Stockwell-like frames for Sobolev spaces

Abstract

We construct a family of frames describing the norm and seminorm of the space $H^s(\mathbb {R}^d)$. We also characterise Besov spaces modeled on $L^2(\mathbb {R}^d)$. Our work is inspired by the discrete orthonormal Stockwell transform introduced by R.G. Stockwell, which provides a time-frequency localised version of the Fourier basis of $L^2([0,1])$. This approach is a hybrid between Gabor and Wavelet frames. We construct explicit and computable examples of these frames, discussing their properties and comparing them with the existing literature.

Appendices

Technical results

We show some technical result needed to prove the frame property.

Lemma A.1

Let $s\ge 0$ and $\varphi$ such that $\varPhi _{j,k}$—cf. (10)—satisfies

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

(37)

for some $\alpha$. Then

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

(38)

Proof

Inequality (38) is trivially verified when $\omega \in I_{j,k}$ since $\omega \asymp 2^j$.

Assume $\omega \notin I_{j,k}$, then we immediately notice that by hypothesis (37)

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

Now, there exists $\overline{\omega }\in I_{j,k}$ such that $d(\omega ,I_{j,k}) = |\omega - \overline{\omega }|$, moreover $|\overline{\omega }|\asymp 2^j$ because the partition is admissible. Hence, by triangular inequality

$$\begin{aligned} \frac{2^{js}}{(1+|\omega |)^s} \left| \varPhi _{j,k}(\omega ) \right|&\lesssim \frac{2^{js}}{(1+|\omega |)^s \left( 1+|\overline{\omega }-\omega | \right) ^s} \frac{2^{jd/2}}{\left( 1+d(\omega ,I_{j,k}) \right) ^{\alpha }}\\&\lesssim \frac{2^{js}}{ \left( 1+|\overline{\omega }| \right) ^s} \frac{2^{jd/2}}{\left( 1+d(\omega ,I_{j,k}) \right) ^{\alpha }}\lesssim \frac{2^{jd/2}}{\left( 1+d(\omega ,I_{j,k}) \right) ^{\alpha }}. \end{aligned}$$

$\square$

With the same argument, one can show the following result.

Lemma A.2

Let $s\ge 0$ and $\varphi$ such that for all $j \in {\mathbb {N}}$, $\varPhi _{j,k}$—cf. (10)—satisfies

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

(39)

for some $\alpha >0$. Then

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

(40)

for all $j \in {\mathbb {N}}$.

As a consequence of Lemma A.1, we have the following result.

Lemma A.3

Assume the hypothesis (37) of Lemma A.1 above and also that $\alpha > d/2$. Then, there exists $b_s \in \mathbb {R}$ such that

$$\begin{aligned} \sum _{j,k\in \varGamma }\frac{2^{2js}}{(1+|\omega |)^{2s}} |\varPhi _{j,k}(\omega )|^2\le b_s, \quad \text{ a.e. } \omega \in \mathbb {R}^d. \end{aligned}$$

(41)

Proof

Given $\omega \in \mathbb {R}^d$, since the partition is admissible, there exists a finite collection of indices $\varGamma _F, |\varGamma _F|<W_1$ such that $\omega \in I_{j,k}, \{j,k\}\in \varGamma _F$. Assume for the moment $W_1 = 1$ and $\omega \in I_{\overline{j},\overline{k}}$. First, we want to estimate

$$\begin{aligned} \sum _{k\in K_j }\frac{2^{2js}}{(1+|\omega |)^{2s}} |\varPhi _{j,k}(\omega )|^2, \end{aligned}$$

for $j=\overline{j}$. Clearly, $\omega \asymp 2^{\overline{j}}$, hence $\frac{2^{2js}}{(1+|\omega |)^{2s}} |\varPhi _{\overline{j},k}(\omega )|^2\lesssim 1$. Moreover, since the number of possible directions is bounded by $C_K$,

$$\begin{aligned} \sum _{k\in K_{\overline{j}}}\frac{2^{2\overline{j}s}}{(1+|\omega |)^{2s}} |\varPhi _{\overline{j},k}(\omega )|^2 \lesssim C_K. \end{aligned}$$

(42)

With this majorisation, we can treat also the cases of the adjacent coronae, i.e. for $j = {\overline{j}\pm 1}$. If $j> \overline{j}+ 1$ or $j < \overline{j}- 1$, then $\mathrm {d}(\omega , I_{j,k})\gtrsim 2^j$, and the result follows as above. Hence, if $W_1=1$, the result follows easily by the requirement on $\alpha$. Indeed, using Lemma A.1, one gets

$$\begin{aligned} \sum _{|j-\overline{j}|>1, k\in K_j }\frac{2^{2js}}{(1+|\omega |)^{2s}} |\varPhi _{j,k}(\omega )|^2 \le C_K\left( 3 + \sum _{j}2^{j(d-2\alpha )} \right) , \end{aligned}$$

(43)

which is clearly bounded. Now we can repeat this argument a finite number of time if $W_1>1$ and the result follows. $\square$

Lemma A.4

Let $s\ge 0$ and $\varphi$ such that $\varPhi _{j,k}(\omega )$ satisfies hypothesis (37). Define

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

(44)

then for $\nu \in (0,1]$ and all j, k the system of functions $\left\{ \widetilde{\varphi _{j,k,\lambda }}(t) \right\} _{\lambda \in \nu {\mathbb {Z}}}$ is a Bessel sequence uniformly in j, k, that is

$$\begin{aligned} \sum _{\lambda \in \nu {\mathbb {Z}}} \left| \left\langle \widetilde{\varphi _{j,k,\lambda }}, f\right\rangle \right| ^2\le C_{\nu } \left\| f \right\| ^2_{L^2(\mathbb {R}^d)} \end{aligned}$$

(45)

with $C_{\nu }$ independent on j, k.

Proof

A well known result (see again e.g. [9, Thm 9.2.5, p.206]) states that the Bessel property (45) of $\left\{ \widetilde{\varphi _{j,k,\lambda }}(t) \right\} _{\lambda \in \nu {\mathbb {Z}}}$, for fixed j, is equivalent to the following condition

$$\begin{aligned} \varXi _{j,k}(\gamma )= \sum _{m \in {\mathbb {Z}}^d} \left| {\text {F}}\left( \widetilde{\varphi _{j,k,0}} \right) \left( \left( \gamma -m \right) \frac{2^j}{\nu } \right) \right| ^2\le C_\nu \frac{\nu ^d}{2^{jd}}, \quad a.e.\; \gamma \in [0,1]^d. \end{aligned}$$

By definition

$$\begin{aligned} \varXi _{j,k}(\gamma )&= \frac{1}{2^{jd}}\sum _{m\in {\mathbb {Z}}^d} \left| \frac{2^{js}}{\left( 1+\left| \left( \gamma -m \right) \frac{2^j}{\nu } \right| \right) ^s} \varPhi _{j,k}\left( \left( \gamma -m \right) \frac{2^j}{\nu } \right) \right| ^2. \end{aligned}$$

Using the hypothesis (37) and relation (38), we can write

$$\begin{aligned} \varXi _{j,k}(\gamma )&\lesssim \frac{1}{2^{jd}}\left( 1+ \sum _{|m|>1} \frac{2^{jd}}{ \left( 1+ d\left( \left( \gamma -m \right) \frac{2^j}{\nu }, I_{j,k} \right) \right) ^{2\alpha }} \right) \nonumber \\&\lesssim \frac{1}{2^{jd}}\left( 1+ \sum _{|m|> 1} \frac{2^{jd}}{ \left( (|m|-1) \frac{2^j}{\nu } \right) ^{2\alpha }} \right) \nonumber \\&\lesssim \frac{1}{2^{jd}}\left( 1+ \frac{\nu ^{2\alpha }}{2^{j(2\alpha -d)}} \sum _{|m|> 1} \frac{1}{ \left( (|m|-1) \right) ^{2\alpha }} \right) , \quad \text{ a.e. } \; \gamma \in [0,1], \end{aligned}$$

(46)

where the second inequality follows from our assumption on the radius. Again, by our hypothesis on $\alpha$, the sum in (46) is convergent, and uniformly bounded with respect to j. $\square$

Lemma A.5

Let $\varPhi _{j,k}$, $\widetilde{\varphi _{j,k,\lambda }}$ as in Lemma A.4 and $E_{j,k}$ as in (13).

If $\nu \in (0,1]$ and ${\text {supp}}\widehat{f} \cap E_{j,k} =\emptyset ,$ then

$$\begin{aligned} \sum _{\lambda \in \nu {\mathbb {Z}}} \left| \left\langle \widetilde{\varphi _{j,k,\lambda }}, f\right\rangle \right| ^2\le \frac{C_{\nu }}{2^{j(2\alpha -d)}} \left\| f \right\| ^2_{L^2(\mathbb {R}^d)}, \end{aligned}$$

(47)

with $C_{\nu }$ as in Lemma A.4, therefore independent on j, k.

Proof

Since $\hat{f}(\omega )=0$ if $\omega \in E_{j,k}$

$$\begin{aligned} \sum _{\lambda \in \nu {\mathbb {Z}}} \left| \left\langle \widetilde{\varphi _{j,k,\lambda }}, f\right\rangle \right| ^2&=\sum _{\lambda \in \nu {\mathbb {Z}}} \left| \left\langle \chi _{\mathbb {R}{\setminus } E_{j,k} }{\text {F}}\left( \widetilde{\varphi _{j,k,\lambda }} \right) , {\text {F}}(f)\right\rangle \right| ^2. \end{aligned}$$

Therefore, using the same property of Lemma A.4, (47) is equivalent to prove that

$$\begin{aligned} \frac{1}{2^{jd}}\sum _{m\in {\mathbb {Z}}} \left| \chi _{\mathbb {R}{\setminus } E_{j,k}} \left( m_{\gamma ,j,\nu } \right) \frac{2^{js}}{\left( 1+\left| m_{\gamma ,j,\nu } \right| \right) ^s} \varPhi _{j,k}\left( m_{\gamma ,j,\nu } \right) \right| ^2\le \frac{1}{2^{j(2\alpha )}}, \end{aligned}$$

(48)

where $m_{\gamma ,j,\nu } = \left( \gamma -m \right) \frac{2^j}{\nu }$. Since $\nu \le 1$, for each j, k, there exist a finite number of consecutive indices m such that $\left( \gamma -m \right) \frac{2^j}{\nu } \in E_{j,k}$ for some $\gamma \in [0,1]$. We set

$$\begin{aligned} M_{j,\nu } = \left\{ m \in {\mathbb {Z}}{:}\, \exists \; \gamma \in [0,1] \text { such that } \left( \gamma -m \right) \frac{2^j}{\nu } \in E_{j,k} \right\} . \end{aligned}$$

We notice that $M_{j,\nu }$ is uniformly bounded with respect to j, by the properties of the partitioning and by the definition of $E_{j,k}$.

If $m\in M_{j,\nu }$ and $m_{\gamma ,j,\nu }\in E_{j,k}$, then

$$\begin{aligned} \left| \chi _{\mathbb {R}{\setminus } E_{j,k}} \left( m_{\gamma ,j,\nu } \right) \frac{2^{js}}{\left( 1+\left| m_{\gamma ,j,\nu } \right| \right) ^s} \varPhi _{j,k}\left( m_{\gamma ,j,\nu } \right) \right| ^2 = 0. \end{aligned}$$

Otherwise $\chi _{\mathbb {R}{\setminus } E_{j,k}} \left( m_{\gamma ,j,\nu } \right) =1$ and, using Lemma A.1,

$$\begin{aligned} \left| \frac{2^{js}}{\left( 1+\left| m_{\gamma ,j,\nu } \right| \right) ^s} \varPhi _{j,k}\left( m_{\gamma ,j,\nu } \right) \right| ^2 \lesssim \left| \frac{2^{jd/2}}{\left( 1+2^j \right) ^{\alpha }} \right| ^2 \lesssim 2^{j(d-2\alpha )}. \end{aligned}$$

Hence, (48) is bounded by

$$\begin{aligned} \left| M_{j,\nu } \right| \left( 2^j \right) ^{-2\alpha } + \frac{1}{2^{jd}}\sum _{m\notin M_{j,\nu }} \left| \chi _{\mathbb {R}{\setminus } E_{j,k}} \left( m_{\gamma ,j,\nu } \right) \frac{2^{js}}{\left( 1+\mathrm {d}\left( m_{\gamma ,j,\nu },I_{j,k} \right) \right) ^s} \varPhi _{j,k}\left( m_{\gamma ,j,\nu } \right) \right| ^2. \end{aligned}$$

(49)

The second term in the equation above may be bounded as follows

$$\begin{aligned}&\frac{1}{2^{jd}}\sum _{m\notin M_{j,\nu }} \left| \chi _{\mathbb {R}{\setminus } E_{j,k}} \left( m_{\gamma ,j,\nu } \right) \frac{2^{js}}{\left( 1+\mathrm {d}\left( m_{\gamma ,j,\nu },I_{j,k} \right) \right) ^s} \varPhi _{j,k}\left( m_{\gamma ,j,\nu } \right) \right| ^2\\&\qquad \lesssim \frac{1}{2^{jd}} \frac{\nu ^{2\alpha } 2^{jd}}{\left( 2^j \right) ^{2\alpha }} \sum _{|m|\ge 2} \frac{1}{ \left( (|m|-1) \right) ^{2\alpha }}\lesssim \frac{\nu ^{2\alpha } }{\left( 2^j \right) ^{2\alpha }} \sum _{|m|\ge 2} \frac{1}{ \left( (|m|-1) \right) ^{2\alpha }}\lesssim \frac{\nu ^{2\alpha } }{\left( 2^j \right) ^{2\alpha }}. \end{aligned}$$

Then the assertion follows as in Lemma A.4. $\square$

Lemma A.6

Let $s\ge 0$ and $\varphi _{\bullet },\varphi$ be a system of functions such that there exists $a>0$ such that, for all $\omega \in \mathbb {R}^d$ then

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

and the constant a does not depend on j, k. Then

$$\begin{aligned} \frac{\left| \varPhi _{\bullet }(\omega ) \right| ^2}{\left( 1+|\omega | \right) ^{2s}}+ \sum _{j,k\in \varGamma } \frac{2^{2js}}{\left( 1+|\omega | \right) ^{2s}} |\varPhi _{j,k}(\omega )|^2 \ge C_s a^2, \quad \text{ a.e. } \omega \in \mathbb {R}^d\;, \end{aligned}$$

(50)

with a constant $C_s$ which depends on s only and is uniform with respect to $\omega$.

Proof

For $s=0$, the statement is trivial while for general s, notice that

$$\begin{aligned} (1+|\omega |)\asymp 2^j, \qquad \omega \in I_{j,k}, \end{aligned}$$

while if $\omega \in I_{\bullet }$, then $(1+|\omega |)\asymp 1$. $\square$

Remark A.1

Inequality (50) could be used as hypothesis on the window function weaker then ours. Since it is quite cumbersome to be checked, we prefer to work with a more transparent assumption.

We state now the counterpart of Lemmas A.4 and A.5 in the framework of seminorm discretisation.

Lemma A.7

Let $s\ge 0$ and $\varphi$ such that $\varPhi _{j,k}(\omega )$ satisfies hypothesis (39). We define

$$\begin{aligned} \widetilde{\varphi _{j,k,\lambda }}(t)&=\frac{1}{2^{jd/2}} T_{2^{-j}\lambda } {\text {F}}^{-1}_{\omega \mapsto t}\left( \frac{2^{js}}{|\omega |^s} \varPhi _{j,k}(\omega ) \right) (t), \quad j\in {\mathbb {N}}\nonumber \\ \widetilde{\varphi _{-j,k,\lambda }}(t)&= \frac{1}{2^{jd/2}} T_{{\lambda }{2^j}} {\text {D}}_{2^{2j}} {\text {F}}^{-1}_{\omega \mapsto t}\left( \frac{2^{js}}{|\omega |^s} \varPhi _{j,k}(\omega ) \right) (t), \quad j\in {\mathbb {N}}{\setminus } \left\{ 0 \right\} \end{aligned}$$

(51)

then, for all $\nu \in (0,1]$ and j, k the system of functions $\left\{ \widetilde{\varphi _{j,k,\lambda }}(t) \right\} _{\lambda \in \nu {\mathbb {Z}}}$ is a Bessel sequence uniformly in j, k, that is

$$\begin{aligned} \sum _{\lambda \in \nu {\mathbb {Z}}} \left| \left\langle \widetilde{\varphi _{j,k,\lambda }}, f\right\rangle \right| ^2\le C_\nu \left\| f \right\| ^2_{L^2(\mathbb {R}^d)} \end{aligned}$$

(52)

with $C_\nu$ independent on j, k.

Lemma A.8

Let $\varPhi _{j,k}$, $\widetilde{\varphi _{j,k,\lambda }}$ as in Lemma A.7 and $E_{j,k}$ as in (13) for $j\in {\mathbb {N}}$ and as

$$\begin{aligned} E_{-j, k}=\left\{ x\in \mathbb {R}\mid 2^{2j} x\in E_{j,k} \right\} \end{aligned}$$

for negative integers. If $\nu \in (0,1]$ and ${\text {supp}}\widehat{f} \cap E_{j,k} =\emptyset ,$ then

$$\begin{aligned} \sum _{\lambda \in \nu {\mathbb {Z}}} \left| \left\langle \widetilde{\varphi _{j,k,\lambda }}, f\right\rangle \right| ^2\le \frac{C_\nu }{\left| 2^j \right| ^{(2\alpha -d)}} \left\| f \right\| ^2_{L^2(\mathbb {R}^d)} \end{aligned}$$

(53)

with $C_\nu$ independent on j, k.

Proof of Lemmas A.7, A.8

If $j\in {\mathbb {N}}$ the proofs for is the same of Lemmas A.4 and A.5. In order to prove Lemma A.7 for $-j$ with $j\in {\mathbb {N}}{\setminus } \left\{ 0 \right\}$, we use the following relation

$$\begin{aligned} \sum _{\lambda \in \nu {\mathbb {Z}}} \left| \left\langle \widetilde{\varphi _{-j,k,\lambda }}, f\right\rangle \right| ^2= \sum _{\lambda \in \nu {\mathbb {Z}}} \left| \left\langle {\text {D}}_{2^{2j}} T_{\frac{\lambda }{ 2^j}} \widetilde{\varphi _{j,k,0 }}, f\right\rangle \right| ^2 =\sum _{\lambda \in \nu {\mathbb {Z}}} \left| \left\langle T_{ \frac{\lambda }{ 2^j} } \widetilde{\varphi _{j,k,0 }}, {\text {D}}_{2^{-2j}}{f}\right\rangle \right| ^2 \end{aligned}$$

and the fact that $\left\| {\text {D}}_{2^{-2j}} f \right\| _{L^2(\mathbb {R}^d)}= \left\| f \right\| _{L^2(\mathbb {R}^d)}$. For Lemma A.8 notice also that ${\text {supp}}\widehat{f} \cap E_{-j,k}=\emptyset$ implies ${\text {supp}}\widehat{{\text {D}}_{2^{-2j}} f}\cap E_{j,k}=\emptyset .$ $\square$

Interpolation techniques

We define a multi-resolution partition using the notation of [8]. Then, we adapt the interpolation result to our specific case.

Set $V=L^2(\mathbb {R}), Z = H^1(\mathbb {R})$ and let

$$\begin{aligned} V_{\overline{j}} = \text {span}\left\{ \varphi _{j,k,\lambda } \right\} _{{j}\le \overline{j}} \end{aligned}$$

(54)

where $\left\{ \varphi _{j,k,\lambda }\mid j, k, \lambda \in \varDelta \right\}$ is the frame defined as in (27). We also define the related projectors

$$\begin{aligned} \begin{aligned} P_{\,\overline{j}}{:}\,L^2&\longrightarrow V_j \\ v&\longmapsto \sum _{j\le {\overline{j}}, k,\lambda }\langle \varphi _{j,k,\lambda },v \rangle \varphi ^D_{j,k,\lambda }, \end{aligned} \end{aligned}$$

(55)

where $\varphi ^D_{j,k,\lambda }$ is the dual window. By definition

$$\begin{aligned} \ldots V_{-j-1} \subseteq V_{-j} \subseteq \cdots \subseteq V_1 \subseteq \cdots \subseteq V_j \subseteq V_{j+1} \cdots \subseteq L^2(\mathbb {R}) \end{aligned}$$

(56)

and also $V_j \subseteq H^s(\mathbb {R}^d), j\in {\mathbb {Z}}$. Finally, we notice that due to our definition

$$\begin{aligned} \lim _{j\rightarrow -\infty }P_j = 0. \end{aligned}$$

(57)

Lemma B.9

Let $r\in {\mathbb {Z}}$, then the following inequalities hold.

$$\begin{aligned} |v|_{r} \lesssim 2^{\overline{j}r}\left\| v \right\| , \quad \forall {v\in V_{\overline{j}}}, \quad \overline{j}\in {\mathbb {Z}}\quad {\text {(Bernstein)}} \end{aligned}$$

(58)

and

$$\begin{aligned} \left\| v-P_{\overline{j}}(v) \right\| \lesssim 2^{-\overline{j}r}\left| v \right| _r \quad \forall {v\in V_{\overline{j}}}, \quad \overline{j}\in {\mathbb {Z}}\quad {\text {(Jackson).}} \end{aligned}$$

(59)

Proof

This follows immediately from Theorem 3.4 and a few observations. We notice that we can write the Sobolev seminorm as

$$\begin{aligned} |v|_r = \left\| v^{(r)} \right\| _2, \end{aligned}$$

where $v^{(r)}$ is the rth derivative of v. We also notice that if $\left\{ \varphi _{j,k,\lambda } \right\} _{j,k,\lambda \in \varGamma }$ is a frame for $H^r$, then $\left\{ 2^{-jr}\varphi _{j,k,\lambda }^{(r)} \right\} _{j,k,\lambda \in \varGamma }$ is a frame for $L^2$. Indeed,

$$\begin{aligned} 2^{-jr}\widehat{\varphi _{j,k,\lambda }^{(r)}}(\xi ) = 2^{-jr}e^{-2\pi \mathrm {i}\,\omega \cdot 2^{-j}\lambda } |\xi |^r \varPhi _{j,k}(\xi ). \end{aligned}$$

and the frequency window

$$\begin{aligned} \widetilde{\varPhi _{j,k}}(\xi ) = 2^{-jr}|\xi |^r \varPhi _{j,k}(\xi ), \end{aligned}$$

clearly satisfies the requirements of Definition 3.3 and thus yields a frame, as claimed.

Hence, we can represent the rth derivative of the function v as

$$\begin{aligned} v^{(r)} = \sum _{j<\overline{j},k,\lambda }2^{jr}\langle v,2^{-jr}\varphi _{j,k,\lambda }^{(r)}\rangle \varphi _{j,k,\lambda }^D. \end{aligned}$$

Finally, using the minimal property of the frame coefficients, see [16, Proposition 5.1.4.],

$$\begin{aligned} |v|^2_s \lesssim \sum _{j<\overline{j},k,\lambda \in \varDelta } 2^{2js}\left| \left\langle v, \varphi _{j,k,\lambda }\right\rangle \right| ^2 \le 2^{2\overline{j}s}\sum _{j<\overline{j},k,\lambda \in \varDelta }\left| \left\langle v, \varphi _{j,k,\lambda }\right\rangle \right| ^2 \le 2^{2\overline{j}s}\left\| v \right\| , \end{aligned}$$

as desired.

To prove Jackson’s inequality, notice that, using again the minimal property of the frame coefficients,

$$\begin{aligned} \left\| v-P_{\overline{j}}(v) \right\| ^2 \lesssim \sum _{j>\overline{j},k,\lambda \in \varDelta }\left| \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right| ^2\le & {} 2^{-2\overline{j}s}\sum _{j>\overline{j},k,\lambda \in \varDelta }2^{2\overline{j}s}\left| \left\langle f, \varphi _{j,k,\lambda }\right\rangle \right| ^2\\\le & {} 2^{-2\overline{j}s}|v|^2_s, \end{aligned}$$

and this concludes the proof. $\square$

We now review Theorem 1 in [8] adapting it to our framework.

Theorem B.1

Consider the spaces $V_j$ and the associated partitions $P_j$ that satisfies (57) and Jackson–Bernstein inequalities. Then, for any $0<\alpha <1, q>1$ one has

$$\begin{aligned} (L^2,H^1)_{\alpha ,q} = \left\{ v\in L^2 \mid \left| v \right| _{\alpha ,q} = \left[ \sum _{j\in {\mathbb {Z}}}2^{j\alpha q}\left( \sum _{k,\lambda \in \varDelta _j}\left| \left\langle \varphi _{j,k,\lambda }, v\right\rangle \right| ^2 \right) ^{q/2} \right] ^{1/q}<\infty \right\} , \end{aligned}$$

and a norm for this space is $\left\| \cdot \right\| = \left\| \cdot \right\| _2 + \left| \cdot \right| _{\alpha ,q}$.

Proof

The hypothesis of [8, Theorem 1] are fulfilled, hence the result follows. $\square$

Remark 9

The same result holds if we interchange the role of the window function and its dual.