:

$$\begin{aligned} \mathbb {N}=\; \text { {\ \textit{d}}} \bigcup _{m\in \mathbb {N}} {\Lambda }_m, \end{aligned}$$
(9)

with \({\Lambda }_m\) an infinite set of natural numbers, e.g. \({\Lambda }_m\) could be the set of natural numbers such that their binary decomposition has exactly m ones.

We define \(\left\{ r_m\right\} _{m\in \mathbb {N}}\) to be a sequence such that:

$$\begin{aligned} {\left\{ r_m\right\} _{m\in \mathbb {N}}} \text { is a dense subset of } \mathrm{{Im}} \left( \frac{S'(\sigma )}{S'(\sigma )-1}\right) \subseteq [0,1]. \end{aligned}$$
(10)

The following set of subscripts will play an important role in Theorem 2.1.

$$\begin{aligned} I = \{(j,k): j\in \Lambda _m \text { and } \left[ \left| k-2^jx_0\right| \right] =[2^{jr_m}], m\in \mathbb {N}\}, \end{aligned}$$
(11)

where \(\left[ x\right] \) denotes the largest integer smaller or equal than x.

If \(S(\sigma )\) is not linear we obtain only sufficient conditions for the prescription on a single point.

Theorem 2.1

Let \( S (\sigma ) \) be a decreasing concave downwards function defined on \(\mathbb {R}\) with \(S'' (\sigma ) <0\). Let \(\mathscr {C}_{j,k} \) and \(\lambda _{j,k}\) be positive sequences such that for any sequence \((k_j)_j\) such that \( |k_j-2^j\;x_0|<2^j\) they verify

  1. (i)

    For any \(C\in \mathbb {R}\),

    $$\begin{aligned} \displaystyle {\varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}{\left( \frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j} +C \; \frac{\log _2\left( \lambda _{j,k_j}\right) }{j}\right) }\;\le 0}. \end{aligned}$$
  2. (ii)

    For \((j,k_j)\in I\) given by (11),

    $$\begin{aligned} \displaystyle {\lim _{\begin{array}{c} j\rightarrow +\infty \end{array}}\frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j}= 0} \text { and } \displaystyle {\lim _{\begin{array}{c} j\rightarrow +\infty \end{array}}\frac{\log _2\left( \lambda _{j,k_j}\right) }{j}= 0}. \end{aligned}$$

Let the coefficients \(c_{j,k}\) be such that

$$\begin{aligned} \left| c_{j,k}\right| \le \mathscr {C}_{j,k}\;. \displaystyle {\inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS(\sigma )}\left( \frac{1+\left| k-2^jx_0\right| }{\lambda _{j,k}}\right) ^{S (\sigma )-\sigma }\right\} }, \end{aligned}$$
(12)

and if \((j,k)\in I\) we require equality.

If \(\psi \) is any wavelet in the Schwartz class with infinitely vanishing moments then the function (or the distribution) f defined by its wavelet expansion as

$$\begin{aligned} f(x)=\sum _{j\ge 0}\;\sum _{\begin{array}{c} k\in \mathbb {Z},\; |k-2^jx_0|<2^j \end{array}}~~c_{j,k} \; \psi (2^{j}x-k) \end{aligned}$$

has \(S(\sigma )\) as its 2-microlocal frontier at \(x_0\).

Proof

Without loss of generality we consider \(x_0=0\). The set of subscripts, in formula (11), is then

$$\begin{aligned} I= \{(j,k): j\in \Lambda _m \text { and } |k|=[2^{jr_m}], m\in \mathbb {N}\}. \end{aligned}$$

Let \(j\ge 0 \). For simplicity we consider the equality

$$\begin{aligned} \left| c_{j,k}\right| = \mathscr {C}_{j,k}\;. \inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS(\sigma )}\left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S (\sigma )-\sigma }\right\} , \end{aligned}$$

for \(j\ge 0 \) and \( |k|<2^j\), although it will be clear, in the proof, that we only need the equality for \((j,k) \in I\), and the inequality if \((j,k)\notin I\).

By using Lemma 2.1 for \(a=2^{-j}\) and \(b=\frac{1+\left| k\right| }{\lambda _{j,k}}\), we compute the wavelet coefficient as:

  • For \( 2^{-j}=1\) and \(\frac{1+\left| k\right| }{\lambda _{j,k}}=1\). \(\left| c_{j,k} \right| = \mathscr {C}_{j,k}= \mathscr {C}_{0,0}\).

  • For \(1\le \frac{1+\left| k\right| }{\lambda _{j,k}}< 2^j\):

$$\begin{aligned} \left| c_{j,k}\right| =\left\{ \begin{array}{lll} \mathscr {C}_{j,k} \; \left( 2^{-j}\right) ^{S(\sigma _{j,k})} \left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S(\sigma _{j,k})-\sigma _{j,k}} &{}\quad \text {if}&{} \quad \exists \,\sigma _{j,k}: \left( 2^{-j}\frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{ S'(\sigma _{j,k})}=\frac{1+\left| k\right| }{\lambda _{j,k}}\\ \\ \\ \mathscr {C}_{j,k} \;\displaystyle {\lim _{\sigma \rightarrow +\infty } (2^{-j})^{S(\sigma )} \left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S(\sigma )-\sigma }} &{}\quad \text { if }&{}\quad \left( 2^{-j}\frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{ S'(\sigma )}<\frac{1+\left| k\right| }{\lambda _{j,k}} \, \forall \sigma \\ \\ \\ \mathscr {C}_{j,k} \;\displaystyle {\lim _{\sigma \rightarrow -\infty } (2^{-j})^{S(\sigma )} \left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S(\sigma )-\sigma }} &{}\quad \text { if }&{} \quad \left( 2^{-j}\frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{ S'(\sigma )}>\frac{1+\left| k\right| }{\lambda _{j,k}} \,\forall \sigma . \end{array} \right. \end{aligned}$$
(13)

  • For any other j, k:

    $$\begin{aligned}\left| c_{j,k}\right| =0.\end{aligned}$$

Let \((\sigma _0, s_0)\) be a point on the graph of \(S(\sigma )\). Our purpose is to prove

$$\begin{aligned} \sup \{s : f \in { C}_{0}^{s,\sigma _0-s}\}=s_0, \end{aligned}$$

that is to prove that A) \(f\notin { C}_{0}^{s_0+\varepsilon , \sigma _0-(s_0+\varepsilon )}\) and B) \(f\in { C}_{0}^{s_0-\varepsilon , \sigma _0-(s_0-\varepsilon )}\), for all \(\varepsilon >0\).

A) Let us show that \(f\notin { C}_{0}^{s_0+\varepsilon , \sigma _0-(s_0+\varepsilon )}\) for all \(\varepsilon >0\):

Let us assume on the contrary that there exists \(\varepsilon >0\) such that \(f\in { C}_{0}^{s_0+\varepsilon , \sigma _0-(s_0+\varepsilon )}\) which means that there exists a constant \(C > 0\) such that

$$\begin{aligned} |c_{j,k}| \le C 2^{-j (s_0 +\varepsilon )}\;(1 + | k | )^{s_0 +\varepsilon -\sigma _0} \end{aligned}$$
(14)

for all j and \(k \in {\mathbb Z}\): \( | k| < 2^j\), and let us show that it is a contradiction.

We will prove that for a given \((\sigma _0, s_0)\) it is possible to construct a sequence \((j_n, k_n)\), with \(j_n\) strictly increasing, such that

$$\begin{aligned} (j_n, k_n)\in I = \{(j,k_j): j\in \Lambda _m \text { and } |k_j|=[2^{jr_m}], m\in \mathbb {N}\}, \end{aligned}$$

and verifies that there exists \(\sigma _n\) such that

$$\begin{aligned} \left( 2^{-j_n}\frac{1+\left| k_n\right| }{\lambda _{j_n,k_n}}\right) ^{S'(\sigma _n)}=\frac{1+\left| k_n\right| }{\lambda _{j_n,k_n}} \;\text { and }\; \lim _{n \rightarrow +\infty } \sigma _n=\sigma _0. \end{aligned}$$
(15)

We construct the sequence \( (j_n, k_n) \) in the following way. Since \(S'(\sigma _0)<0\), \(\frac{S'(\sigma _0)}{S'(\sigma _0)-1} \in \left[ 0,1\right) \). Therefore, there exists a subsequence \((r_{m_n})_{n\in \mathbb {N}}\) of \(\{r_m\}_{m\in \mathbb {N}}\) which is dense in \(\mathrm{{Im}} \left( \frac{S'(\sigma )}{S'(\sigma )-1}\right) \), such that

$$\begin{aligned} \lim _{n \rightarrow +\infty } {r_{m_n}}= \frac{S'(\sigma _0)}{S'(\sigma _0)-1}. \end{aligned}$$

Since the sets \({\Lambda }_{m_n}\) are infinite we can choose \(j_n\in {\Lambda }_{m_n}\) a strictly increasing sequence and \(k_n=k_{j_n}=[2^{j_n r_{m_{n}}}]\). Hence \((j_n, k_n)\in I\) and

$$\begin{aligned} 2^{j_n r_{m_n}}\le 1+ |k_n| \le 1+2^{j_n r_{m_n}}\le 2 \;2^{j_n r_{m_n}}, \end{aligned}$$

i.e. \(1+ |k_n| = K_n \;2^{j_n r_{m_n}}\) with \(1\le K_n\le 2\).

Therefore, taking into account the hypothesis \(\displaystyle {\lim _{n\rightarrow +\infty }\frac{\log _2\left( \lambda _{j_n,k_n}\right) }{j_n}= 0},\) we obtain

$$\begin{aligned} \lim _{n \rightarrow +\infty } {\frac{\log _2 \left( \frac{1+\left| k_n\right| }{\lambda _{j_n,k_n}} \right) }{\log _2 \left( 2^{-j_n}\;\frac{1+\left| k_n\right| }{\lambda _{j_n,k_n}}\right) }}= & {} \lim _{n \rightarrow +\infty } {\frac{\log _2 \left( 1+\left| k_n\right| \right) -\log _2(\lambda _{j_n,k_n}) }{-j_n+\log _2 \left( 1+\left| k_n\right| \right) -\log _2(\lambda _{j_n,k_n}) }} \\= & {} \lim _{n \rightarrow +\infty } {\frac{j_n\left( \frac{K_n}{j_n}+ r_{m_n}-\frac{\log _2\left( \lambda _{j_n,k_n}\right) }{j_n}\right) }{j_n \left( \frac{K_n}{j_n}+ (r_{m_n}-1)-\frac{\log _2\left( \lambda _{j_n,k_n}\right) }{j_n}\right) }} \\ \nonumber= & {} S'(\sigma _0). \end{aligned}$$

Since S is strictly concave downwards, the function \(S':\mathbb {R}\longrightarrow \mathrm{{Im}} (S')\) is strictly decreasing and thus bijective. Then there exists \(\sigma _n \) such that

$$\begin{aligned} \lim _{n \rightarrow +\infty } \sigma _n=\sigma _0 \;\; and\;\;S'(\sigma _n)=\frac{\log _2 \left( \frac{1+\left| k_n \right| }{\lambda _{j_n,k_n}} \right) }{\log _2 \left( 2^{-j_n}\;\frac{1+\left| k_n\right| }{\lambda _{j_n,k_n}}\right) } \end{aligned}$$

for \(n\ge n_0\), \(n_0\in \mathbb {N}\), which is equivalent to (15).

Furthermore, we can take \(n_0\) such that, for all \(n\ge n_0\),

$$\begin{aligned} 1\le \frac{1+\left| k_n\right| }{\lambda _{j_n,k_n}}\le 2^{j_n}, \end{aligned}$$

and thus taking \(\log _2(\cdot )\) and dividing by \(j_n\), we have

$$\begin{aligned} 0 \le \lim _{n \rightarrow +\infty } {\frac{\log _2 \left( \frac{1+\left| k_n \right| }{\lambda _{j_n,k_n}} \right) }{j_n}}= \lim _{n \rightarrow +\infty }{\left( \frac{K_n}{j_n}+ r_{m_n}-\frac{\log _2\left( \lambda _{j_n,k_n}\right) }{j_n}\right) }=\frac{S'(\sigma _0)}{S'(\sigma _0)-1}<1. \end{aligned}$$

In short, the first equality in the formula (13) holds with \(\sigma _{j_n,k_n}=\sigma _n\) and therefore

$$\begin{aligned} |c_{j_n,k_n}|= \mathscr {C}_{j_n,k_n} \; \left( 2^{-j_n}\right) ^{S(\sigma _n)} \left( \frac{1+\left| k_n\right| }{\lambda _{j_n,k_n}}\right) ^{S(\sigma _n)-\sigma _n}. \end{aligned}$$

In consequence, the inequality in formula (14) can be reformulated, for \(n\ge n_0\), \(j=j_n\) and \(k=k_n\), as

$$\begin{aligned} \mathscr {C}_{j_n,k_n} \; \left( 2^{-j_n}\right) ^{S(\sigma _n)} \left( \frac{1+\left| k_n\right| }{\lambda _{j_n,k_n}}\right) ^{S(\sigma _n)-\sigma _n} \le C 2^{-j_n (s_0 +\varepsilon )}\;(1 + | k_n| )^{s_0 +\varepsilon -\sigma _0}. \end{aligned}$$

Or equivalently,

$$\begin{aligned} \mathscr {C}_{j_n,k_n} \le C 2^{-j_n (s_0-S(\sigma _n))} 2^{-j_n\varepsilon } \; (1 + | k_n| )^{s_0 -\sigma _0-S(\sigma _n)+\sigma _n} \; (1 + | k_n| )^\varepsilon \; {\left( \lambda _{j_n,k_n}\right) }^{S(\sigma _n)-\sigma _n}. \end{aligned}$$

Applying \(\log _2(\cdot )\) and dividing by \(j_n\) we obtain

$$\begin{aligned} \frac{\log _2(\mathscr {C}_{j_n,k_n})}{j_n}&\le \frac{\log _2(C)}{j_n} - \left( s_0-S(\sigma _n)\right) -\;\varepsilon \; \\&\quad + \;\frac{\log _2(1 + | k_n| )}{j_n} \left( s_0 -\sigma _0-S(\sigma _n)+\sigma _n\right) \\&\quad + \; \frac{\log _2(1 + | k_n| )}{j_n}\varepsilon +\frac{ \log _2(\lambda _{j_n,k_n})}{j_n}(S(\sigma _n)-\sigma _n). \nonumber \end{aligned}$$

Recalling that \(j_n\) and \(k_n\) were selected such that \((j_n,k_n)\in I\), that is \(1+|k_n|=K_n 2^{j_n r_{m_n}}\) for \(1\le K_n\le 2\), we obtain

$$\begin{aligned} \frac{\log _2(C_{j_n,k_n})}{j_n}&\le \frac{\log _2(C)}{j_n}-\left( s_0-S(\sigma _n)\right) -\;\varepsilon \nonumber \\&\quad + \left( \frac{\log _2(K_n)}{j_n} +r_{m_n}\right) \left( s_0 -\sigma _0-S(\sigma _n) +\sigma _n\right) \nonumber \\&+\quad \left( \frac{\log _2(K_n)}{j_n} +r_{m_n} \right) \; \varepsilon +\frac{ \log _2(\lambda _{j_n,k_n})}{j_n}(S(\sigma _n)-\sigma _n).\;\; \end{aligned}$$
(16)

Therefore, by hypothesis, \(\mathscr {C}_{j,k_j}\) and \(\lambda _{j,k_j}\) satisfy for \((j_n,k_n)\in I\),

$$\begin{aligned} \displaystyle {\lim _{\begin{array}{c} j\rightarrow +\infty \\ (j,k_j)\in I \end{array}}\frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j}= 0} \text { and } \displaystyle {\lim _{\begin{array}{c} j\rightarrow +\infty \\ (j,k_j)\in I \end{array}}\frac{\log _2\left( \lambda _{j,k_j}\right) }{j}= 0}. \end{aligned}$$

Taking limit for \(n\rightarrow +\infty \) in (16), we obtain

$$\begin{aligned} 0\le \varepsilon \left( \frac{S'(\sigma _0)}{S'(\sigma _0)-1}-1\right) = \frac{\varepsilon }{S'(\sigma _0)-1}, \end{aligned}$$

which is a contradiction because \(\frac{\varepsilon }{S'(\sigma _0)-1}<0\).

B) Let us show that \(f\in { C}_{0}^{(s_0-\varepsilon , \sigma _0-(s_0-\varepsilon ))}\) for all \(\varepsilon >0\):

By Definition 1.1 we need to prove that there exists \(C>0\) such that

$$\begin{aligned} |c_{j,k}| \le C 2^{-j (s_0-\varepsilon )}\;(1 + | k | )^{s_0-\varepsilon -\sigma _0} \quad \forall \; j ,\; k \in {\mathbb Z}:\; j\ge 0,\; | k| < 2^j. \end{aligned}$$
(17)

In fact, it is enough to prove (17) for \(j\ge n_0\), \(n_0\in \mathbb {N}\), since C can be adjusted for a finite set of j. In other words, it will be enough to find \(n_0\) and C such that

$$\begin{aligned} \dfrac{|c_{j,k}|}{ 2^{-j (s_0-\varepsilon )}\;(1 + | k | )^{s_0-\varepsilon -\sigma _0}}\le C \quad \text { for all } j\ge n_0,\; | k| < 2^j. \end{aligned}$$
(18)

We use formula (13) to compute the wavelet coefficients \(c_{j,k}\) for different cases and will show that the boundedness of \(\dfrac{|c_{j,k}|}{ 2^{-j (s_0-\varepsilon )}\;(1 + | k | )^{s_0-\varepsilon -\sigma _0}}\) can be obtained in analogous ways for all cases.

  • For

    $$\begin{aligned} |c_{j,k}|= & {} \mathscr {C}_{j,k} \; \left( 2^{-j}\right) ^{S(\sigma _{j,k})} \left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S(\sigma _{j,k})-\sigma _{j,k}}, \text { where } \sigma _{j,k} \text { is such that }\nonumber \\&\left( 2^{-j}\frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{ S'(\sigma _{j,k})}=\frac{1+\left| k\right| }{\lambda _{j,k}}, \text { that is }\;\; 1+\left| k\right| =\lambda _{j,k}\;2^{j\;\frac{S'(\sigma _{j,k})}{S'(\sigma _{j,k})-1}},\qquad \quad \end{aligned}$$
    (19)

    it is enough to show that, for \(j\ge n_0\) and \( | k|<2^j\),

    $$\begin{aligned} \frac{\mathscr {C}_{j,k} \; \left( 2^{-j}\right) ^{S(\sigma _{j,k})} \left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S(\sigma _{j,k})-\sigma _{j,k}}}{2^{-j (s_0-\varepsilon )}\;(1 + | k| )^{s_0-\varepsilon -\sigma _0}} \text { is bounded.} \end{aligned}$$

    If we replace \(1 + | k|\) by formula (19) and reformulate the last quotient, we have to prove that

    $$\begin{aligned} \mathscr {C}_{j,k} \; 2^{-j(S(\sigma _{j,k})-(s_0-\varepsilon ))} \left( 2^{j \frac{S'(\sigma _{j,k})}{S'(\sigma _{j,k})-1}}\right) ^{S(\sigma _{j,k})-\sigma _{j,k}+\sigma _0-s_0+\varepsilon }\;(\lambda _{j,k})^{\sigma _0-s_0+\varepsilon } \end{aligned}$$
    (20)

    is bounded.

  • The case

    $$\begin{aligned} |c_{j,k}|=\mathscr {C}_{j,k} \;\displaystyle {\lim _{\sigma \rightarrow +\infty } (2^{-j})^{S(\sigma )} \left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S(\sigma )-\sigma }} \end{aligned}$$

    is valid if

    $$\begin{aligned} \left( 2^{-j}\frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S'(\sigma )}<\frac{1+\left| k\right| }{\lambda _{j,k}}\;\; \forall \sigma , \;\;\text { i.e. }\;\; \left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S'(\sigma )-1}<2^{jS'(\sigma )}\;\; \forall \sigma . \end{aligned}$$

    As \(S'(\sigma )-1<0\), this is equivalent to \(1+\left| k\right| > \lambda _{j,k} 2^{j\frac{S'(\sigma )}{S'(\sigma )-1}}\;\; \forall \sigma .\) On the other hand, since \(S(\sigma )- \sigma \longrightarrow -\infty \) when \(\sigma \longrightarrow +\infty \) we can select \(\overline{\sigma }\), sufficiently large, such that

    $$\begin{aligned}S(\overline{\sigma })-\overline{\sigma }-s_0+\sigma _0+\varepsilon <0.\end{aligned}$$

    In particular

    $$\begin{aligned}&\left| c_{j,k}\right| =\mathscr {C}_{j,k}\;. \displaystyle {\inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS(\sigma )}\left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S (\sigma )-\sigma }\right\} }\\&\le \mathscr {C}_{j,k} \; \left( 2^{-j}\right) ^{S(\overline{\sigma })}\left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S(\overline{\sigma })-\overline{\sigma }}. \end{aligned}$$

    Therefore, to prove (18) we only need to show that

    $$\begin{aligned}&\frac{\mathscr {C}_{j,k} \; \left( 2^{-j}\right) ^{S(\overline{\sigma })}\left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S(\overline{\sigma })-\overline{\sigma }} }{2^{-j (s_0-\varepsilon )}\;(1 + | k| )^{s_0-\varepsilon -\sigma _0}} \nonumber \\&\quad = \mathscr {C}_{j,k} \; \left( 2^{-j}\right) ^{S(\overline{\sigma })-(s_0-\varepsilon )} \left( 1+\left| k\right| \right) ^{S(\overline{\sigma })-\overline{\sigma }-s_0+\sigma _0+\varepsilon } {\lambda _{j,k}}^{\overline{\sigma }-S(\overline{\sigma })}, \end{aligned}$$
    (21)

    is bounded for \(j\ge n_0\) and \( | k|<2^j\).

    Since \( 1+\left| k\right| > \lambda _{j,k} 2^{j\frac{S'(\overline{\sigma })}{S'(\overline{\sigma })-1}}\) and \(S(\overline{\sigma })-\overline{\sigma }-s_0+\sigma _0+\varepsilon <0,\) formula (21) is bounded by

    $$\begin{aligned} \mathscr {C}_{j,k} \; 2^{-j(S(\overline{\sigma })-(s_0-\varepsilon ))} \left( 2^{j \frac{S'(\overline{\sigma })}{S'(\overline{\sigma })-1}}\right) ^{S(\overline{\sigma })-\overline{\sigma }+\sigma _0-s_0+\varepsilon }\;(\lambda _{j,k})^{\sigma _0-s_0+\varepsilon }. \end{aligned}$$
    (22)

  • Finally, let

    $$\begin{aligned} |c_{j,k}|=\mathscr {C}_{j,k} \;\displaystyle {\lim _{\sigma \rightarrow -\infty } (2^{-j})^{S(\sigma )} \left( \frac{1+\left| k\right| }{\lambda _{j,k}}\right) ^{S(\sigma )-\sigma }} \text { with } 1+\left| k\right| < \lambda _{j,k} 2^{j\frac{S'(\sigma )}{S'(\sigma )-1}}\;\; \forall \sigma . \end{aligned}$$

    Similarly as before we can select \(\overline{\sigma }\), negatively large, such that

    $$\begin{aligned}&S(\overline{\sigma })-\overline{\sigma }-s_0+\sigma _0+\varepsilon >0, \text { and thus }\\&\quad \left( 1+\left| k\right| \right) ^{S(\overline{\sigma })-\overline{\sigma }-s_0+\sigma _0+\varepsilon }< \left( \lambda _{j,k} 2^{j\frac{S'(\overline{\sigma })}{S'(\overline{\sigma })-1}}\right) ^{S(\overline{\sigma })-\overline{\sigma }-s_0+\sigma _0+\varepsilon }. \end{aligned}$$

    Therefore, \( \dfrac{|c_{j,k}|}{ 2^{-j (s_0-\varepsilon )}\;(1 + | k | )^{s_0-\varepsilon -\sigma _0}}\) is bounded by

    $$\begin{aligned} \mathscr {C}_{j,k} \; 2^{-j(S(\overline{\sigma })-(s_0-\varepsilon ))} \left( 2^{j \frac{S'(\overline{\sigma })}{S'(\overline{\sigma })-1}}\right) ^{S(\overline{\sigma })-\overline{\sigma }+\sigma _0-s_0+\varepsilon }\;(\lambda _{j,k})^{\sigma _0-s_0+\varepsilon }. \end{aligned}$$
    (23)

Therefore, we have to show that (20), (22) and (23) are bounded. Since (22) and (23) are equivalent to (20), with \(\sigma _{j,k}= \overline{\sigma }\) for all j, k because \(\overline{\sigma }\) only depends on \(\sigma _0\), \(s_0\) and \(\varepsilon \), in each case, we have to prove that, for all \(j\ge n_0\) and \( | k|<2^j\), formula (20) or its equivalent

$$\begin{aligned}\mathscr {C}_{j,k} \;\;(\lambda _{j,k})^{(\sigma _0-s_0+\varepsilon )}\; 2^{j\left[ \frac{S'(\sigma _{j,k})(\sigma _0-\sigma _{j,k})+S(\sigma _{j,k})-s_0+\varepsilon }{S'(\sigma _{j,k})-1}\right] }, \text { is bounded.}\end{aligned}$$

Taking \(\log _2(\cdot )\) and factoring out j we obtain

$$\begin{aligned} j\left[ \frac{\log _2(\mathscr {C}_{j,k})}{j} +\frac{\log _2(\lambda _{j,k})}{j}\;(\sigma _0-s_0+\varepsilon )+\frac{S'(\sigma _{j,k})(\sigma _0-\sigma _{j,k})+S(\sigma _{j,k})-s_0+\varepsilon }{S'(\sigma _{j,k})-1}\right] . \end{aligned}$$
(24)

Since \(|k|<2^j\), by hypothesis

$$\begin{aligned} \displaystyle {\varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}{\left( \frac{\log _2\left( \mathscr {C}_{j,k}\right) }{j} +(\sigma _0-s_0+\varepsilon ) \; \frac{\log _2\left( \lambda _{j,k}\right) }{j}\right) }\;\le 0}. \end{aligned}$$

Further, Lemma 2.2 gives

$$\begin{aligned} \frac{S'(\sigma _{j,k})(\sigma _0-\sigma _{j,k})+S(\sigma _{j,k})-s_0+\varepsilon }{S'(\sigma _{j,k})-1}<B <0 \;\;\text { for all } \sigma _{j,k}. \end{aligned}$$

Hence,

$$\begin{aligned} \varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}\log _2\left( \mathscr {C}_{j,k} \;\;(\lambda _{j,k})^{(\sigma _0-s_0+\varepsilon )}\; 2^{j\left[ \frac{S'(\sigma _{j,k})(\sigma _0-\sigma _{j,k})+S(\sigma _{j,k})-s_0+\varepsilon }{S'(\sigma _{j,k})-1}\right] }\right) =-\infty \end{aligned}$$

which implies that there exists \(M>0\) such that

$$\begin{aligned} 0<\mathscr {C}_{j,k} \;(\lambda _{j,k})^{(\sigma _0-s_0+\varepsilon )}\; 2^{j\left[ \frac{S'(\sigma _{j,k})(\sigma _0-\sigma _{j,k})+S(\sigma _{j,k})-s_0+\varepsilon }{S'(\sigma _{j,k})-1}\right] }<M, \end{aligned}$$

for j and \(k \in {\mathbb Z}\) such that, \(j\ge 0, \; | k| < 2^j\). \(\square \)

Remark 2.2

The previous Theorem 2.1 also holds if the domain of S is a half-line, \( Dom(S)=(-\infty , b)\) for \(b\in \mathbb {R}\), instead of the whole line, \(Dom(S)=\mathbb {R}\).

For the case that \(S(\sigma )\) is a line we have a general theorem, giving necessary and sufficient conditions for a function (or distribution) to have that line as its prescribed 2-microlocal frontier at \(x_0\), whereas the previous Theorem only yields sufficient conditions. However, since every smooth curve can be approximated by linear functions, it may be feasible to extend the necessity to any piecewise smooth curve.

Theorem 2.2

Let \(S(\sigma )\) be the line \(S(\sigma )= \alpha + \frac{\gamma }{1-\gamma } (\alpha - \sigma )\), with \(0 \le \gamma < 1\). Let f be the function (or the distribution) defined by its wavelet expansion as

$$\begin{aligned} f(x)=\sum _{j\ge 0}\;\sum _{\begin{array}{c} k\in \mathbb {Z},\; |k-2^jx_0|<2^j \end{array}}~~c_{j,k} \; \psi (2^{j}x-k), \end{aligned}$$

where \(\psi \) is any wavelet in the Schwartz class with infinitely vanishing moments.

The 2-microlocal frontier of f at \(x_0\) is \(S(\sigma )\) if and only if for \(j\ge 0\) and \(|k-2^jx_0|<2^j\),

$$\begin{aligned} |c_{j,k}|\le \mathscr {C}_{j,k}\;. \displaystyle {\inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS(\sigma )}\left( \frac{1+\left| k-2^jx_0\right| }{\lambda _{j,k}}\right) ^{S (\sigma )-\sigma }\right\} } \end{aligned}$$
(25)

with \(\mathscr {C}_{j,k} \) and \(\lambda _{j,k} \) positive sequences such that:

  1. (i)

    For any \(C\in \mathbb {R}\) and for all \(k_j\) such that \(|k_j-2^jx_0|<2^j\),

    $$\begin{aligned} \displaystyle {\varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}{\left( \frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j} +C \; \frac{\log _2\left( \lambda _{j,k_j}\right) }{j}\right) }\;\le 0}. \end{aligned}$$
  2. (ii)

    There exists a sequence \((j_n, k_n)\), with \(j_n\) strictly increasing, such that for \((j_n,k_n)\) the equality holds in (25), with \(c_{j_n,k_n}\ne 0\), and

    $$\begin{aligned} \displaystyle {\lim _{\begin{array}{c} n\rightarrow +\infty \end{array}}\frac{\log _2\left( \mathscr {C}_{j_n,k_{n}}\right) }{j_n}= 0} \;\;\text { and } \;\;\displaystyle {\lim _{\begin{array}{c} n\rightarrow +\infty \end{array}}\frac{\log _2\left( \lambda _{j_n,k_{n}}\right) }{j_n}= 0}. \end{aligned}$$

Proof of Theorem 2.2

Without loss of generality we consider \(x_0=0\).

\(\Longleftarrow )\) The proof is similar and even simpler than the one given in Theorem 2.1. Since we have the hypothesis that \(c_{j_n,k_n}\ne 0\) and

$$\begin{aligned} \left| c_{j_n,k_n}\right| =\displaystyle {\inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-j_nS(\sigma )}\left( \frac{1+\left| k_n\right| }{\lambda _{j_n,k_n}}\right) ^{S (\sigma )-\sigma }\right\} }, \end{aligned}$$

by formula (7), we have \(|c_{j_n,k_n}| = \mathscr {C}_{j_n,k_n}\; 2^{-j\alpha }\) with \( 1+|k_n| = \lambda _{j_n,k_n} 2^{j_n \gamma }.\) The equality \(1+|k_n| = \lambda _{j_n,k_n} 2^{j_n \gamma }\) is equivalent to

$$\begin{aligned} \left( 2^{-j_n}\frac{1+\left| k_n\right| }{\lambda _{j_n,k_n}}\right) ^{S'(\sigma _0)}=\frac{1+\left| k_n\right| }{\lambda _{j_n,k_n}}, \end{aligned}$$

and thus formula (15), in the proof of Theorem 2.1, is satisfied for \(\sigma _n= \sigma _0\).

Also in this case the arguments given in Theorem 2.1 for the set I, defined in formula (11), can be adapted to the set \(\left\{ (j_n, k_n)\right\} _{n\in \mathbb {N}}\) of (ii).

\(\Longrightarrow \)) For the necessity we first have to prove that the wavelet coefficients of the function f can be described as in (25). For simplicity we consider the equality, that is to reformulate the given \(| c_{j,k}|\) as

$$\begin{aligned} \mathscr {C}_{j,k}. \inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS(\sigma )}\left( \frac{1+|k|}{\lambda _{j,k}}\right) ^{S (\sigma )-\sigma }\right\} = \left\{ \begin{array}{lll} \mathscr {C}_{j,k} \;2^{-j \alpha } &{}\quad \text { for }&{} \quad 1+|k| = \lambda _{j,k} 2^{j \gamma } \\ \\ 0 &{}\quad \text { for }&{} \quad 1+|k| \ne \lambda _{j,k} 2^{j \gamma }. \end{array} \right. \end{aligned}$$

For \(|c_{j,k}|\ne 0\), it is enough to select

$$\begin{aligned} \lambda _{j,k}=\frac{1+|k|}{2^{j\gamma }} \quad \text { and }\quad \mathscr {C}_{j,k}=\frac{|c_{j,k}|}{2^{-j\gamma }}. \end{aligned}$$
(26)

And, for \(|c_{j,k}|=0\) it is sufficient to choose \(\lambda _{j,k}\ne \frac{1+|k|}{2^{j\gamma }} \) to be zero in (25), e.g.

$$\begin{aligned} \lambda _{j,k}=2^{j} \quad \text { and }\quad \mathscr {C}_{j,k}=2^{-j^2}. \end{aligned}$$
(27)

Let us prove that \(\mathscr {C}_{j,k} \) and \(\lambda _{j,k} \) satisfy conditions (i) and (ii).

(i) For \(c_{j,k}= 0\) the condition is trivial.

For \(c_{j,k}\ne 0\), since the 2-microlocal frontier of the function f is \(S(\sigma )\) we have \(S(\sigma _0)= \sup \{s : f \in {C}_{x_{0}}^{s,\sigma _0-s}\}\) for all \(\sigma _0\in \mathbb {R}\). This means that for all \(\varepsilon >0\) and \((\sigma _0, s_0)\in \text {Graph}(S)\),

1) \(f\notin { C}_{0}^{s_0+\varepsilon , \sigma _0-(s_0+\varepsilon )}\)    and    2) \(f\in { C}_{0}^{s_0-\varepsilon , \sigma _0-(s_0-\varepsilon )}\).

By 2), for \(j\ge 0\), \(|k|<2^j\), there exists \(C>0\) such that

$$\begin{aligned} \frac{|c_{j,k|} }{2^{-j (s_0-\varepsilon )}\;(1 + | k| )^{s_0-\varepsilon -\sigma _0}}\le C. \end{aligned}$$

Thus, by the definitions of (26),

$$\begin{aligned} \frac{\mathscr {C}_{j,k}2^{-j\alpha }}{2^{-j (s_0-\varepsilon )}\;{(\lambda _{j,k}2^{j\gamma })} ^{s_0-\varepsilon -\sigma _0}}\le C. \end{aligned}$$
(28)

Since \(s_0= S(\sigma _0 )= \alpha + \frac{\gamma }{1-\gamma } (\alpha - \sigma _0)\) i.e. \((1-\gamma )s_0-\alpha +\gamma \sigma _0=0\), (28) can be reformulated as

$$\begin{aligned} \mathscr {C}_{j,k}\;{\lambda _{j,k}} ^{-s_0+\varepsilon +\sigma _0}\;\; 2^{j\varepsilon (\gamma -1)} \le C. \end{aligned}$$

Taking \(\log _2(\cdot )\) and dividing by j, we obtain

$$\begin{aligned} \frac{\log _2(\mathscr {C}_{j,k})}{j}+\frac{\log _2(\lambda _{j,k})}{j} (\sigma _0-s_0+\varepsilon )+ \varepsilon (\gamma -1) \le \frac{\log _2(C)}{j}. \end{aligned}$$

And thus, taking \(\varepsilon \longrightarrow 0\) and letting \(j\longrightarrow +\infty \), we have

$$\begin{aligned} \displaystyle {\varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}{\left( \frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j} + \; (\sigma _0-s_0)\frac{\log _2\left( \lambda _{j,k_j}\right) }{j}\right) }\;\le 0}, \end{aligned}$$

and since \(\sigma _0-s_0\) can take any real value, we obtain condition (i).

(ii) Let \(J:=\{(j,k):1+|k| = \lambda _{j,k} 2^{j \gamma }, j\ge 0, |k|<2^j \}\).

By (i), taking \(C=0\), we have

$$\begin{aligned} \displaystyle {\varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}{\frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j}} \le 0},\;\; \text { for } (j,k_j)\in J. \end{aligned}$$

We need to construct a sequence \((j_n,k_n) \in J\) such that \(j_n\) is strictly increasing and

$$\begin{aligned} \displaystyle {\lim _{\begin{array}{c} n\rightarrow +\infty \end{array}}\frac{\log _2\left( \mathscr {C}_{j_n,k_{n}}\right) }{j_n}= 0} \;\;\text { and } \;\;\displaystyle {\lim _{\begin{array}{c} n\rightarrow +\infty \end{array}}\frac{\log _2\left( \lambda _{j_n,k_{n}}\right) }{j_n}= 0}. \end{aligned}$$

For this we will show that there is a sequence \((j,k_j)\in J\) such that,

$$\begin{aligned} \displaystyle {\varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}{\frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j} }}=0. \end{aligned}$$

For, assume that for all \((k_j)_{j\in \mathbb {N}}\) such that \((j,k_j)\in J\) there exists \(\delta <0 \) such that

$$\begin{aligned} \displaystyle {\varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}{\frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j} }}= \delta \left( (k_j)_{j\in \mathbb {N}}\right)<\delta <0. \end{aligned}$$

Taking \(\varepsilon >0\) such that

$$\begin{aligned} \delta<\varepsilon (\gamma -1)<0 \quad i.e.\quad \varepsilon < \dfrac{\delta }{\gamma -1}, \end{aligned}$$

we have

$$\begin{aligned} \frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j}< \delta< \varepsilon (\gamma -1)\quad i.e. \quad \mathscr {C}_{j,k_j} < 2^{j \varepsilon (\gamma -1)}, \end{aligned}$$
(29)

for all \(j\ge j_0\) and all \((k_j)_{j\in \mathbb {N}}\) such that \((j,k_j)\in J\).

Since \((1-\gamma )s_0=\alpha -\gamma \sigma _0\) we have

$$\begin{aligned} \varepsilon (\gamma -1)=\alpha -(s_0+\varepsilon )+\gamma (s_0+\varepsilon -\sigma _0), \end{aligned}$$

and (29) can be reformulated as

$$\begin{aligned} \mathscr {C}_{j,k_j}2^{-j\alpha } < \;2^{-j(s_0+\varepsilon )}\;(2^{j\gamma })^{s_0+\varepsilon -\sigma _0}. \end{aligned}$$
(30)

Hence, for \( c_{j,k}\ne 0\) i.e. \(|c_{j,k}| = \mathscr {C}_{j,k} \;2^{-j \alpha }\) and \( 1+|k_j| = \lambda _{j,k_j} 2^{j \gamma }\), formula (30) is equivalent to

$$\begin{aligned} |c_{j,k_j}|<2^{-j(s_0+\varepsilon )}\;{\left( \frac{1+\left| k_j\right| }{\lambda _{j,k_j}}\right) }^{ s_0+\varepsilon -\sigma _0}= 2^{-j(s_0+\varepsilon )}\;{\left( 1+\left| k_j\right| \right) }^{ s_0+\varepsilon -\sigma _0} \lambda _{j,k_j}^{ \sigma _0-s_0-\varepsilon }. \end{aligned}$$

Thereby, for \(\varepsilon < \dfrac{\delta }{\gamma -1}\) and \((\sigma _0,s_0)\) such that \(\sigma _0-s_0-\varepsilon =0\) we have

$$\begin{aligned} |c_{j,k_j}|< 2^{-j(s_0+\varepsilon )}\;{\left( 1+\left| k_j\right| \right) }^{ s_0+\varepsilon -\sigma _0}, \end{aligned}$$

for all \(j\ge j_0\) and all sequences \((k_j)_{j\in \mathbb {N}}\), \((j,k_j)\in J\).

In the case \( c_{j,k}=0\) the last inequality is obvious. And thus, there exists \(C>0\) such that

$$\begin{aligned} |c_{j,k}| \le C 2^{-j (s_0 +\varepsilon )}\;(1 + | k | )^{s_0 +\varepsilon -\sigma _0}, \end{aligned}$$

for all \( j\ge 0 \) and \(k \in {\mathbb Z}\): \( | k| < 2^j\), which means that \(f\in { C}_{0}^{s_0+\varepsilon , \sigma _0-(s_0+\varepsilon )}\). This fact contradicts the hypothesis stated in 1) and the contradiction arises because we assumed that \(\displaystyle {\varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}{\frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j} }}= \delta \left( (k_j)_{j\in \mathbb {N}}\right)<\delta <0,\) for all \((k_j)_{j\in \mathbb {N}}\), \((j,k_j)\in J\). Consequently, we can choose \((j_m,k_m)\in J\) such that

$$\begin{aligned} \displaystyle {\lim _{\begin{array}{c} m\rightarrow +\infty \end{array}}\frac{\log _2\left( \mathscr {C}_{j_m,k_{m}}\right) }{j_m}= 0}. \end{aligned}$$

Furthermore, since we have

$$\begin{aligned} \displaystyle {\varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}{\left( \frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j} + \; C\;\frac{\log _2\left( \lambda _{j,k_j}\right) }{j}\right) }\;\le 0},\end{aligned}$$

for all \((k_j)_{j\in \mathbb {N}}\), \((j,k_j)\in J\) and for all C, we obtain

$$\begin{aligned} \displaystyle {\varlimsup _{\begin{array}{c} m\rightarrow +\infty \end{array}}\left( \frac{\log _2\left( \mathscr {C}_{j_m,k_{m}}\right) }{j_m}+ C\;\frac{\log _2\left( \lambda _{j_m,k_{m}}\right) }{j_m}\right) \le 0} \quad \text { for all } C. \end{aligned}$$

Thus,

$$\begin{aligned} \displaystyle {\varliminf _{\begin{array}{c} m\rightarrow +\infty \end{array}} C\;\frac{\log _2\left( \lambda _{j_m,k_{m}}\right) }{j_m}\;\le 0} \quad \text { for all } C,\end{aligned}$$

which is only valid if \(\displaystyle {\varliminf _{\begin{array}{c} m\rightarrow +\infty \end{array}}\;\frac{\log _2\left( \lambda _{j_m,k_{m}}\right) }{j_m}= 0}.\) Hence, there exists a subsequence \((j_{m_n}, k_{m_n})\in J\) such that

$$\begin{aligned} {\lim _{\begin{array}{c} n\rightarrow +\infty \end{array}}\;\frac{\log _2\left( \lambda _{j_{m_n},k_{m_n}}\right) }{j_{m_n}}= 0}. \end{aligned}$$

Therefore, choosing \((j_n,k_n)=(j_{m_n},k_{m_n})\) both conditions of (ii) are satisfied. \(\square \)

Remark 2.3

Note that the case when the 2-microlocal frontier is a vertical line \(\sigma =\alpha \) formula (25) cannot be applied. For this case one has to consider an alternate formula to replace \(\inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS(\sigma )}\left( \frac{1+\left| k-2^jx_0\right| }{\lambda _{j,k}}\right) ^{S (\sigma )-\sigma }\right\} \). The result will then be bounded by \(2^{-j\alpha }\).

Example 2.1

In [11] the authors define a function f such that the 2-microlocal frontier at \(x_0\) is a predetermined non linear function \(S(\sigma )\). This function satisfies that the supremum in the formula

$$\begin{aligned} S(\sigma ) = \sup \{s : f \in {C}_{x_{0}}^{s,\sigma -s} \}, \end{aligned}$$

is in fact a maximum for all \(\sigma \).

Our more general formula (25) allows to find functions (or distributions) for which \(\sup \{s : f \in {C}_{x_{0}}^{s,\sigma -s} \}\) is not a maximum for all \(\sigma \).

Specifically, let \(x_0 = 0\) and \(S(\sigma )= \alpha + \frac{\gamma }{1-\gamma } (\alpha - \sigma )\) with \(\alpha >0\), and \(\gamma \in \mathbb {Q}\) such that \(0 \le \gamma < 1\). By our formula, the wavelet coefficients for the function f, that has \(S(\sigma )\) as its 2-microlocal frontier at 0, satisfy:

$$\begin{aligned} c_{j,k} = \mathscr {C}_{j,k} \;2^{-j \alpha }, \quad \text { for } j,k: j\ge 0 \;\text { and }\; 1+|k| = \lambda _{j,k} 2^{j \gamma }, \end{aligned}$$

and the remaining ones are 0.

If we choose \(\lambda _{j,k}=1\) and \(\mathscr {C}_{j,k}= 2^{\sqrt{j}}\) for \(j\ge 0\), we obtain a function whose 2-microlocal frontier at 0 is \(S(\sigma ) = \sup \{s : f \in {C}_{x_{0}}^{s,\sigma -s} \}\) and for all \(\sigma \) this supremum is not a maximum (see [23]).

2.3 Connections with the Previous Results About 2-Microlocal Frontier Prescription

In this subsection we will briefly outline how the examples previously constructed in [11, 21, 22] are special cases of our general formula (12).

Since our method was directly inspired by the construction in [11], their function is obtained from our formula by choosing in (12) \(\mathscr {C}_{j,k}= 1 \) and \(\;\lambda _{j,k}=1 \) and considering only equality.

To see that the constructions given in [22] and [21] are special cases of our general formula, requires some technicalities. In this section we will just show how to choose the sequences \(\mathscr {C}_{j,k}\) and \(\;\lambda _{j,k}\) and give the details in the Appendix.

Meyer [22] develops his result in the \((s, s')\) plane and considers the 2-microlocal frontier as the set of \((s, s')\) such that \( s = A (s') \), \(A:\mathbb {R}\longrightarrow \mathbb {R}\) be a concave downwards Lipschitz function which is decreasing with \(-1\le \frac{dA}{dt}(t)\le 0\).

In our setting, in the \((\sigma ,s)\) plane, the 2-microlocal frontier can be written as

$$\begin{aligned} S(\sigma )=s \quad \text {such that}\quad s=A(\sigma -s). \end{aligned}$$

Using Theorem 2.1, we obtain the function f proposed by Y. Meyer, by selecting on one hand \(\mathscr {C}_{j,k}=1\) and \(\lambda _{j,k}\) such that

$$\begin{aligned} \lambda _{j,k}= \left\{ \begin{array}{lll} \frac{1+\left| k\right| }{2^{jp_m}}&{}\quad \text { if }&{}\quad j\in {\Lambda }_m \,\text { and } k=\left[ 2^{jp_m}\right] \\ \\ 1&{}\quad \text { otherwise, }&{} \end{array} \right. \end{aligned}$$

with \(\{p_m\}_{m \in \mathbb {N}}\) dense in the set \(\{y \in \mathbb {R}: \exists t \text { with }\frac{dA}{dt}(t)=y \}\) and \({\Lambda }_m\) defined as in (9). And on the other hand, in formula (12) we only consider equality for \(j\in {\Lambda }_m\) and \(k=\left[ 2^{jp_m}\right] \), and we choose \(c_{j,k}=0\) for any other j, k.

Lévy Véhel and Seuret [21] work in the \((s', \sigma )\) plane and thus the 2-microlocal frontier is an increasing function defined by \(\sigma =g(s')\), where \(g:\mathbb {R}\longrightarrow \mathbb {R}\) is concave downwards, with slope between 0 and 1. If g is strictly increasing with \(g''(s')<0\) or is a (non-vertical) line, by selecting appropriate sequences \(\mathscr {C}_{j,k} \) and \(\lambda _{j,k}\) we can also describe the wavelet coefficients defined in [21], via the formula (12) stated in Theorem 2.1. We characterize \(\sigma =g(s')\), in the \((\sigma ,s)\) plane, by defining \(S(\sigma )\) as \(S(\sigma )=s\) such that \(\sigma =g(\sigma -s).\)

In this case, we have to consider the equality in formula (12) for \(j>0\) and \(k\ge 0\), and \(c_{j,k}=0\) for \(j=0\) or \(k<0\).

Therefore, by choosing:

$$\begin{aligned} \mathscr {C}_{j,k}&=\left\{ \begin{array}{lll} 1 &{}\quad \text { if }&{}\quad j> -g^*(\rho _{j,k}) \\ 2^{-j g^*(\rho _{j,k})}2^{-j^2} &{}\quad \text { if }&{}\quad j\le -g^*(\rho _{j,k}) \text { and }-g^*(\rho _{j,k})\, \text { is finite} \\ 2^{-j g^*(g'(0))}2^{-j^2} &{} \quad \text { if }&{}\quad j\le -g^*(\rho _{j,k})\, \text { and }-g^*(\rho _{j,k})=+\infty , \end{array} \right. \end{aligned}$$
(31)
$$\begin{aligned} \text {and}&\nonumber \\ \lambda _{j,k}&=\left\{ \begin{array}{ccc} \frac{1+|k|}{2^{j(1-\rho _{j,k})}} &{} \quad \text { if }&{}\quad -g^*(\rho _{j,k})\, \text { is finite} \\ \frac{1+|k|}{2^{j(1-g'(0))}} &{} \quad \text { if }&{}\quad -g^*(\rho _{j,k})=+\infty , \end{array} \right. \end{aligned}$$
(32)

where \(g^*\) is the Legendre transform of g and \(-g^{*}(\rho _{j,k})= \inf \{-g^*(\rho ):\rho \in E_{j,k}\}\) with \(E_{j,k}=\{\rho : 0\le \rho \le 1 ~y~k= [ 2^{j(1-\rho )}] \}, \) we have the construction proposed in [21]. (For additional details see the Appendix.)

3 A Generalization of the 2-Microlocal Frontier Prescription on a Countable Dense Set

In this section we will extend our Theorem  2.1 so that we can prescribe the 2-microlocal frontier not only at a single point, but on a countable dense set in [0, 1]. This result is inspired by the analogous result of Lévy Véhel and Seuret [21] where they use an iterative construction to obtain their result, imposing some restrictions on the family of curves that are the frontiers at the different points.

Again, we determine a wide class of functions that have the prescribed 2-microlocal frontiers at each of the points \(\{x_n\}_{n\in \mathbb {N}} \subset [0,1]\), where \(\{x_n\}_{n\in \mathbb {N}} \) is a countable dense set in [0, 1].

3.1 Preliminary Results

For each \(x_n\) we choose

$$\begin{aligned} \Gamma _{x_n}= \left\{ (j,k): j>1, k\in \mathbb {Z}, \left| k-2^jx_n \right| \le 2^{j(1-\frac{1}{\log (j)})} \right\} . \end{aligned}$$
(33)

Iteratively we define a subset \(\Omega _n\subset \Gamma _{x_n}\) on which we will determine the wavelet coefficients \(c_{j,k}\) for \((j,k)\in \Omega _n\). For \(x_1\) define \(\Omega _1:= \Gamma _{x_1}\); for \(x_2\) let \( \Omega _2:=\Gamma _{x_2}-\Gamma _{x_1}\), and having defined \(\Omega _{n-1}\), for \(x_n\),

$$\begin{aligned} \Omega _n= \Gamma _{x_n}-\bigcup _{i=1}^{n-1}\Gamma _{x_i}. \end{aligned}$$
(34)

Remark 3.1

Note that \(\Omega _n\) is not empty, and moreover there exists \(j_n\) such that if \(j\ge j_n\) and (j, k) satisfy \(\left| \frac{k}{2^j}-x_n \right| \le 2^{-\frac{j}{\log (j)}}\) then \((j,k)\in \Omega _n\) (See Lemma 6.1 in [21]).

In addition, for each \(x_n\), as in (11), we consider the index set \(I_n\) as

$$\begin{aligned} I_n= \{(j,k): j\in \Lambda _m \text { and } \left[ \left| k-2^jx_n\right| \right] =[2^{jr_m}], \text { with } m\in \mathbb {N}\}. \end{aligned}$$
(35)

With these definitions we can prove the following lemma, which we will use to prove the existence of a function (or a set of functions) with prescribed 2-microlocal frontiers at a countable dense set of points \(\{x_n\}\).

Lemma 3.1

For \(m\in \mathbb {N}\) there exists \(j_m\) such that if \(j\ge j_m\), \(j\in \Lambda _m\) and \((j,k)\in I_{n}\) then \((j,k)\in \Omega _n\).

Proof

If \(j\in \Lambda _m\) and \((j,k)\in I_{n}\) then \(\left[ \left| k-2^jx_n\right| \right] =[2^{jr_m}]\).

Since \(r_m<1\) there exists \(j_m\) such that

$$\begin{aligned} r_m< 1-\dfrac{1}{\log (j)}-\frac{1}{j} \quad \quad \forall j\ge j_m, \end{aligned}$$

and hence

$$\begin{aligned} 2^{jr_m +1}<2^{j\left( 1-\frac{1}{\log (j)}\right) } \quad \quad \forall j\ge j_m, \end{aligned}$$

which, using that \(\left[ \left| k-2^jx_n\right| \right] =[2^{jr_m}]\) yields

$$\begin{aligned} \left| k-2^jx_n\right| \le 2^{jr_m}+1<2^{j\left( 1-\frac{1}{\log (j)}\right) } \quad \forall j\ge j_m. \end{aligned}$$

Hence, if \(j\ge j_m\) and \(j\in \Lambda _m\) and \((j,k)\in I_{n}\) it follows that \((j,k)\in \Gamma _{x_n}\). We can the use Remark 3.1 to pick \(j_m\) in such a way that \((j,k)\notin \Gamma _{x_i}\) for \(i=1, \ldots ,n-1\). \(\square \)

We are now ready to state our theorem.

Theorem 3.1

Let \((x_n)_{n\in \mathbb {}N}\) be a dense countable subset of [0, 1] and for each \(x \in [0,1]\) let \(S_x: (-\infty ,b_x)\longrightarrow \mathbb {R}\), \(b_x \in \mathbb {R}\cup +\infty \), be such that:

  1. (1)

    For each \(n \in \mathbb {N}\), \(S_n(\sigma ) = S_{x_n}(\sigma )\) is a downwards concave function with \(S_n''(\sigma )<0\) or \(S_n(\sigma )\) is a line,

  2. (2)

    If \(\sigma _x\) is the fixed point of \(S_x\), i.e. \(S_x(\sigma _x)= \sigma _x\) we have that

    $$\begin{aligned} b_x\le \varliminf _{y \rightarrow x} \sigma _y. \end{aligned}$$
    (36)

Let \(\mathscr {C}_{j,k} \) y \(\lambda _{j,k}\) positive sequences such that for any sequence \((k_j)_j\) such that \((j,k_j)\in \bigcup _{n\in \mathbb {N}}\Omega _n\), we have that

(i):

For any \(C\in \mathbb {R}\),

$$\begin{aligned} \displaystyle {\varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}{\left( \frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j} +C \; \frac{\log _2\left( \lambda _{j,k_j}\right) }{j}\right) }\;\le 0}. \end{aligned}$$
(ii):

If \((j,k_j)\in \bigcup _{n\in \mathbb {N}} I_n\), with \(I_n\) as in (35),

$$\begin{aligned} \displaystyle {\lim _{\begin{array}{c} j\rightarrow +\infty \end{array}}\frac{\log _2\left( \mathscr {C}_{j,k_j}\right) }{j}= 0} ~\text { and } \displaystyle {\lim _{\begin{array}{c} j\rightarrow +\infty \end{array}}\frac{\log _2\left( \lambda _{j,k_j}\right) }{j}= 0}. \end{aligned}$$

Let \(c_{j,k}\) be numbers such that for \((j,k)\in \Omega _n\)

$$\begin{aligned} \left| c_{j,k}\right| \le \mathscr {C}_{j,k}\;. \displaystyle {\inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS_n(\sigma )}\left( \frac{1+\left| k-2^jx_n\right| }{\lambda _{j,k}}\right) ^{S_n (\sigma )-\sigma }\right\} }, \end{aligned}$$
(37)

and equality holds if \((j,k)\in I_n\cap \Omega _n\), with \(c_{j,k}\ne 0\) for infinitely many j, with \((j,k)\in I_n\cap \Omega _n.\)

If \(\psi \) is any wavelet in the Schwartz class, with infinitely many zero moments, then the function (or distribution)

$$\begin{aligned} f(x)=\sum _{n\in \mathbb {N}}\sum _{(j,k) \in \Omega _n}\;~~c_{j,k} \; \psi (2^{j}x-k) \end{aligned}$$

has \(S_n(\sigma )\) as the 2-microlocal frontier at \(x_n\), for \(n\in \mathbb {N}\).

Proof

Let \(x_n\) be fixed. To see that \(S_n\) is indeed the 2-microlocal frontier of f at \(x_n\), let \((\sigma _0, s_0)\), be any point on the curve \((\sigma , S_n(\sigma ))\). We will show that \(\sup \{s : f \in { C}_{x_n}^{s,\sigma _0-s}\}=s_0,\) meaning that, for each \(\varepsilon >0\), we have

A) \(f\notin { C}_{x_n}^{s_0+\varepsilon , \sigma _0-(s_0+\varepsilon )}\) and B) \(f\in { C}_{x_n}^{s_0-\varepsilon , \sigma _0-(s_0-\varepsilon )}\).

A) We proceed by contradiction:

Assume there exists \(\varepsilon >0\) such that \(f\in { C}_{x_n}^{s_0+\varepsilon , \sigma _0-(s_0+\varepsilon )}\). Given \((\sigma _0, s_0)\), as in the proof of Theorem 2.1, we construct a sequence \((j_l, k_l)\), with \(j_l\) strictly increasing and \((j_l, k_l)\in I_n \cap \Omega _n\) which satisfies that there exists \(\sigma _l\) such that

$$\begin{aligned} \left( 2^{-j_l}\frac{1+\left| k_l\right| }{\lambda _{j_l,k_l}}\right) ^{S_n'(\sigma _l)}=\frac{1+\left| k_l\right| }{\lambda _{j_l,k_l}} \;\text { and }\; \lim _{n \rightarrow +\infty } \sigma _l=\sigma _0. \end{aligned}$$
(38)

The difference of this construction with the one of Theorem 2.1 is that we need that \((j_l, k_l)\in \Omega _n\) on top of \((j_l, k_l)\in I_n\).

If \(S_n\) is linear, (15) is satisfied by hypothesis, picking \(\sigma _l = \sigma _0 \forall l\), since equality holds in (37) and \(c_{j,k}\ne 0\) for infinitely many j, with \((j, k)\in I_n \cap \Omega _n\). In the non-linear case, we consider a subsequence \((r_{m_l})_{l\in \mathbb {N}}\) of \((r_m)_{n\in \mathbb {N}}\) which is dense in the set \(\{y\in \mathbb {R}: \exists \sigma \in (-\infty , b_{x_n}), \text { with } y = \frac{ \mathrm {S}_{\mathrm {n}}^{\prime }(\sigma )}{\mathrm {S}_{\mathrm {n}}^{\prime }(\sigma )-1}\}\) such that

$$\begin{aligned} \lim _{n \rightarrow +\infty } r_{m_{l}}=\frac{S_{n}^{\prime }\left( \sigma _{0}\right) }{S_{n}^{\prime }\left( \sigma _{0}\right) -1}. \end{aligned}$$

Using Lemma 3.1, given \(m_1\) it is possible to choose \(j_1\in \Lambda _{m_1}\); \(k_1\), \([|k_1-2^{j_1}x_n|]=[2^{j_1r_{m_1}}]\) and \((j_1,k_1)\in \Omega _n\). Now, for \(m_2\) we choose \(j_2 > j_1\) such that \(j_2 \in \Lambda _{m_2}\); \( k_2\), \([|k_2- 2^{j_2}x_n|] = [2^{j_2r_{m_2} }]\) and \((j_2,k_2)\in \Omega _n\). So iteratively we construct \((j_l, k_l)\) with \(j_l\) strictly increasing, \(j_l\in \Lambda _{m_l}\) and \((j_l, k_l)\in I_n \cap \Omega _n\).

By hypothesis (ii) and reasoning as in Theorem 2.1 we obtain a contradiction when \(l \longrightarrow +\infty \).

B) We need to show that \(f\in { C}_{x_n}^{s_0-\varepsilon , \sigma _0-(s_0-\varepsilon )}\) for all \(\varepsilon > 0\). That is, we need to show that there exists \(C>0\) such that

$$\begin{aligned} |c_{j, k}| \le C 2^{-j\left( s_{0}-\varepsilon \right) }\left( 1+\left| k-2^{j} x_{n}\right| \right) ^{s_{0}-\varepsilon -\sigma _{0}}, \quad \forall j, k \in \mathbb {Z}: j \ge 1,\left| k-2^{j} x_{n}\right| <2^{j}. \end{aligned}$$
(39)

Or equivalently, since there are only finitely many j, k such that \( \eta<\left| \frac{k}{2^{j}}- x_{n}\right| <1\), we need to see that (39) is satisfied \(\forall j, k \in \mathbb {Z}: j \ge 1,\left| k-2^{j} x_{n}\right| <\eta 2^{j}\), with \(0<\eta <1\).

If (j, k) is such that \(\left| k-2^{j} x_{n}\right| < 2^{j(1- \frac{1}{\log (j)})}\), by Remark 3.1 there is \(j_n\) such that \((j, k)\in \Omega _n\) for all \(j \ge j_n\). Therefore we have

$$\begin{aligned} \left| c_{j,k}\right| \le \mathscr {C}_{j,k}\;. \displaystyle {\inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS_n(\sigma )}\left( \frac{1+\left| k-2^jx_n\right| }{\lambda _{j,k}}\right) ^{S_n (\sigma )-\sigma }\right\} }, \end{aligned}$$

and again, reasoning as in B) of Theorem 2.1 we can prove (39).

What is left to show is the case \(j> 1\) and \(\left| k-2^{j} x_{n}\right| < \eta \;2^{j} \) but \((j, k)\notin \Omega _n\), or \(2^{j(1-\frac{1 }{\log (j)} )}<\left| k-2^{j} x_{n}\right| <\eta \;2^{j}\). It is here where (36) comes into play:

Given \(\varepsilon >0\) we fix \(\eta \) such that for all y with \(0<|y-x_n|<2\eta \) we have that: If \(\delta >0\) is such that \(\sigma _0 +\delta <b_{x_n}\) and \(\sigma _y\) is such that \(S_y(\sigma _y)= \sigma _y\), then

$$\begin{aligned} \sigma _0+\delta <\sigma _y. \end{aligned}$$
(40)

Now, since \((j, k)\in \Omega _\mathbf{m} \), with \(\mathbf{m} \ne n\), \(\left| c_{j,k}\right| \) satisfies

$$\begin{aligned} \left| c_{j,k}\right| \le \mathscr {C}_{j,k}\;. \displaystyle {\inf _{\sigma \in \mathbb {R}} \left\{ ~2^{-jS_\mathbf{m} (\sigma )}\left( \frac{1+\left| k-2^jx_\mathbf{m }\right| }{\lambda _{j,k}}\right) ^{S_\mathbf{m} (\sigma )-\sigma }\right\} }. \end{aligned}$$

Recalling that \(s_0=S_n(\sigma _0)\), the inequality (39) will hold if we prove that for appropriate \(\sigma _{j,k}\), the expression

$$\begin{aligned} \frac{{\mathscr {C}}_{j, k} 2^{-j S_\mathbf{m }\left( \sigma _{j, k}\right) }\left( \frac{1+\left| k-2^{j} x_\mathbf{m }\right| }{\lambda _{j, k}}\right) ^{S_\mathbf{m }\left( \sigma _{j, k}\right) -\sigma _{j, k}}}{2^{-j\left( S_n(\sigma _0)-\varepsilon \right) }\left( 1+\left| k-2^{j} x_{n}\right| \right) ^{S_n(\sigma _0)-\varepsilon -\sigma _{0}}} \end{aligned}$$
(41)

is bounded for \(j\ge n_0\).

In particular, choosing \(\sigma _{j, k}= \sigma _\mathbf{m} \), with \( \sigma _\mathbf{m} \) such that \(S_\mathbf{m} (\sigma _\mathbf{m} ) = \sigma _\mathbf{m} \), we have that (41) is

$$\begin{aligned} \frac{{\mathscr {C}}_{j, k} 2^{-j S_\mathbf{m }\left( \sigma _\mathbf{m } \right) }}{2^{-j\left( S_n(\sigma _0)-\varepsilon \right) }\left( 1+\left| k-2^{j} x_{n}\right| \right) ^{S_n(\sigma _0)-\varepsilon -\sigma _{0}}}. \end{aligned}$$
(42)

To prove the boundedness of this quotient, we consider 2 cases: 1) \(S_n(\sigma _0) -\varepsilon -\sigma _0 \le 0\) and 2) \(S_n(\sigma _0) -\varepsilon -\sigma _0 > 0\).

1) If \(S_n(\sigma _0) -\varepsilon -\sigma _0 \le 0\):

Since \(\left| k-2^{j} x_{n}\right|<\eta \;2^{j}<2^j\) and \(\sigma _0-S_n(\sigma _0) +\varepsilon \ge 0\) we can bound (42) by

$$\begin{aligned} {\mathscr {C}}_{j, k} 2^{-j (S_\mathbf{m }\left( \sigma _\mathbf{m } \right) -\sigma _0)} \; 2^{\sigma _{0}-S_n(\sigma _0)+\varepsilon }. \end{aligned}$$
(43)

Taking \(\log _2(\cdot )\) and factoring out j we have

$$\begin{aligned} j \left[ \frac{\log _2({\mathscr {C}}_{j, k})}{j}-(S_\mathbf{m }\left( \sigma _\mathbf{m } \right) -\sigma _0)+\frac{\sigma _{0}-S_n(\sigma _0)+\varepsilon }{j} \right] . \end{aligned}$$
(44)

By taking \(j\ge j_0\) such that

$$\begin{aligned} \left| k-2^{j} x_\mathbf{m }\right|< 2^{j(1-\frac{1 }{\log (j)})}<\left| k-2^{j} x_{n}\right| < \eta 2^j, \end{aligned}$$

we have that \(\left| x_\mathbf{m }-x_n\right| <2 \eta \), and so, by (40) we know that \(S_\mathbf{m }\left( \sigma _\mathbf{m } \right) -\sigma _0=\sigma _\mathbf{m }-\sigma _0>\delta >0\). Hence (44) is bounded by

$$\begin{aligned} j \left[ \frac{\log _2({\mathscr {C}}_{j, k})}{j}-\delta +\frac{\sigma _{0}-S_n(\sigma _0)+\varepsilon }{j} \right] . \end{aligned}$$
(45)

By hypothesis (i), we have

$$\begin{aligned} \varlimsup _{\begin{array}{c} j\rightarrow +\infty \end{array}}{j \left[ \frac{\log _2({\mathscr {C}}_{j, k})}{j}-\delta +\frac{\sigma _{0}-S_n(\sigma _0)+\varepsilon }{j} \right] }=-\infty , \end{aligned}$$

this limit is independent of \(\mathbf{m} \) for \(\left| x_\mathbf{m }-x_n\right| <2 \eta .\)

Therefore there exists \(n_0>j_0\), (\(n_0\) independent on \(\mathbf{m} \) ) such that (43) is bounded for \(j> n_0\) and \(\left| k-2^{j} x_{n}\right| < \eta \;2^{j} \) but \((j, k)\notin \Omega _n\).

2) If \(S_n(\sigma _0) -\varepsilon -\sigma _0 > 0\):

Since \(\; 2^{j(1-\frac{1 }{\log (j)} )} <\left| k-2^{j} x_{n}\right| \) we can bound (42) by

$$\begin{aligned} \mathscr {C}_{j, k} 2^{-j \left( S_\mathbf{m }\left( \sigma _\mathbf{m } \right) -\sigma _0\right) } \left( 2^{\frac{j}{\log (j)} }\right) ^{S_n(\sigma _0)-\varepsilon -\sigma _{0}}. \end{aligned}$$
(46)

Again, taking \(\log _2(\cdot )\) and factoring out j we have

$$\begin{aligned} j \left[ \frac{\log _2({\mathscr {C}}_{j, k})}{j}-(S_\mathbf{m }\left( \sigma _\mathbf{m } \right) -\sigma _0)+\frac{S_n(\sigma _0)-\sigma _{0}-\varepsilon }{\log (j)} \right] . \end{aligned}$$
(47)

As in 1) the upper limit of (47) is \(-\infty \), independently of \( \mathbf{m} \).

Therefore, there also exists \(n_0>j_0\) such that (46) is bounded if \(j> n_0\) and \(\left| k-2^{j} x_{n}\right| < \eta \;2^{j} \) but \((j, k)\notin \Omega _n\).

This completes the proof of this Theorem. \(\square \)