1 Introduction

The discretisation of function spaces is an interesting problem both from a pure and applied perspective. One of the leading ideas is that the smoothness or regularity must be characterised via decay or sparsity properties of the associated discrete expansions. For example, it is well known that \(H^s([0,1])\) can be characterised by the decay properties of the Fourier coefficients. Similarly, Sobolev spaces \(H^s(\mathbb {R})\), and more generally inhomogeneous Besov spaces, can be described using suitable wavelet expansions, see for example [10]. See [6, 7, 12,13,14,15, 18, 21, 24] for other discrete time-frequency techniques and representations of function spaces related to our research.

Here, we focus on the Stockwell frames associated to a dyadic partition of the frequency domain. In [2], the Stockwell frames were first introduced in connection to the so called \(\alpha \)-partitioning, see e.g. [14]. It is well known that \(\alpha \)-partitioning and \(\alpha \)-modulation spaces with \(\alpha =1\), i.e. with pure dyadic scaling factor \(2^j, j\in {\mathbb {Z}}\), lead to wavelet analysis and Sobolev spaces, see [13, 14]. In particular, it can be shown that wavelets can be used to characterise these spaces, see [10]. Hence, we investigate the conditions under which our Stockwell frame characterises Sobolev spaces representing their norms and seminorms and also show how to characterise Besov spaces. We introduce a dyadic partition of the frequency domain with a fixed number of possible directions at any scale, similarly to what it is usually done with multi-dimensional wavelets, which is suited for both Sobolev and Besov spaces. Our theory can even cover a wider class of decomposition spaces, see [5, 27] and reference therein. In particular, we later introduce a parameter \(\delta \) which regulates the number of directions we have at each step of the decomposition and describes the different partitions associated with it. This captures the case of finite directions described above while also allowing the number of directions to grow; in this way, one obtains partitions related with different kind of parabolic molecules, see Remark 7 below. We also show that, in some cases, the Stockwell frame detects higher regularity than wavelets. In particular, it is known that \(H^s(\mathbb {R}), s<\frac{1}{2}\) can be characterised using Haar wavelets. Our frame associated with the characteristic function improves the result to \(s<1\).

The paper is organised as follows. In Sect. 2 we introduce the concept of admissible multi-dimensional partition; roughly, a dyadic partition which covers the whole of \(\mathbb {R}^d\) (seen as frequency space) and has the uniformly finite intersection property, and a finite number of possible directions in frequency, see Definition 2.1 below. Giving a suitable function \(\varphi \), we define the following system of functions

$$\begin{aligned} \varphi _{\bullet ,\lambda }= T_{\lambda } \varphi (x), \, \varphi _{j,k,\lambda }(x)=T_{2^{-j}\lambda } \left( \frac{1}{ 2^{jd/2}} \sum _{\eta \in Z_{j,k}} e^{2\pi \mathrm {i}\,\eta t } \varphi (x) \right) , \end{aligned}$$
(1)

depending on a certain number of suitable parameters. The parameter \(\lambda \) represents the space translation, j represents the (parabolic) scaling, while the parameter k the “direction” as we shall show later on. For example, when \(d=1\), we have just two admissible directions \(k=\pm \) and we can set

$$\begin{aligned} Z_{j,+}= 2^{j}, \ldots , 2^{j+1}-1 \quad \text{ and } \quad Z_{j,-}=-2^{j+1}+1, \ldots , -2^{j}. \end{aligned}$$
(2)

It is worth explaining why we choose (1) as a candidate to be an \(L^2\)-frame.

Consider dimension \(d=1\) and let \(Z_{j,\pm }\) be as in (2). In [1] it is proved that the so called DOST-functions,

$$\begin{aligned} \begin{aligned} P_{j,k,\tau }(t)&= T_{\frac{\tau }{2^{j}}}\frac{1}{2^{j/2}} \sum _{\eta \in Z_{j,k}} e^{2\pi \mathrm {i}\,\eta t}, \quad j\in {\mathbb {N}}, \tau =0, \ldots , 2^{j}-1, k=\pm ,\\ P_{\bullet }(t)&=1, \end{aligned} \end{aligned}$$
(3)

form an orthonormal basis of \(L^2([0,1])\); these functions (3) were first introduced in [25] as a discretisation of the S-transform, defined in [26]. This basis was further studied by Wang and Orchard [28] and then extended to dimension \(d=2\) by Drabycz, Stockwell, Mitchell, Brown, Lauzon and Frayne [31], see also [11]. A first step into the direction of the Stockwell frame development was done by Yan and Zhu [30], where the authors studied general and flexible tiling in time-frequency space in dimension d \(=\) 1. See also [28] for a numerical perspective and [17, 23, 29] for a more abstract analysis of the S-transform.

Then, the naive idea of the definition of the Stockwell frames comes from the interpretation of Gabor frames

$$\begin{aligned} T_{\lambda } \left( e^{2\pi \mathrm {i}\,m t} \varphi (t) \right) , \quad \lambda \in \nu {\mathbb {Z}}, m\in {\mathbb {Z}}, \end{aligned}$$

as the uniform translation of \(\left\{ e^{2\pi \mathrm {i}\,mt} \right\} _{m\in {\mathbb {Z}}}\), the usual Fourier basis of \(L^2([0,1])\), localised via a window function \(\varphi \). Using the DOST basis instead of the Fourier one, and a natural non-stationary translation, we are led exactly to the system of functions defined in (1). Notice that the non-stationary translation is related to the frequency parameter j. Roughly, we refine the space translation as the frequency increases.

In Sect. 3, we study the system of functions (1) in detail. In particular we prove our main result, Theorem 3.2, i.e. the system (1) is a frame of \(L^2(\mathbb {R}^d)\) which characterises the \(H^s\)-Sobolev norm, that is

$$\begin{aligned} A\left\| f \right\| ^2_s \le \sum _{\lambda \in \nu {\mathbb {Z}}^d}| \left\langle f, \varphi _{\bullet ,\lambda }\right\rangle |^2 + \sum _{j\in {\mathbb {N}}, k\in K, \lambda \in \nu {\mathbb {Z}}^d}2^{2js}\left| \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right| ^2 \le B\left\| f \right\| ^2_s. \end{aligned}$$
(4)

The requirements, see Definition 3.1, involve the decay and non vanishing properties of suitable linear combinations of translations of \(\widehat{\varphi }\), which is the Fourier Transform of \(\varphi \). Given \(s\ge 0 \), it is not difficult to find functions which satisfy such conditions, as we show at the end of the section in Theorem 3.6. For example, the Gaussian function works for all s.

In Theorem 3.4, we generalise the construction and prove a similar result for seminorms. As expected, the conditions are stronger than those of the \(H^s\)-norm characterisation. As an application, we show that this frame can also characterise Besov spaces \(B^{s}_{2,q}\), with \(s\in (0,1)\) and \(q>1\). We end the section by stating some sufficient conditions for the existence of our frames.

In Sect. 4, we give some explicit examples of Stockwell-like frames and, in a particular case, we show that the system of functions (1) is indeed an orthonormal basis of \(L^2(\mathbb {R})\), characterizing all Sobolev spaces. Moreover, as anticipated, we show how the Stockwell frame compares with Haar-wavelet in terms of Sobolev regularity, showing how the former improves the wavelet result.

In Sect. 5, we extend the result to isotropic admissible partitions with a growing number of frequency directions.

2 Notations

We choose the following normalisation for the Fourier transform

$$\begin{aligned} \left( {\text {F}}u \right) (\omega )=\int e^{-2 \pi i x\cdot \omega } u(x){d}x. \end{aligned}$$

We denote by \(\left\| \cdot \right\| _s\) and \(\left| \cdot \right| _{s}\) the \(H^s\)-norm and seminorm, respectively. We write \( f\lesssim g, \) if there exists a constant \(C>0\) such that \( f\le C\, g. \) We write \(f\asymp g\) if \(f\lesssim g\) and \(g \lesssim f\). Finally, \(\mathrm {d}(\cdot ,\cdot )\) represents the Euclidean distance. We also use \(\lceil \cdot \rceil \) to represent the “ceiling” function.

3 Stockwell-like frames

First, we define a class of suitable tiling of \(\mathbb {R}^d\) in the frequency domain; then we construct frames for the \(H^s\)-norm associated to these admissible partitions.

3.1 Admissible partitions of \(\mathbb {R}^d\): the finite directions case

We state here a first definition in arbitrary dimension with finitely many frequency directions; then we give some examples of admissible partitions. Similar concepts are used in [13, 14], and we refer also to the bibliography presented there.

Definition 2.1

The family \(\left\{ I_{j,k} \right\} _{j\in {\mathbb {N}}, k\in K_{j}} \cup I_{\bullet }\), where \(K_{j}\) is an index set, is called admissible partition if

  1. (i)

    \(I_{j,k}\) and \(I_{\bullet }\) are non empty connected subsets of \(\mathbb {R}^d\);

  2. (ii)

    \(\bigcup _{j\in {\mathbb {N}}, k\in K_{j} } I_{j,k} \cup I_{\bullet }= \mathbb {R}^d\);

  3. (iii)

    there exists \(N\in {\mathbb {N}}\) such that for each \(\left( \bar{j}, \bar{k} \right) \) there are at most N indices (j, k) such that \(I_{\bar{j},\bar{k}}\cap I_{j,k}\ne \emptyset \) (uniformly finite intersection);

  4. (iv)

    \(I_\bullet \) is a neighbourhood of the origin and, for each \(j\in {\mathbb {N}}\) and \(k\in K_j\), \(I_{j,k}\cap {\mathbb {Z}}^d\ne \emptyset \);

  5. (v)

    there exist \(c_{\min },c_{\max }>0\) such that for all \(\omega \in I_{j,k}\)

    $$\begin{aligned} c_{\min }<\frac{|\omega |}{2^j}<c_{\max }, \quad \text{ uniformly } \text{ in } j,k; \end{aligned}$$
  6. (vi)

    \(|I_{j,k}|\asymp |I_{j,k'}|, k,k' \in K_j\), where \(|I_{j,k}|\) is the Lebesgue measure of \(I_{j,k}\);

  7. (vii)

    there exists \(C_K\) such that \(|K_{j}| \lesssim C_K\), uniformly in j.

Remark 1

An admissible partition covers \(\mathbb {R}^d\) with no holes—(i), (ii)—with finitely many intersections—(iii)—and each set contains at least some point with integer coordinates—(iv). The distance from the origin of each elements of the set \(I_{j,k}\) is of order \(2^j\)—(v). In (vi), at the scale j, each set has comparable size. Finally—(vii)—the constant \(C_K\) represents the possible number of frequency direction at any scale j, which we assume to be uniformly bounded.

We now describe some examples of admissible partition in different dimensions.

Dimension \(d=1\) In this case we just have two possible directions represented by \(k = \pm \). We set

$$\begin{aligned} \begin{aligned} I_{j,+}&= \left\{ \omega \in \mathbb {R}, \omega \in \left[ 2^j-\frac{1}{2}, 2^{j+1}-\frac{1}{2} \right) \right\} ,\\ I_{j,-}&= \left\{ \omega \in \mathbb {R}, \omega \in \left( -2^{j+1}+\frac{1}{2}, -2^j +\frac{1}{2}\right] \right\} ,\\ I_{\bullet }&=\left( -\frac{1}{2},\frac{1}{2} \right) . \end{aligned} \end{aligned}$$
(5)

One can easily see that this partition is admissible. With this particular choice, we show in Sect. 4 that it is possible to define a Stockwell-like frame which is also an orthonormal basis of \(L^2(\mathbb {R})\).

Dimension \(d=2\) We express \(\omega \in \mathbb {R}^2\) using polar coordinates \(\left( \rho ,\theta \right) \) and set

Set

$$\begin{aligned} \begin{aligned} I_{j,k}&= \left\{ \omega \in \mathbb {R}^2{:}\,2^j -\frac{1}{2}\le \rho<2^{j+1} -\frac{1}{2},\, \frac{k\pi }{4}\le \theta<\frac{(k+1)\pi }{4} \right\} , \, k=0,\ldots ,7\\ I_{\bullet }&=\left\{ \omega \in \mathbb {R}^2{:}\,\rho <\frac{1}{2} \right\} . \end{aligned} \end{aligned}$$
(6)

As before, one can easily check that this is an admissible partition; we refer to Fig. 1 for a plot.

Fig. 1
figure 1

An example of admissible partition in \(d=2\)

Full size image

The last example represents a partition which naturally extends to arbitrary dimensions.

3.2 Stockwell-like frames

Consider functions \(\varphi , \varphi _{\bullet }\in L^2(\mathbb {R}^d)\), an admissible partition \(\left\{ I_{j,k} \right\} _{j \in {\mathbb {N}}, k \in K_{j}} \cup I_\bullet \) and

$$\begin{aligned} \varGamma = \left\{ (j, k, \lambda ) \mid j\in {\mathbb {N}}, k\in K_{j}, \lambda \in \nu {\mathbb {Z}}^d \right\} . \end{aligned}$$
(7)

Let \(Z_{j,k} = I_{j,k}\cap {\mathbb {Z}}^d\), then the Stockwell frame is defined as

$$\begin{aligned} \varphi _{\bullet ,\lambda }(x) = \varphi _{\bullet }(x - \lambda ),\, \varphi _{j,k,\lambda }(x)=T_{2^{-j}\lambda } \left( \frac{1}{2^{jd/2}} \sum _{\eta \in Z_{j,k}} e^{2\pi \mathrm {i}\,\eta \cdot x } \varphi (x) \right) , \end{aligned}$$

with \((j,k,\lambda ) \in \varGamma ,\, x\in \mathbb {R}^d\). We choose \(2^{jd/2}\) as \(\ell ^2({\mathbb {Z}}^d)\) normalisation, since \(|Z_{j,k}| \asymp 2^{jd}\). Finally, we define the Stockwell-like system as

$$\begin{aligned} \mathcal {S}(\varphi _{\bullet },\varphi ,\varGamma ) = \left\{ \varphi _{\bullet ,\lambda }, \lambda \in \nu {\mathbb {Z}}^d \right\} \cup \left\{ \varphi _{j,k,\lambda }, \, j,k,\lambda \in \varGamma \right\} . \end{aligned}$$
(8)

Applying the Fourier transform one gets

$$\begin{aligned} \begin{aligned} \widehat{{\varphi }_{\bullet ,\lambda }} (\omega )&= e^{-2\pi \mathrm {i}\,\omega \cdot \lambda } \widehat{\varphi _{\bullet }}( \omega ), \\ \widehat{\varphi _{j,k,\lambda }}(\omega )&= e^{-2\pi \mathrm {i}\,\omega \cdot 2^{-j}\lambda } \left( \frac{1}{2^{jd/2}} \sum _{\eta \in Z_{j,k}} \widehat{\varphi }(\omega -\eta ) \right) .\\ \end{aligned} \end{aligned}$$
(9)

If \(\varphi \) is a localisation window, the sum over the integer in \(Z_{j,k}\) determines, in the space domain, a higher localisations near the point \(2^{-j}\lambda \) as the scale j increases.

In frequency, it implies that the frame element \(\varphi _{j,k,\lambda }\) has a Fourier transform \(\widehat{\varphi _{j,k,\lambda }}\) with the support “essentially” in the set \(I_{j,k}\). In Fig. 2 we plot some frame elements with different windows \(\varphi \), in the dimension \(d=1\) case. In Fig. 3, we plot a frame element and its Fourier transform in the dimension \(d=2\) case using the partition defined above, see (6).

Fig. 2
figure 2

Real parts of the frame element \(\varphi _{j,k,\lambda }(t)\), \(\varphi = {\text {sinc}}^n(t)\), in dimension \(d=1\), both in time and frequency domain. Here, the scale is \(2^j, j=4\). We observe that the frame window in frequency works as a characteristic function of \(\left[ 2^j-\frac{1}{2}, 2^{j+1}-\frac{1}{2} \right] \). a Time, \(\varphi _{4,+,0}(t) \), \(n=3\), b frequency, \(\widehat{\varphi _{4,+,0}}(\omega )\), \(n=3\), c time, \(\varphi _{4,+,0}(t)\), \(n=15\), d frequency, \(\widehat{\varphi _{4,+,0}}(\omega )\), \(n=15\)

Full size image
Fig. 3
figure 3

A frame element obtained using the partition of Fig. 1, the scale parameter is \(j=4\) the angular parameter is the same of Fig. 1, that is \(k=1\). As localizing window we consider \(\varphi (x,y)={\text {sinc}}^3(x)\cdot {\text {sinc}}^3(y)\). On the left is plotted the real part of the frame element and on the right the absolute value of its Fourier transform

Full size image

4 Frames of \(H^s(\mathbb {R}^d)\)

In this section, we introduce conditions under which the system of functions \(\left\{ \varphi _{j,k,\lambda } \right\} _{j,k,\lambda \in \varGamma }\) is a frame representing the \(H^s\)-norm. Later, we analyse the \(H^s\)-seminorm’s case as well.

4.1 \(H^s\)-norm

Before going into specific details, let us give a brief explanation of which conditions we need to impose. We require that the frame elements cover in the frequency domain the whole \(\mathbb {R}^d\), and we require decay properties at infinity in the frequency domain, in order to obtain Sobolev regularity. Precisely, we have the following definition.

Definition 3.1

(s-admissibility for \(H^s\)-norms, finite case) Consider an admissible partition \(\left\{ I_{j,k} \right\} _{j,k \in \varGamma } \cup I_\bullet \) and \(s\ge 0\); we say that a pair of functions \(\varphi _{\bullet },\varphi \) is s-admissible with respect to the partition if, given

$$\begin{aligned} \varPhi _{\bullet }(\omega ) = \widehat{\varphi _{\bullet }}(\omega ),\qquad \varPhi _{j,k}(\omega )=\sum _{\eta \in Z_{j,k}} \hat{\varphi }(\omega - \eta ), \end{aligned}$$
(10)

then

  1. (i)

    there exists \(\alpha > d/2\)

    $$\begin{aligned} |\varPhi _{\bullet }(\omega )| \lesssim \frac{1}{(1+|\omega |)^{\alpha }},\,\text { and }\, |\varPhi _{j,k}(\omega )|\lesssim \left\{ \begin{array}{ll} \frac{2^{jd/2}}{\left( 1+d(\omega , I_{j,k}) \right) ^{\alpha +s}},&{} \, \omega \notin I_{j,k}\\ 1, &{}\, \omega \in \mathbb {R}^d \end{array} \right. ; \end{aligned}$$
    (11)
  2. (ii)

    there exists \(a>0\) such that, for all \(\omega \in \mathbb {R}^d\) then

    $$\begin{aligned} \left| \varPhi _{\bullet }(\omega ) \right| \ge a, \text{ if } \omega \in I_{\bullet }, \qquad \left| \varPhi _{j,k}(\omega ) \right| \ge a, \text{ if } \omega \in I_{j,k} \end{aligned}$$
    (12)

    (the constant a does not depend on j, k).

Remark 2

The decay hypothesis on \(\varphi _{\bullet }\) implies

$$\begin{aligned} \sum _{\lambda \in \nu {\mathbb {Z}}^d} | \left\langle T_{\lambda } \varphi _{\bullet }, f\right\rangle |^2\lesssim \left\| f \right\| ^2_{L^2(\mathbb {R}^d)}, \end{aligned}$$

see e.g. [9, Thm 9.2.5 p.206].

We state the main result for our Stockwell-like system. The \(L^2\)-case, hereby represented by \(s=0\), has been proved already in [2] under a different and stronger set of hypotheses.

Theorem 3.2

Let \(s\ge 0\) and consider an s-admissible couple of functions \(\varphi _{\bullet },\varphi \). Then there exists \(\nu _0>0\) such that for each \(\nu \in (0,\nu _0)\) the system \(\mathcal {S}\left( \varphi _{\bullet }, \varphi ,\varGamma \right) \) defined in (8) is a frame representing the \(H^s(\mathbb {R}^d)\) norm. Precisely, there exist \(A,B>0\) such that for each \(f\in H^s(\mathbb {R}^d)\)

$$\begin{aligned} A\left\| f \right\| ^2_s \le \sum _{\lambda \in \nu {\mathbb {Z}}^d}| \left\langle f, T_{\lambda } \varphi _{\bullet }\right\rangle |^2 + \sum _{j,k,\lambda \in \varGamma }2^{2js}\left| \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right| ^2 \le B\left\| f \right\| ^2_s. \end{aligned}$$

Proof

The result follows by Property 1 (upper bound) and Property 2 (lower bound), stated and proved below. \(\square \)

In order to show the boundedness of the coefficients with respect to the Sobolev norm, we introduce the following family of sets

$$\begin{aligned} E_{\bullet } = I_{\bullet }, \quad E_{j,k}= \left\{ \omega \in \mathbb {R}^d{:}\,\mathrm {d}\left( \omega ,I_{j,k} \right) \le 2^{j-1} \right\} . \end{aligned}$$
(13)

Proposition 1

(Upper bound) In the hypothesis of Theorem 3.2, for any \(s\ge 0\) there exist a positive constant C such that

$$\begin{aligned} \sum _{j,k,\lambda \in \varGamma }2^{2js}\left| \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right| ^2 \le C \left\| f \right\| ^2_s, \end{aligned}$$
(14)

where C depends on \(\nu \), the lattice constant defined in (7).

Proof

For each index j set

$$\begin{aligned} f_{j,k,1}(t)= {\text {F}}^{-1} \left( \chi _{E_{j,k}(\omega )} \widehat{f}(\omega ) \right) ,\quad f_{j,k,2}(t)= 1- f_{j,k,1}(t), \end{aligned}$$

where \(E_{j,k}\) is as in (13). Notice that \( E_{j,k} \cap E_{j',k}=\emptyset , \quad \text{ if } |j-j'|\ge 2. \) By Plancherel Theorem,

$$\begin{aligned}&\sum _{j,k,\lambda \in \varGamma }2^{2js}\left| \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right| ^2 = \sum _{j,k,\lambda \in \varGamma }2^{2js}\left| \left\langle {\text {F}}(f), {\text {F}}(\varphi _{j,k,\lambda })\right\rangle \right| ^2\nonumber \\&\quad =\sum _{j,k,\lambda \in \varGamma }\left| \left\langle {\text {F}}(f) (1+|\omega |)^s, \frac{2^{js}}{(1+|\omega |)^s}{\text {F}}(\varphi _{j,k,\lambda })\right\rangle \right| ^2\nonumber \\&\quad =\sum _{j,k,\lambda \in \varGamma }\left| \left\langle {\text {F}}(f) (1+|\omega |)^s, \frac{1}{2^{jd/2}} e^{-2\pi \mathrm {i}\,2^{-j}\lambda (\cdot )}\varPhi _{j,k}(\cdot ) \frac{2^{js}}{(1+|\omega |)^s} \right\rangle \right| ^2\nonumber \\&\quad =\sum _{j,k,\lambda \in \varGamma }\left| \left\langle {\text {F}}(f) (1+|\omega |)^s, {\text {F}}( \widetilde{\varphi _{j,k,\lambda }) }\right\rangle \right| ^2, \end{aligned}$$
(15)

where—cf. Lemma A.4—

$$\begin{aligned} \widetilde{\varphi _{j,k,\lambda }}(t)=\frac{1}{2^{jd/2}} T_{2^{-j}\lambda } {\text {F}}^{-1}\left( \frac{2^{js}}{(1+|\omega |)^s} \varPhi _{j,k}(\omega ) \right) (t). \end{aligned}$$

Splitting \(f = f_{j,k,1}+f_{j,k,2}\) for each j we get

$$\begin{aligned} \sum _{j,k,\lambda \in \varGamma }2^{2js}\left| \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right| ^2&\lesssim \sum _{j,k,\lambda \in \varGamma }\left| \left\langle {\text {F}}(f_{j,k,1}) (1+|\omega |)^s, {\text {F}}( \widetilde{\varphi _{j,k,\lambda }) }\right\rangle \right| ^2\\&\quad \,\,+ \sum _{j,k,\lambda \in \varGamma }\left| \left\langle {\text {F}}(f_{j,k,2})(1+|\omega |)^s, {\text {F}}( \widetilde{\varphi _{j,k,\lambda }) }\right\rangle \right| ^2. \end{aligned}$$

Notice that \(f_{j,k,1},f_{j,k,2}\) satisfy the hypothesis of Lemmas A.4 and A.5 respectively, therefore

$$\begin{aligned} \sum _{j,k,\lambda \in \varGamma } 2^{2js} \left| \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right| ^2&\lesssim \frac{1}{\nu } \sum _{j,k}\left( \left\| f_{j,k,1} \right\| _s^2 +2^{j(d-2\alpha )} \left\| {\text {F}}\left( f_{j,k,2} \right) (1+|\cdot |)^s \right\| ^2 \right) \\&\lesssim \nu ^{-1}\Vert f\Vert ^2_s, \end{aligned}$$

as desired. Indeed, since the partition \(\left\{ I_{j,k}, I_\bullet \right\} \) is admissible then \(E_{j,k}\) have the (uniform) finite intersection property as well, then

$$\begin{aligned} \sum _{j,k}\left\| f_{j,k,1} \right\| _s^2 \le \sum _{j,k}\left\| f \right\| ^2_{H^s(E_{j,k})}\lesssim \left\| f \right\| ^2_s, \end{aligned}$$

and

$$\begin{aligned} \sum _{j,k}2^{j(d-2\alpha )} \left\| {\text {F}}\left( f_{j,k,2} \right) (1+|\omega |)^s \right\| ^2 \le \left\| f \right\| ^2_s\sum _{j,k}2^{j(d-2\alpha )}\lesssim \left\| f \right\| ^2_s. \end{aligned}$$

In the last inequality, we used that the series

$$\begin{aligned} \sum _{j,k}2^{j(d-2\alpha )} \end{aligned}$$
(16)

is convergent, since \(d-2\alpha <0\). \(\square \)

Corollary 1

The analysis operator

$$\begin{aligned} C{:}\,L^2(\mathbb {R}^d)&\longrightarrow \ell ^2(\varGamma )\\ f&\longmapsto \left\{ \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right\} _{j,k,\lambda \in \varGamma } \end{aligned}$$

is continuous. Hence, the same is true for the frame operator \(S=C^{*}C\), where \(C^*\) is the adjoint operator.

Using the hypothesis on the window functions, we show that there exists a (uniform) lower bound for the \(H^s\)-norm.

Proposition 2

(Lower bound) In the hypothesis of Theorem 3.2, for any \(s\ge 0\) there exist \(\nu _0>0\) and \(C>0\) such that for every \(\nu \in (0,\nu _0)\), we have

$$\begin{aligned} \sum _{\lambda \in \nu {\mathbb {Z}}^d}| \left\langle T_{\lambda } \varphi _{\bullet }, f\right\rangle |^2 + \sum _{j,k,\lambda \in \varGamma }2^{2js}\left| \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right| ^2 \ge C\left\| f \right\| ^2_s, \end{aligned}$$
(17)

where the constant C depends on \(\nu \).

Proof

Consider the (modified) frame operator

$$\begin{aligned} S^{s}f(x) = S^s_{\varphi _{\bullet }} f(x) + S_{\varphi }^s f(x), \end{aligned}$$

where

$$\begin{aligned} S^s_{\varphi _{\bullet }} f(x) {}= \sum _{\lambda } \left\langle f, \varphi _{\bullet ,\lambda }\right\rangle \varphi _{\bullet ,\lambda }, \quad S_{\varphi }^s f(x) {}= \sum _{j,k,\lambda \in \varGamma }2^{2js} \left\langle f, \varphi _{j,k,\lambda }\right\rangle \varphi _{j,k,\lambda }(x). \end{aligned}$$

We use the representation formula presented in [2, Lemma 4.6], to rewrite

$$\begin{aligned} \left\langle S^s f(x), f(x)\right\rangle&= \left\langle \sum _{\sigma }\sum _{m\in {\mathbb {Z}}^d}4^{\sigma s} T_{{\frac{m}{\nu }\beta _{\sigma }}}\left( \overline{\varPhi _{\sigma } \widehat{f}} \right) \varPhi _{\sigma }, \widehat{f}\right\rangle _{L^2(\mathbb {R}^d)}, \end{aligned}$$
(18)

where \(\sigma \in \left\{ \bullet , \left( j,k \right) \right\} _{j,k\in \varGamma }\) and, with an abuse of notation, we set

$$\begin{aligned} \beta _{\sigma }= 2^j, \text { for } \sigma = \left( j,k \right) , \qquad \beta _{\sigma }= 1, \text { for } \sigma = \bullet . \end{aligned}$$

First, for \(m=0\) we apply (50) and obtain

$$\begin{aligned}&\left\langle \sum _{\sigma }4^{\sigma s}\frac{\left| \varPhi _{\sigma } \right| ^2}{(1+|\omega |)^{2s}} (1+|\omega |)^{s}\widehat{f}, (1+|\omega |)^{s}\widehat{f}\right\rangle _{L^2(\mathbb {R}^d)}\\&\quad = \left\langle \left[ \frac{|\varPhi _{\bullet }|^2}{(1+|\omega |)^{2s}} + \sum _{j,k} 4^{\sigma s}\frac{|\varPhi _{j,k}|^2}{(1+|\omega |)^{2s}} \right] (1+|\omega |)^s \hat{f}, (1+|\omega |)^s \hat{f}\right\rangle _{L^2(\mathbb {R}^d)}\\&\quad \ge C_s a^2 \left\| f \right\| _s^2. \end{aligned}$$

Hence,

$$\begin{aligned} \left\langle S^s f(x), f(x)\right\rangle \ge a^2 \left\| f \right\| _s^2 +\left\langle \sum _{\sigma }4^{\sigma s}\sum _{m\ne 0}T_{\frac{m}{\nu }\beta _{\sigma }}\left( \overline{\varPhi _{\sigma } \widehat{f}} \right) \varPhi _{\sigma }, \widehat{f}\right\rangle _{L^2(\mathbb {R}^d)}. \end{aligned}$$
(19)

We study the last term of the above sum by splitting the different components. Precisely,

$$\begin{aligned} \left\langle \sum _{\sigma }4^{\sigma s} \sum _{m\ne 0} T_{\frac{m}{\nu }\beta _{\sigma }}\left( \overline{\varPhi _{\sigma } \widehat{f}} \right) \varPhi _{\sigma }, \widehat{f}\right\rangle _{L^2(\mathbb {R}^d)}= R_{1,1}+R_{2,1}+R_{1,2}+R_{2,2} \end{aligned}$$

where

$$\begin{aligned} R_{i,j}= \left\langle \sum _{\sigma } 4^{\sigma s}\sum _{m\ne 0} T_{\frac{m}{\nu }\beta _{\sigma }}\left( \overline{\varPhi _{\sigma } \widehat{f_{\sigma ,i}}} \right) \varPhi _{\sigma }, \widehat{f_{\sigma ,j}} \right\rangle _{L^2(\mathbb {R}^d)}, \quad i,j = 1,2. \end{aligned}$$
(20)

and

$$\begin{aligned} f = f_{\sigma ,1} + f_{\sigma ,2}, \quad \text {and} \quad \widehat{f}_{\sigma ,1} = \widehat{f}\chi _{E_{\sigma }}, \end{aligned}$$

\(E_{\sigma }\) is defined in (13). We want to show that the terms \(R_{1,1}, R_{1,2}, R_{2,1}\) in (20) go to zero as \(\nu \) does. Precisely, we show that there exists \(\nu _0>0\) such that for each \(\nu <\nu _0\) one has

$$\begin{aligned} \frac{C_s a^2}{2}\left\| f \right\| ^2_s \ge R_{1,1}+R_{2,1}+R_{1,2}. \end{aligned}$$
(21)

Also, we know that \(R_{2,2}\ge 0\). Indeed, we can rewrite \(R_{2,2}\) in (20) as

$$\begin{aligned} \sum _{\sigma } 2^{2s\sigma }\left| \left\langle f_{\sigma ,2}, \varphi _{\sigma ,\lambda }\right\rangle _{L^2(\mathbb {R}^d)} \right| ^2, \end{aligned}$$

which is quadratic hence positive, as desired.

Step 1 Show that (20), with \(i=j = 1\) is identically zero.

Equation (20) vanishes for all \(m\ne 0\) if \(\nu \) is small, since the support of \(\widehat{f}_{\sigma ,1}\) is compact.

Indeed, assuming \(\nu <\frac{1}{2}\), if \(\omega \in E_{\sigma }\), then

$$\begin{aligned} \omega - \frac{m}{\nu }\beta _{\sigma }\notin E_{\sigma }, \quad m\in {\mathbb {Z}}^d{\backslash }\left\{ 0 \right\} , \end{aligned}$$

since the diameter of \(E_{\sigma }\) is bounded by \(2^{\sigma +1}\).

Step 2 Show that the term in (20), with \(i=2,j=1\) goes to zero as \(\nu \) does.

We rearrange the sum as

$$\begin{aligned}&\sum _{\sigma } 4^{s\sigma }\sum _{m\ne 0} \left\langle \overline{\widehat{f_{\sigma ,2}}}\left( \cdot -m \frac{\beta _{\sigma }}{\nu } \right) \varPhi _{\sigma }, {\varPhi _{\sigma }}\left( \cdot -m \frac{\beta _{\sigma }}{\nu } \right) \widehat{f_{\sigma ,1}}\right\rangle _{L^2(\mathbb {R}^d)}. \end{aligned}$$

Multiply and dividing by \((1+\left| \omega \right| )^{s},\left( 1+\left| \omega -m \frac{\beta _{\sigma }}{\nu } \right| \right) ^{s} \), since \(f_{\sigma ,1}\) vanishes outside \(E_\sigma \), we are led to

$$\begin{aligned}&\sum _{\sigma }4^{\sigma s}\sum _{m\ne 0}\int _{E_{\sigma }}\left( 1+\left| \omega -m \frac{\beta _{\sigma }}{\nu } \right| \right) ^s\overline{\widehat{f_{\sigma ,2}}}\left( \omega -m \frac{\beta _{\sigma }}{\nu } \right) \frac{\varPhi _{\sigma }(\omega )}{(1+|\omega |)^{s}}\\&\qquad \frac{\overline{\varPhi _{\sigma }}\left( \omega -m \frac{\beta _{\sigma }}{\nu } \right) }{\left( 1+\left| \omega -m \frac{\beta _{\sigma }}{\nu } \right| \right) ^s}(1+|\omega |)^{s}\overline{\widehat{f_{\sigma ,1}(\omega )}}\mathrm {d}\omega . \end{aligned}$$

Then, by Cauchy–Schwartz inequality,

$$\begin{aligned} \left\langle \sum _{\sigma } 4^{\sigma s}\sum _{m\ne 0} T_{m \frac{\beta _{\sigma }}{\nu }}\left( \overline{\varPhi _{\sigma } \widehat{f_{\sigma ,2}}} \right) \varPhi _{\sigma }, \widehat{f_{\sigma ,1}}\right\rangle _{L^2(\mathbb {R}^d)} \le \sum _{\sigma }\sum _{m\ne 0} c^{1/2}_{\sigma ,m}d^{1/2}_{\sigma ,m}, \end{aligned}$$
(22)

where

$$\begin{aligned} c_{\sigma ,m}&= \int _{E_{\sigma }}\left| \overline{\widehat{f_{\sigma ,2}}}\left( \omega -m \frac{\beta _{\sigma }}{\nu } \right) \right| ^2\left( 1+\left| \omega -m \frac{\beta _{\sigma }}{\nu } \right| \right) ^{2s} 4^{\sigma s}\frac{\left| \varPhi _{\sigma }(\omega ) \right| ^2}{(1+|\omega |)^{2s}}\mathrm {d}\omega \\&\lesssim \int _{E_{\sigma }}\left| \overline{\widehat{f_{\sigma ,2}}}\left( \omega -m \frac{\beta _{\sigma }}{\nu } \right) \right| ^2\left( 1+\left| \omega -m \frac{\beta _{\sigma }}{\nu } \right| \right) ^{2s}\mathrm {d}\omega \end{aligned}$$

and

$$\begin{aligned} d_{\sigma ,m}&= \int _{E_{\sigma }}\left| \overline{\widehat{f_{\sigma ,1}}}\left( \omega \right) \right| ^2\left( 1+\left| \omega \right| \right) ^{2s} 4^{\sigma s}\frac{\left| \varPhi _{\sigma }\left( \omega -m \frac{\beta _{\sigma }}{\nu } \right) \right| ^2}{\left( 1+\left| \omega -m \frac{\beta _{\sigma }}{\nu } \right| \right) ^{2s}}\mathrm {d}\omega \\&\le \int _{E_{\sigma }}\left| \overline{\widehat{f_{\sigma ,1}}}\left( \omega \right) \right| ^2\left( 1+\left| \omega \right| \right) ^{2s} \frac{|\beta _{\sigma }|^d}{\left( 1+\mathrm {d}\left( \omega -m \frac{\beta _{\sigma }}{\nu },I_{\sigma } \right) \right) ^{2\alpha }}\mathrm {d}\omega \\&\le \sup _{\omega \in E_{\sigma }}\left\{ \frac{|\beta _{\sigma }|^d}{\left( 1+\mathrm {d}\left( \omega -m \frac{\beta _{\sigma }}{\nu },I_{\sigma } \right) \right) ^{2\alpha }} \right\} \int _{E_{\sigma }}\left| \overline{\widehat{f_{\sigma ,1}}}\left( \omega \right) \right| ^2\left( 1+\left| \omega \right| \right) ^{2s}\mathrm {d}\omega . \end{aligned}$$

Notice that we have used inequality (38). Since

$$\begin{aligned} \mathrm {d}\left( \omega -m \frac{\beta _{\sigma }}{\nu },I_{\sigma } \right) \ge \frac{\beta _{\sigma }}{\nu }\left( |m|-2\nu \right) ,\quad \omega \in E_{\sigma }, \end{aligned}$$

if \(\nu <\frac{1}{2}\), then \(\left( |m|-2\nu \right) >0\), for any \(m\ne 0\). Therefore,

$$\begin{aligned} d_{\sigma ,m} \le \nu ^{2\alpha }\beta _{\sigma }^{(d-2\alpha )}\frac{1}{\left( |m|-2\nu \right) ^{2\alpha }} \left\| f_{\sigma ,1} \right\| ^2_s. \end{aligned}$$

Hence, using the properties of the scalar product and the norm in the space \(\ell ^2\) with parameters \((\sigma ,m)\), we can write

$$\begin{aligned} \sum _{\sigma , m\ne 0} c^{1/2}_{\sigma , m}d^{1/2}_{\sigma , m}&\lesssim \sum _{\sigma , m\ne 0 }c^{1/2}_{\sigma , m} \nu ^\alpha \beta _{\sigma }^{\frac{1}{2}(d-2\alpha )} \frac{1}{\left( |m|-2\nu \right) ^\alpha }\left\| f_{\sigma ,1} \right\| _s\nonumber \\&\lesssim \nu ^{\alpha } \left\| c^{1/2}_{\sigma , m} \beta _{\sigma }^{\frac{1}{2}(d-2\alpha )} \right\| _{\ell ^2} \left\| \frac{1}{\left( |m|-2\nu \right) ^\alpha } \left\| f_{\sigma ,1} \right\| _s \right\| _{\ell ^2}\,. \end{aligned}$$
(23)

Since \(\left\{ I_{j,k}, I_{\bullet } \right\} \) is an admissible partition, \(\left\{ E_\sigma \right\} \) is a partition of \(\mathbb {R}^d\) as well, with the uniformly finite intersection property, and since \(\alpha >d/2\)

$$\begin{aligned} \left\| \frac{1}{\left( |m|-2\nu \right) ^\alpha } \left\| f_{\sigma ,1} \right\| _s \right\| ^2_{\ell ^2}&\le \left( \sum _{m\ne 0}\frac{1}{\left( |m|-2\nu \right) ^{2\alpha }} \right) \sum _{\sigma }\left\| f_{\sigma ,1} \right\| ^2_s\nonumber \\&\lesssim \left\| f \right\| ^2_s. \end{aligned}$$
(24)

Moreover,

$$\begin{aligned} \left\| c^{1/2}_{\sigma , m} \beta _{\sigma }^{\frac{1}{2}(d-2\alpha )} \right\| ^2_{\ell ^2}&= \sum _{\sigma } \left( \beta _{\sigma }^{(d-2\alpha )} \left( \sum _{m\ne 0} c_{\sigma ,m} \right) \right) \lesssim \left\| f \right\| _s^2. \end{aligned}$$
(25)

Step 3 Show that the term in (20) with \(i=1,j=2\) goes to zero as \(\nu \) does.

This follows from the previous step using a change of variable of integration. \(\square \)

4.2 \(H^s\)-seminorm

In Theorem 3.2 we proved that \(F(\varphi _{\bullet },\varphi ,\varGamma )\) is a frame that describes the \(H^s\)-norm, provided the parameter \(\nu \) is small enough. Going through the proof it is clear that, under the hypothesis of the previous section, it is not possible to describe the \(H^s\)-seminorm as well. The problem arises, near the point \(\omega =0\), in the frequency domain. The main reason is that the partition (5) is too coarse near the origin, in particular, the set \(I_\bullet \) is too large. Therefore we need sufficiently many vanishing moments in the origin, see [20]. In Fig. 4, we show an example of window with such properties.

Fig. 4
figure 4

Frame element \(\varphi _{j,k,\lambda }(t)\) in both time and frequency. Here, \(\varphi \) is the convolution between \(\chi _{\left[ -\frac{1}{2},\frac{1}{2} \right] }\) and itself, the scale is \(j=4\). a Time domain, b frequency domain, c frequency domain, zoom in the origin

Full size image

We first introduce the dilation operator \(({\text {D}}_a f)(x)= a^{-d/2} f\left( \frac{x}{a} \right) , a\in \mathbb {R}{\setminus }\left\{ 0 \right\} ,\) which is unitary on \(L^2(\mathbb {R}^d)\). Let \(\left\{ I_{j,k} \cup I_\bullet \right\} \) be an admissible partition in the sense of Definition 2.1, we consider a new set of indices [compared to \(\varGamma \) in (7)], defined as follows:

$$\begin{aligned} \varSigma = \left\{ (j, k, \lambda ) \mid j\in {\mathbb {Z}}, k\in K_{j}, \lambda \in \nu {\mathbb {Z}}^d \right\} , \end{aligned}$$

and the sets

(26)

Remark 3

These new sets are just contractions of the original ones. We define these sets in order to refine the analysis on \(I_\bullet \). The union of these covers \(I_{\bullet }\) entirely, except the point \(\left\{ 0 \right\} \) which has zero measure.

For each \(\varphi \), we define the new frame

$$\begin{aligned} \varphi _{j,k,\lambda }(x)&=\varphi _{j,k,\lambda }(x), \quad j\in {\mathbb {N}}, k\in K_{j},\nonumber \\ \varphi _{-j,k,\lambda }(x)&= D_{2^{2j}}\varphi _{j,k,\lambda }(x) = {\text {D}}_{2^{2j}} T_{2^{-j}\lambda }\varphi _{j,k,0}(t), \quad j \in {\mathbb {N}}{\setminus } \left\{ 0 \right\} , \; k\in K_{j}. \end{aligned}$$
(27)

Recalling (10), the natural expression of \(\varPhi _{-j,k}(\omega )\) for \(j\in {\mathbb {N}}{\setminus } \left\{ 0 \right\} \) is: \( \varPhi _{-j,k}(\omega )= \varPhi _{j,k}(2^{2j}\omega ). \) Finally, we define the Stockwell-like system for the seminorm as

$$\begin{aligned} \mathcal {G}(\varphi ,\varSigma ) = \left\{ \varphi _{j,k,\lambda }, \, j,k,\lambda \in \varSigma \right\} . \end{aligned}$$
(28)

We introduce now the admissibility criteria for the seminorm characterisation.

Definition 3.3

(s-admissibility for \(H^s\)-seminorms, finite case) Let \(s\ge 0\), we say that \(\varphi \) is s-admissible for the Sobolev seminorm with respect to the partition \(\left\{ I_{j,k} \right\} _{j\in {\mathbb {Z}}, k\in K_{j}}\) if

  1. (i)

    there exists \(\alpha > d/2\) such that for all \(j\in {\mathbb {N}}\)

    $$\begin{aligned} |\varPhi _{j,k}(\omega )|\lesssim \left\{ \begin{array}{ll} \frac{2^{jd/2}\min \left( 1,|\omega | \right) ^s }{\left( 1+d(\omega , I_{j,k}) \right) ^{\alpha +s}},&{} \quad \omega \notin I_{j,k}\\ 1, &{}\quad \omega \in \mathbb {R}^d\,; \end{array} \right. \end{aligned}$$
    (29)
  2. (ii)

    there exists a constant \(a>0\) such that

    $$\begin{aligned} |\varPhi _{j,k}(\omega )|\ge a, \quad \omega \in I_{j,k}. \end{aligned}$$

    (the constant a does not depend on j, k).

We state the counterpart of Theorem 3.2 for seminorms.

Theorem 3.4

Let \(s\ge 0\) and consider an s-admissible couple of functions \(\varphi _{\bullet },\varphi \). Then there exists \(\nu _0>0\) such that for each \(\nu \in (0,\nu _0)\), the system \(\mathcal {G}\left( \varphi ,\varSigma \right) \) defined in (28) is a frame representing the \(H^s(\mathbb {R}^d)\) seminorm. That is, there exist constants \(A, B>0\) such that

$$\begin{aligned} A \left| f \right| ^2_s\le \sum _{j,k,\lambda \in \varSigma } 2^{2js}\left| \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right| ^2 \le B \left| f \right| ^2_s. \end{aligned}$$
(30)

Proof

The proof is a consequence of the upper bound and of the lower bound as in Theorem 3.2. We avoid an explicit proof here. Nonetheless, in the “Appendix”, we provide the counterparts of the properties used in the first part so that the entire proof can be reproduced. \(\square \)

4.3 Characterisation of Besov spaces

We discuss here an important consequence of Theorem 3.4; precisely we show how to characterise Besov spaces of type \(B^s_{2,q}\) with \(s\in (0,1]\) and \(q>1\), using (real) interpolation techniques. We recall that a function f belongs to the Besov space \(B^s_{2,q}(\mathbb {R}^d)\subseteq L^2(\mathbb {R}^d)\), if the following seminorm is finite

$$\begin{aligned} \left| f \right| _{B^s_{2,q}} = \left( \int _0^{+\infty }[t^{-s}\omega ^{(r)}_2(f,t)]^q \frac{\text {d}t}{t} \right) ^{\frac{1}{q}}, \end{aligned}$$

with \(r\in {\mathbb {N}}, r>s\), where

$$\begin{aligned} \omega ^{(r)}_2(f,t) = \sup _{\left| h \right| <t}\left\| \varDelta ^r_h f(\cdot ) \right\| _2, \end{aligned}$$

and \(\varDelta ^r_h \) is the difference of order r and step h

$$\begin{aligned} \varDelta ^r_h f(x) = \sum _{j=1}^r(-1)^{r+j}\left( {\begin{array}{c}r\\ j\end{array}}\right) f(x+jh). \end{aligned}$$

A norm can be defined as follows

$$\begin{aligned} \left\| f \right\| _{B^s_{2,q}} = \left\| f \right\| _2+\left| f \right| _{B_{2,q}^s}. \end{aligned}$$

Theorem 3.5

Consider the Stockwell frame \(\left\{ \varphi _{j,k,\lambda } \right\} _{j,k,\lambda \in \varSigma }\) as defined in Sect. 3.2 and let \(s\in \left( 0,1 \right) , q>1\); then for any \(v\in B^s_{2,q}\) one has

$$\begin{aligned} \left| v \right| _{B^{s}_{2,q}}\asymp \left( \sum _{j\in {\mathbb {Z}}}2^{js q}\left( \sum _{k,\lambda \in \varDelta _j}\left| \left\langle \varphi _{j,k,\lambda }, v\right\rangle \right| ^2 \right) ^{q/2} \right) ^{1/q}. \end{aligned}$$

Proof

The result follows from Theorem B.1 and the fact that for \(\alpha \in (0,1)\) and \(q\in (1,+\infty ]\),

$$\begin{aligned} B^{s}_{2,q}(\mathbb {R}) = (L^2(\mathbb {R}),H^1(\mathbb {R}))_{s,q} \end{aligned}$$

(see e.g. [19]). \(\square \)

4.4 Sufficient conditions for Sobolev-like frames

The definition of s-admissible window function involves properties of \(\varPhi _{j,k}\) rather than of the window function \(\varphi \). In general, it is very difficult to check the properties of \(\varPhi _{j,k}\) since it is a sum of translations (recall Definition 3.1). Nevertheless, it is possible to provide sufficient conditions on \(\varphi \) which guarantee the s-admissibility. In the following we will always consider \(\varphi =\varphi _{\bullet }\).

Theorem 3.6

Let \(\varphi \) be a function such that

$$\begin{aligned} \left| \widehat{\varphi }(\omega ) \right| \lesssim \frac{1}{\left( 1+\omega \right) ^{s+d+\varepsilon }}, \quad \varepsilon >0, \end{aligned}$$
(31)

for a certain \(s\in \mathbb {R}\) and such that \(\widehat{\varphi }\) has definite sign and \(|\widehat{\varphi }(\omega )|\ge a,\) for \( \omega \in I_{\bullet }\) then the pair \((\varphi ,\varphi )\) is s-admissible for \(H^s\)-norm.

Proof

Condition (31) implies that \(\widehat{\varphi }\) belongs to \(L^1(\mathbb {R}^d)\), moreover

$$\begin{aligned} \left| \varPhi _{j,k}(\omega ) \right|&\le \sum _{\eta \in Z_{j,k}} \left| \widehat{\varphi }(\omega -\eta ) \right| \lesssim \sum _{\eta \in Z_{j,k}} \frac{1}{\left( 1+|\omega -\eta | \right) ^{s+d+\varepsilon }}\\&\lesssim \int \frac{1}{\left( 1+|\omega | \right) ^{d+\varepsilon }} d\omega \lesssim 1. \end{aligned}$$

Therefore, \(\left| \varPhi _{j,k}(\omega ) \right| \) is uniformly bounded.

In order to prove the decay property, notice again that the issue is to have a uniform bound with respect to j. Since \(\varPhi _{j,k}(\omega )\) is uniformly bounded, we can prove the inequality when \(d(\omega , I_{j,k}) = \varepsilon >0\). Under this hypothesis, recalling that, \(|Z_{j,k}|\asymp 2^{jd}\) we can write

$$\begin{aligned} \left| \varPhi _{j,k}(\omega ) \right|&\le \sum _{\eta \in Z_{j,k} }\left| \widehat{\varphi }(\omega -\eta ) \right| \lesssim \sum _{\eta \in Z_{j,k}} \frac{1}{\left( 1+|\omega -\eta | \right) ^{s+d+\varepsilon }}\\&\lesssim \sum _{|r|<2^{j}} \frac{1}{\left( 1+ \varepsilon +|r| \right) ^{s+d+\varepsilon }}. \end{aligned}$$

If \(\varepsilon < 2^{j-1}\), we notice that this sum may be bounded by the corresponding integral and, after a change of variable,

$$\begin{aligned} \left| \varPhi _{j,k}(\omega ) \right|&\lesssim \frac{1}{(1+\varepsilon )^{s+d+\varepsilon }} \int _{|x|<2^j}\frac{1}{\left( 1+\frac{x}{1+\varepsilon } \right) ^{s+d/2+\varepsilon }}\\&\lesssim \frac{(1+\varepsilon )^d}{(1+\varepsilon )^{s+d+\varepsilon }} = \frac{(1+\varepsilon )^{d/2}}{(1+\varepsilon )^{s+d/2+\varepsilon }} \end{aligned}$$

due to the convergence of the integral on the whole \(\mathbb {R}^d\) space. Thus, upon noticing that \((1+\varepsilon )^{d/2} \lesssim 2^{jd/2}\), one gets

$$\begin{aligned} \left| \varPhi _{j,k}(\omega ) \right| \lesssim \frac{2^{jd/2}}{(1+\varepsilon )^{s+d/2+\varepsilon }} \le \frac{2^{jd/2}}{(1+d(\omega , I_{j,k}))^{s+d/2+\varepsilon }} \end{aligned}$$

as desired.

On the contrary, if \(\varepsilon > 2^{j-1}\), then

$$\begin{aligned} \left| \varPhi _{j,k}(\omega ) \right| \lesssim \sum _{|r|<2^{j}} \frac{1}{\left( 1+ \varepsilon +|r| \right) ^{s+d+\varepsilon }} \le \frac{2^{jd}}{\left( 1+\varepsilon \right) ^{s+d+\varepsilon }} \lesssim \frac{2^{jd/2}}{\left( 1+d(\omega , I_{j,k}) \right) ^{s+d/2+\varepsilon }}, \end{aligned}$$

since

$$\begin{aligned} \frac{2^{jd/2}}{\left( 1+d(\omega , I_{j,k}) \right) ^{d/2}}\lesssim 1. \end{aligned}$$

To conclude, we need to show that \(\varPhi _{j,k}\) satisfies condition (12), since for \(\varPhi _{\bullet }\) there is nothing to prove. Notice that, by hypothesis,

$$\begin{aligned} |\varPhi _{j,k}(\omega )|=\sum _{\eta \in Z_{j,k}}\left| \widehat{\varphi }(\omega -\eta ) \right| \end{aligned}$$

and that for all \(\omega \in I_{j,k}\) there exists \(\bar{\eta } \in Z_{j,k}\) such that \(\omega -\bar{\eta }\in I_{\bullet }\), therefore

$$\begin{aligned} |\varPhi _{j,k}(\omega )|=\sum _{\eta \in Z_{j,k} }\left| \varphi (\omega -\eta ) \right| \ge \left| \varphi (\omega -\bar{\eta }) \right| \ge a. \end{aligned}$$

\(\square \)

Fig. 5
figure 5

Time and frequency outlook of a frame element with normalised Gaussian window. On the right the real part in the time domain, on the left the absolute value in the frequency domain. a Time domain, \(j=4\), b frequency domain, \(j=4\)

Full size image

The easiest example of a function satisfying the conditions of Theorem 3.6 is the Gaussian, see Fig. 5.

We can obtain similar sufficient conditions for the seminorm by adding a vanishing condition on non-zero integers. Indeed, summing integer translations of window functions will generate vanishing moments at the origin as required by the s-admissibility condition. We show an example using \({\text {sinc}}(\cdot )\) in Sect. 4.2 below.

5 Explicit examples of \(H^s\)-frames

In this section we provide some explicit examples of frames which discretise the \(H^s\)-norm and seminorm.

5.1 Shannon-like basis

Let us consider \(\varphi _{\bullet }(t)=\varphi (t) ={\text {sinc}}(t)=\left( {\text {F}}^{-1}\chi _{\left( -\frac{1}{2},\frac{1}{2} \right) } \right) (t)\) and the partition introduced in (5); see Fig. 6. Since the characteristic function has compact support, the couple \(\varphi _{\bullet }, \varphi \) is trivially s-admissible for all \(s\ge 0\) both for the Sobolev norm and seminorm, hence a frame for \(\nu \) small enough.

Fig. 6
figure 6

Frame element \(\varphi _{j,k,\lambda }(t)\) in both time and frequency, \(\varphi = {\text {sinc}}(t)\) and \(j=4\). a Time domain, b frequency domain

Full size image

This example is indeed very peculiar. This system is not only a frame for \(\nu \) small enough, but it is an orthonormal basis of \(L^2(\mathbb {R})\) if we set \(\nu =1\). Indeed, notice that, using the notations of Sect. 2

$$\begin{aligned} \varPhi _{\bullet }(\omega )=\chi _{(-\frac{1}{2},\frac{1}{2})}(\omega ), \quad \varPhi _{j,\pm }(\omega )= \sum _{\eta = \pm 2^j}^{\pm (2^{j}-1)} \chi _{\left( -\frac{1}{2}, \frac{1}{2} \right) }(\omega -\eta )= \chi _{I_{j,\pm }}(\omega ). \end{aligned}$$

Therefore, if \(j\ne j'\) or \(k\ne k'\), by Plancherel Theorem

$$\begin{aligned} \left\langle \varphi _{j,k,\lambda }, \varphi _{j',k',\lambda '}\right\rangle&= \left\langle \frac{1}{2^{j/2}} e^{-2\pi \mathrm {i}\,\cdot \lambda /2^{j}} \chi _{I_{j,k}}(\cdot ) , \frac{1}{2^{j'/2}} e^{-2\pi \mathrm {i}\,\cdot \lambda '/2^{j'}} \chi _{I_{j',k'}} (\cdot )\right\rangle =0 \end{aligned}$$

for all \(\lambda , \lambda '\in {\mathbb {Z}}\). Moreover if \(j=j'\) and \(k=k'\), by well known properties of Fourier series,

$$\begin{aligned} \left\langle \varphi _{j,k,\lambda }, \varphi _{j,k,\lambda '}\right\rangle = \left\langle \frac{1}{2^{j/2}} e^{-2\pi \mathrm {i}\,\cdot \lambda /2^{j}} \chi _{I_{j,k}}(\cdot ) , \frac{1}{2^{j/2}} e^{-2\pi \mathrm {i}\,\cdot \lambda '/2^{j}} \chi _{I_{j,k}} (\cdot )\right\rangle = \delta _0, \end{aligned}$$

where \(\delta _0\) is the Dirac delta. The same holds for \(\varphi _{\bullet ,\lambda }\) which is also orthogonal to \(\varphi _{j,k,\lambda }\) for all \(j,k,\lambda \).

Remark 4

The orthonormal basis system we introduced is strictly related to the Shannon basis. As expected the localisation properties in the time domain of this frame are not so strong, due to the lack of such properties of the \({\text {sinc}}\) function. Nevertheless, in our setting, we can gain localisation considering powers of \({\text {sinc}}\). That is, we can consider \(\varphi (t)={\text {sinc}}(t)^n\) and this new window has increasing localisation as n increases, moreover it is always s-admissible since its Fourier transform has compact support (see Fig. 2).

5.2 Haar-like frame

In this subsection we describe in detail an example in dimension \(d=1\) which shows that the conditions of Theorem 3.6 are sufficient, but not necessary.

Let us consider

$$\begin{aligned} \varphi (t)=\chi _{\left( -\frac{1}{2},\frac{1}{2} \right) }(t)=\left( {\text {F}}^{-1} {\text {sinc}}(\cdot ) \right) (t)=\left( {\text {F}}^{-1} \frac{\sin (\pi \cdot )}{\pi \cdot } \right) (t). \end{aligned}$$
(32)

It is clear that \(\varphi \) defined in (32) does not satisfy the sufficient conditions (31). Nevertheless, it will provide a frame. See Fig. 7 for the plots of the frame with this particular window function. One can prove that the pair \((\varphi ,\varphi )\) is s-admissible for the Sobolev seminorm for each \(s\in \left[ 0, 1\right) \).

Fig. 7
figure 7

Frame element \(\varphi _{j,k,\lambda }(t)\) in both time and frequency. Here, \(\varphi = \chi _{\left[ -\frac{1}{2},\frac{1}{2} \right] }\) and \(j=4\). a Time domain, b frequency domain

Full size image

It is enough to prove the decay property for \(j\ge 0\).

Let us suppose \(k=+\), the case \(k=-\) is equivalent. By definition

$$\begin{aligned} \varPhi _{j,+}(\omega )&=\sum _{\eta =2^{j}}^{2^{j+1}-1} {\text {sinc}}(\omega -\eta )=\sum _{\eta =2^{j}}^{2^{j+1}-1} \frac{\sin \left( \pi \left( \omega -\eta \right) \right) }{\pi \left( \omega -\eta \right) }. \end{aligned}$$

If \(\omega \in I_{j,+}\) then, by construction, there exists \(\bar{\eta }\in Z_{j,+}\) such that \(|\omega -\bar{\eta }|\le \frac{1}{2}\), hence

$$\begin{aligned} \left| \frac{ \sin \left( \pi (\omega -\bar{\eta }) \right) }{\pi \left( \omega -\bar{\eta } \right) } \right| \le 1. \end{aligned}$$

We can write, using trigonometric inequalities,

$$\begin{aligned} \left| \varPhi _{j,+}(\omega ) \right|&=\left| \sum _{\eta =2^{j}}^{2^{j+1}-1} \frac{\sin \left( \pi (\omega -\eta ) \right) }{\pi (\omega -\eta )} \right| \\&\le \left| \sum _{m =1}^{\bar{\eta }-2^{j}} \frac{(-1)^m}{\pi (\omega -\eta +m)} \right| +\left| \sum _{m =1}^{2^{j+1}-\bar{\eta }} \frac{(-1)^m}{\pi (\omega -\eta -m)} \right| +1. \end{aligned}$$

Since the alternate harmonic series is convergent, we can obtain a uniform bound with respect to j for \(\varPhi _{j,+}(\omega )\), if \(\omega \in I_{j,+}\). Notice that, if \(\omega \not \in I_{j,+}\) the above inequality still holds; actually one could improve the bound, but this is not important for our purpose. In order to prove (29), in view of the uniform bound of \(\varPhi _{j,+}\) we can suppose \(d(w,I_{j,+})>2^{j-1}\), as shown above. Therefore

$$\begin{aligned} \left| \varPhi _{j,+}(\omega ) \right|&= \left| \sum _{m =0}^{2^{j}-1} \frac{\sin \left( \pi \omega \right) (-1)^m}{\pi (\omega -m-2^{j})} \right| \\&\le \left| \sin \left( \pi \omega \right) \sum _{m =0}^{2^{j-1}-1} \frac{ 1 }{\pi (\omega -m-2^{j})(\omega -m-1-2^{j})} \right| \\&\le 2^{j} \left| \sin \left( \pi \omega \right) \frac{ 1 }{\pi d(\omega , I_{j,+})^2} \right| \lesssim 2^{j/2} \left| \sin \left( \pi \omega \right) \frac{ 1 }{\pi d(\omega , I_{j,+})^{\frac{3}{2}}} \right| , \end{aligned}$$

which implies (29) for each \(s\in [0,1)\).

With the same notation as above, notice that \( \displaystyle \left| \frac{ \sin \left( \pi (\omega -\bar{\eta }) \right) }{\pi \left( \omega -\bar{\eta } \right) } \right| \ge \frac{2}{\pi }.\) Hence

$$\begin{aligned} \left| \varPhi _{j,+}(\omega ) \right|&\ge \frac{2}{\pi } -\left| \sin \left( \pi (\omega -\bar{\eta }) \right) \left( \sum _{m =1}^{\bar{\eta }-2^{j}} \frac{(-1)^m}{\pi (\omega -\eta +m)} + \sum _{m =1}^{2^{j+1}-\bar{\eta }} \frac{(-1)^m}{\pi (\omega -\eta -m)} \right) \right| \\ {}&\ge \frac{2}{\pi }- \frac{1}{2}>0. \end{aligned}$$

Remark 5

This example is closely related to the standard Haar basis. It is well known that the Haar basis is suited to represent Sobolev spaces \(H^s(\mathbb {R})\) with \(s<\frac{1}{2}\), essentially due to the lack of continuity in the time domain which is related to the slow decay at infinity of the function \({\text {sinc}}\). With our approach we can get rid of this problem and reach all \(s<1\), the main reason is that performing sums instead of dilation we are able to exploit the oscillation behavior of the \({\text {sinc}}\) function and increase the decay rate of the functions \(\varPhi _{j,k}\).

6 Extension to a wider class of partitions

In this section we discuss the extension of the Stockwell-like frame defined in Sect. 3 to more general partitions. Specifically, so far we only considered a fixed amount of direction, while here we relax that requirement.

We start with a preliminary definition.

Definition 5.1

Let \(\left\{ X_\ell \right\} _{\ell \in L}\), be a family of subsets of \(\mathbb {R}^d\) with L index sets at most countable. Let us denote by \(v_\ell \) the volume of each \(X_{\ell }\) and define two constants \(c_1,c_2\) such that \(0< c_1 <c_2\). Then \(\left\{ X_\ell \right\} _{\ell \in L}\) is an almost-isotropic family of sets if for each \(\ell \in L\) there exist a point \(x_{\ell }\in X_\ell \) such that

$$\begin{aligned} B_{c_1 v_{\ell }^{1/d}}(x_{\ell }) \subseteq X_{\ell } \subseteq B_{c_2 v_{\ell }^{1/d}}(x_{\ell }), \end{aligned}$$

uniformly in \(\ell \), where \(B_{r}(x)\) is the ball of radius r and center x.

See Fig. 8 for a simple example of isotropic set.

Fig. 8
figure 8

A simple example of almost-isotropic set. In blue and red the inner and outer ball, respectively; in black the set \(X_{\ell }\) (colour figure online)

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Remark 6

With this definition, we guarantee that every point on the boundary of \(X_{\ell }\) has a comparable distance from the centroid of the set itself, and this distance is roughly \(v_{\ell }^{1/d}\) modulo uniform constants. This requirement is crucial in order to prove technical results such as the generalization of Lemmas A.4 and A.5.

We now state here a general definition and later give some examples of admissible partitions with growing number of frequency directions.

Definition 5.2

The family \(\left\{ I_{j,k} \right\} _{j\in {\mathbb {N}}, k\in K_{j}} \cup I_{\bullet }\), where \(K_{j}\) is an index set, is an admissible partition if it satisfies (i)–(vi) of Definition 2.1 and moreover:

  1. (vii)

    there exists \(\delta \), with \(0\le \delta \le \min {\left( 1,d-1 \right) }\), such that \(|K_{j}| \asymp 2^{|j|\delta (d-1)}\);

  2. (viii)

    \(\left\{ I_{j,k} \right\} _{j,k}\) is a family of almost isotropic sets, see Definition 5.1 below.

Let us give an example of such partitions which is similar to the one presented in [4]. For \(j \ge 0\) we let \(\left\{ c_{j,k}{:}\,j>0, k\le n_j \right\} \subseteq \mathbb {R}^d, d>1\), for some \(n_j>2\), be a set of points such that:

  1. 1.

    the elements of the family \(\left\{ B_{2^{j/2}}(c_{j,k}), k=1,\ldots , n_j \right\} \) are pairwise disjoint sets and are contained in the corona \(C_j = B_{2^{j+1}}(0) {\setminus } B_{2^{j}}(0)\);

  2. 2.

    \(C_j \subseteq \bigcup _{k=1}^{n_j} B_{2^{(j+1)/2}}(c_{j,k})\).

We set \(I_{j,k} =B_{2^{(j+1)/2}}(c_{j,k})\), see Fig. 9. Then, from a volume estimate, it is easy to see that the number of sets \(n_j\) inside each corona is \(n_j \asymp 2^{jd/2}\). This is clearly an admissible partition which satisfies the requirements with \(\delta = \frac{d}{2(d-1)}\).

Fig. 9
figure 9

Isotropic tailing of the frequency plane related to isotropic frames

Full size image

6.1 Frame generalisation

Let \(Z_{j,k} = I_{j,k}\cap {\mathbb {Z}}^d\) and let \(\beta _{j,k} \in \mathbb {R}^d\) be the diameter of \(I_{j,k}\). Then the Stockwell system depending on the parameters \((j,k,\lambda ) \in \varGamma ,\, x\in \mathbb {R}^d\) is defined to be

$$\begin{aligned} \varphi _{\bullet ,\lambda }(x) = \varphi _{\bullet }(x - \lambda ),\, \varphi _{j,k,\lambda }(x)=T_{\beta _{j,k}^{-1}\lambda } \left( \frac{1}{\left| \beta _{j,k} \right| ^{d/2} } \sum _{\eta \in Z_{j,k}} e^{2\pi \mathrm {i}\,\eta \cdot x } \varphi (x) \right) , \end{aligned}$$

which can be written in compact form as

$$\begin{aligned} \mathcal {S}(\varphi _{\bullet },\varphi ,\varGamma ) = \left\{ \varphi _{\bullet ,\lambda }, \lambda \in \nu {\mathbb {Z}}^d \right\} \cup \left\{ \varphi _{j,k,\lambda }, \, j,k,\lambda \in \varGamma \right\} , \end{aligned}$$
(33)

with

$$\begin{aligned} \begin{aligned} \widehat{{\varphi }_{\bullet ,\lambda }} (\omega )&= e^{-2\pi \mathrm {i}\,\omega \cdot \lambda } \widehat{\varphi _{\bullet }}( \omega ), \\ \widehat{\varphi _{j,k,\lambda }}(\omega )&= e^{-2\pi \mathrm {i}\,\omega \cdot \beta _{j,k}^{-1}\lambda } \left( \frac{1}{\left| \beta _{j,k} \right| ^{d/2}} \sum _{\eta \in Z_{j,k}} \widehat{\varphi }(\omega -\eta ) \right) .\\ \end{aligned} \end{aligned}$$
(34)

Remark 7

Consider a scale j, we want to determine the number of points with integer coordinates contained in \(I_{j,k}\), i.e. \( |Z_{j,k}|\). The number of such points contained in the union \(\bigcup _{k \in K_j} I_{j,k}\) is approximately \(|K_{j}|\left| I_{j,k} \right| \asymp 2^{jd}\). Thus

$$\begin{aligned} \left| Z_{j,k} \right| \asymp \frac{1}{|K_{j}|}2^{dj} \asymp 2^{jd - j\delta (d-1)}= 2^{jd(1-\delta ) +j\delta }. \end{aligned}$$

Definition 3.1 generalises to the following one.

Definition 5.3

(s-admissibility for \(H^s\)-norms—infinite case) Consider an admissible partition \(\left\{ I_{j,k} \right\} _{j,k \in \varGamma } \cup I_\bullet \) and \(s\ge 0\), we say that a pair of functions \(\varphi _{\bullet },\varphi \) is s-admissible with respect to the partition if, given \(\varPhi _{\bullet },\varPhi _{j,k}\) as in (10), then

  1. (i)

    there exists \(\alpha > \frac{d}{2(1 - \delta \frac{d-1}{d})}\)

    $$\begin{aligned} |\varPhi _{\bullet }(\omega )| \lesssim \frac{1}{(1+|\omega |)^{\alpha }},\quad \text { and }\quad |\varPhi _{j,k}(\omega )|\lesssim \left\{ \begin{array}{ll} \frac{(\beta _{j,k})^{d/2}}{\left( 1+d(\omega , I_{j,k}) \right) ^{\alpha +s}},&{} \quad \omega \notin I_{j,k}\\ 1, &{}\quad \omega \in \mathbb {R}^d \end{array} \right. ; \end{aligned}$$
    (35)
  2. (ii)

    there exists \(a>0\) such that, for all \(\omega \in \mathbb {R}^d\) then

    $$\begin{aligned} \left| \varPhi _{\bullet }(\omega ) \right| \ge a, \text{ if } \omega \in I_{\bullet }, \qquad \left| \varPhi _{j,k}(\omega ) \right| \ge a, \text{ if } \omega \in I_{j,k} \end{aligned}$$
    (36)

    (the constant a does not depend on j, k).

Remark 8

Notice that the decay rate increases with the number of directions allowed, in particular, we ask \(\alpha > d/2\) for \(\delta =0\), \(\alpha > \frac{2d^2}{d+1}\) for \(\delta =\frac{1}{2}\) and \(\alpha >d^2/\) if \(\delta =1\).

With these new definitions, we can generalise Theorem 3.2.

Theorem 5.4

Let \(s\ge 0\) and consider an s-admissible set of functions \(\varphi _{\bullet },\varphi \)—cf. Definition 5.3. Then there exists \(\nu _0>0\) such that for each \(\nu \in (0,\nu _0)\) the system \( \mathcal {S}\left( \varphi _{\bullet }, \varphi ,\varGamma \right) \) defined in (33) is a frame representing the \(H^s(\mathbb {R}^d)\) norm. Precisely, there exist \(A,B>0\) such that for each \(f\in H^s(\mathbb {R}^d)\)

$$\begin{aligned} A\left\| f \right\| ^2_s \le \sum _{\lambda \in \nu {\mathbb {Z}}^d}| \left\langle f, T_{\lambda } \varphi _{\bullet }\right\rangle |^2 + \sum _{j,k,\lambda \in \varGamma }2^{2js}\left| \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right| ^2 \le B\left\| f \right\| ^2_s. \end{aligned}$$

Proof

The proof of this result follows the one of Theorem 3.2, since all the technical results extend to the general definition of isotropic partition and s-admissibility. The main difference, which is also the key point of the proof, is to show that the summability of the series in (16) is preserved. This can be shown as follows.

Since \(\beta _{j,k}^d= |Z_{j,k}|\lesssim \frac{2^{jd}}{|K_{j}|}\), then

$$\begin{aligned} \sum _{j,k}\beta _{j,k}^{d-2\alpha } \lesssim \sum _{j}|K_{j}|\beta _{j,k}^{d-2\alpha } \lesssim \sum _{j}2^{jd}\left( \beta _{j,k} \right) ^{-2\alpha }. \end{aligned}$$

Notice that, making \(\delta \) explicit

$$\begin{aligned} \beta _{j,k} \asymp 2^{j - \delta j\frac{d-1}{d}},\qquad \left( \beta _{j,k} \right) ^{2\alpha } \asymp 2^{ 2\alpha j(1 - \delta \frac{d-1}{d})}. \end{aligned}$$

Finally, the sum converges if

$$\begin{aligned} 2\alpha \left( 1 - \delta \frac{d-1}{d}\right)> d,\qquad \text { i.e. } \alpha > \frac{d}{2\left( 1 - \delta \frac{d-1}{d}\right) }, \end{aligned}$$

as assumed.

Similarly, one can extend the case of the seminorm using the same pattern.

7 Conclusion

In this paper we focused on the Sobolev properties of the Stockwell-like frame in arbitrary dimension obtaining a characterisation of these spaces. Although this is a standard property that many other types of frames share, we believe that the extreme flexibility of this frame opens interesting research paths. Specifically, this frame can be adapted to a variety of different partitions. Hence, we claim that our approach can be used to describe decomposition spaces associated to different partitions of the frequency plane.

It is clear from the result on the Sobolev seminorms, that this frame is very close to the wavelet one. With our frame, it is possible to construct a structure close to the wavelet Multi Resolution Analysis and use that to represent interpolation spaces. As an example, we showed the characterisation of Besov spaces.

In Sect. 4 we proposed several examples of frame windows; in particular, we considered the powers of the \({\text {sinc}}\) function, that gives arbitrary high regularity while providing great localisation in space and being just a perturbation of the characteristic function of the frequency band in the Fourier domain.

The example of the Haar-like Stockwell frame presented in Sect. 4, shows that even if we start with the same characteristic window as in the Haar wavelets, we can reach higher regularity.

In Sect. 5, we provided a generalisation to a frequency tiling that allows isotropic sets with a (parabolic) number of frequency directions paired with the scale j. It is clear that the machinery presented here is not tailored to treat anisotropic partitions; this is a possible topic for future researches.

Among other interesting open problems on this frame, it would be challenging to investigate the density of the Stockwell frame with Gaussian window and compare it with well known results on Gabor frames.

Concerning numerics, we implemented a new algorithm for Stockwell-like frames. We aim to test this on numeric problems and compare it to other existing frames. The code is available for tests and academic use upon request to the authors.

One of the possible applications of our frame is medical imaging, since the Stockwell transform is widely used in that research area, see [3, 22] for recent results on this matter.