Anisotropic multiscale systems on bounded domains

Abstract

We establish a construction of hybrid multiscale systems on a bounded domain ${\Omega } \subset \mathbb {R}^{2}$ consisting of shearlets and boundary-adapted wavelets, which satisfy several properties advantageous for applications to imaging science and the numerical analysis of partial differential equations. More precisely, we construct hybrid shearlet-wavelet systems that form frames for the Sobolev spaces $H^{s}({\Omega }),~s\in \mathbb {N} \cup \{0\}$ with controllable frame bounds and admit optimally sparse approximations for functions which are smooth apart from a curve-like discontinuity. Per construction, these systems allow to incorporate boundary conditions.

Appendices

Appendix A: $H^{s}(\mathbb {R}^{2})$ Frame property of a reweighted shearlet system

This section shall be concerned with the proof of Theorem 3.3. The proof is split into Lemma A.2 for the upper frame bound and Lemma A.3 for the lower frame bound.

We use the following lemma to estimate weighted $L^{2}(\mathbb {R}^{2})$ - frame coefficients by the $H^{s}(\mathbb {R}^{2})$ norm of a function f.

Lemma A.1 (40, Lemma 2.4.9.)

Let$s \in \mathbb {N}$, $\tilde {C}>0,$β_sh > 3 and α_sh > β_sh + s. Further, let$\phi , \psi , \widetilde {\psi } \in L^{2}(\mathbb {R}^{2})$such that for almost all$\xi \in \mathbb {R}^{2}$there holds

$$ \begin{array}{@{}rcl@{}} |\hat \phi(\xi)| &\leq& \tilde{C} \min\{1, |{\xi}_{1} |^{-{\beta}_{\text{sh}}}\}\min\{1, |{\xi}_{2} |^{-{\beta}_{\text{sh}}}\}\\ |\hat \psi(\xi)| &\leq& \tilde{C} \min\{1, |{\xi}_{1} |^{{\alpha}_{\text{sh}}}\}\min\{1, |{\xi}_{1} |^{-{\beta}_{\text{sh}}}\}\min\{1, |{\xi}_{2} |^{-{\beta}_{\text{sh}}}\} \\ |\hat {\tilde{\psi}}(\xi)| &\leq& \tilde{C} \min\{1, | {\xi}_{2} |^{{\alpha}_{\text{sh}}}\}\min\{1, |{\xi}_{1} |^{-{\beta}_{\text{sh}}}\}\min\{1, |{\xi}_{2} |^{-{\beta}_{\text{sh}}}\}. \end{array} $$

Then there exists a constant C > 0 depending only on s, α_shand β_shsuch that$\mathcal {SH}(\phi , \psi , \tilde {\psi }, c)$satisfies

$$ \left\|\left( 2^{js}\langle f, {\psi}_{j,k,m,\iota}\rangle_{L^{2}(\mathbb{R}^{2})}\right)_{(j,k,m,\iota)\in {\Lambda}} \right\|_{\ell^{2}({\Lambda})}\leq \frac{C \tilde{C} }{\sqrt{|\det(M_{c})|}} \|f\|_{H^{s}(\mathbb{R}^{2})}, \quad \text{ for all } f \in H^{s}(\mathbb{R}^{2}). $$

We continue to give the upper frame bound for the weighted shearlet system in $H^{s}(\mathbb {R}^{2})$.

Lemma A.2

For$s\in \mathbb {N}$let$\phi ^{1}, {\psi }^{1} \in L^{2}(\mathbb {R})$be such that for some$\tilde {C} \geq 0$, α_sh − s > β_sh > 3, and all 0 ≤ r₁, r₂ ≤ 2s,

$$ \begin{array}{@{}rcl@{}} \left|\widehat{D^{r_{1}}\phi^{1}}(\xi)\right| &\leq& \sqrt{\tilde{C}} \min \{1,|\xi|^{-{\beta}_{\text{sh}}} \}, \text{ for all } \xi \in \mathbb{R}, \\ \left|\widehat{D^{r_{2}}{\psi}^{1}}(\xi)\right| &\leq& \sqrt{\tilde{C}} \min \{1,|\xi|^{{\alpha}_{\text{sh}}} \} \min \{1,|\xi|^{-{\beta}_{\text{sh}}} \}, \text{ for all } \xi \in \mathbb{R}. \end{array} $$

(12)

We further denote ϕ = ϕ¹ ⊗ ϕ¹ and ψ = ψ¹ ⊗ ϕ¹and$\tilde {\psi }(x_{1}, x_{2}) := \psi (x_{2}, x_{1})$for all$x\in \mathbb {R}^{2}$. Then, there exists some B > 0 depending only on α_sh, β_shand ssuch for$({\psi }_{j,k,m,\iota })_{(j,k,m,\iota )\in {\Lambda }} = \mathcal {SH}(\phi , \psi , \tilde {\psi }, (c_{1},c_{2}))$we have the estimate

$$ \begin{array}{@{}rcl@{}} \sum\limits_{(j,k,m,\iota)\in {\Lambda}} |\langle f, 2^{-js} {\psi}_{j,k,m,\iota} \rangle_{H^{s}(\mathbb{R}^{2})} |^{2} \leq B \frac{\tilde{C}}{|\det(M_{c})|} \|f\|_{H^{s}(\mathbb{R}^{2})}^{2} \end{array} $$

(13)

for all$f \in H^{s}(\mathbb {R}^{2})$.

Proof

We define for $\mathbf {r}= (r_{1},r_{2})$: $\phi ^{(\mathbf {r})} = D^{r_{1}}\phi ^{1} \otimes D^{r_{2}}\phi ^{1}$, ${\psi }^{(\mathbf {r})} = D^{r_{1}}{\psi }^{1}\otimes D^{r_{2}}\phi ^{1}$, and $\tilde {\psi }^{(\mathbf {r})}(x_{1},x_{2})={\psi }^{(\mathbf {r})}(x_{2},x_{1})$. By Lemma A.1 there exists a constant C, dependent only on ${\alpha }_{\text {sh}}, {\beta }_{\text {sh}},$ such that for all $0\leq r_{1},r_{2} \leq 2s$ and for $({\psi }_{j,k,m,\iota }^{(\mathbf {r})})_{(j,k,m,\iota )\in {\Lambda }} = \mathcal {SH}(\phi ^{(\mathbf {r})}, {\psi }^{(\mathbf {r})}, \tilde {\psi }^{(\mathbf {r})}, c)$ we have that

$$ \begin{array}{@{}rcl@{}} \sum\limits_{(j,k,m,\iota)\in {\Lambda}} 2^{2js}|\langle f, {\psi}_{j,k,m,\iota}^{(\mathbf{r})}\rangle_{L^{2}(\mathbb{R}^{2})} |^{2} \leq C \frac{\tilde{C}}{|\det(M_{c})|} \|f\|_{H^{s}(\mathbb{R}^{2})}^{2}, \text{ for all } f\in H^{s}(\mathbb{R}^{2}). \end{array} $$

(14)

From the assumptions (12) we have that $\phi , \psi , \tilde {\psi } \in H^{2s}(\mathbb {R}^{2})$ and thus we can estimate the left-hand side of (13).

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{(j,k,m,\iota) \in {\Lambda}} | 2^{-js}\left \langle f, {\psi}_{j,k,m,\iota}\right \rangle_{H^{s}(\mathbb{R}^{2})} |^{2} \\ &=& \sum\limits_{(j,k,m,\iota) \in {\Lambda}} | \sum\limits_{|\mathbf{a}|\leq s} 2^{-js} \left \langle D^{\mathbf{a}}f, D^{\mathbf{a}}{\psi}_{j,k,m,\iota}\right \rangle_{L^{2}(\mathbb{R}^{2})} |^{2}\\ &\lesssim& \sum\limits_{(j,k,m,\iota) \in {\Lambda}} \sum\limits_{|\mathbf{a}|\leq s}|2^{-js} \left \langle f, D^{2\mathbf{a}}{\psi}_{j,k,m,\iota}\right \rangle_{L^{2}(\mathbb{R}^{2})} |^{2} = \mathrm{I}. \end{array} $$

(4)

By using the product rule we calculate

$$ \begin{array}{@{}rcl@{}} D^{2\mathbf{a}}{\psi}_{j,k,m,\iota} = \sum\limits_{0\leq r_{1},r_{2} \leq 2s} c_{\mathbf{r}}^{(\mathbf{a})} {\psi}^{(\mathbf{r})}_{j,k,m,\iota}, \end{array} $$

(16)

with coefficients $c_{\mathbf {r}}^{\mathbf {a}}$ bounded in absolute value by 2^2js. After applying (16) to (15) we estimate

$$ \begin{array}{@{}rcl@{}} \mathrm{I} &\lesssim& \sum\limits_{(j,k,m,\iota) \in {\Lambda}} \sum\limits_{|\mathbf{a}|\leq s} \sum\limits_{0\leq r_{1}, r_{2} \leq 2s} 2^{2js} | \langle f, {\psi}_{j,k,m,\iota}^{(\mathbf{r})}\rangle_{L^{2}(\mathbb{R}^{2})}|^{2}\\ &\lesssim& \sup\limits_{0\leq r_{1},r_{2} \leq 2s}\sum\limits_{(j,k,m,\iota) \in {\Lambda}} 2^{2js} | \langle f, {\psi}_{j,k,m,\iota}^{(\mathbf{r})} \rangle_{L^{2}(\mathbb{R}^{2})} |^{2} =\text{II}. \end{array} $$

Invoking (14), there exists some B > 0 such that

$$ \begin{array}{@{}rcl@{}} \text{II} \leq B \frac{\tilde{C}}{|\det(M_{c})|} \|f\|_{H^{s}(\mathbb{R}^{2})}^{2}, \text{ for all } f\in H^{s}(\mathbb{R}^{2}). \end{array} $$

This yields the result. □

As a next step we provide the lower frame bound.

Lemma A.3

For$s\in \mathbb {N}$, α_sh − s > β_sh > 4 and some$\tilde {C}>0$let$\phi ^{1}, {\psi }^{1} \in L^{2}(\mathbb {R})$be such that for all 0 ≤ r₁, r₂ ≤ 2s,

$$ \begin{array}{@{}rcl@{}} \left|\widehat{D^{r_{1}}\phi^{1}}(\xi)\right| &\leq& \sqrt{\tilde{C}}\min \{1,|\xi|^{-{\beta}_{\text{sh}}} \}, \text{ for all } \xi \in \mathbb{R}, \\ \left|\widehat{D^{r_{2}}{\psi}^{1}}(\xi)\right| &\leq& \sqrt{\tilde{C}} \min \{1,|\xi|^{{\alpha}_{\text{sh}}} \} \min \{1,|\xi|^{-{\beta}_{\text{sh}}} \}, \text{ for all } \xi \in \mathbb{R}. \end{array} $$

Further, let$\phi , \psi , \widetilde {\psi }, \theta , \widetilde {\theta }, \mu$be as in (1). Assume that there exists$\bar c>0$such that for all$c_{1},c_{2}\leq \bar {c}$we have that$\mathcal {SH}(\mu ,\theta ,\tilde {\theta },(c_{1},c_{2}))$forms a frame for$L^{2}(\mathbb R^{2})$with lower frame bound, which can be bounded from below independently ofc₁, c₂. Then there exists$\tilde {c}>0$such that for all$c_{1}=c_{2} \leq \tilde {c}$and c = (c₁, c₂) the system$({\psi }_{j,k,m,\iota })_{(j,k,m,\iota )\in {\Lambda }} = \mathcal {SH}(\mu ,\theta ,\tilde {\theta },c)$, obeys

$$ \|f\|_{H^{s}(\mathbb{R}^{2})}^{2}\lesssim \sum\limits_{(j,k,m,\iota)\in {\Lambda}} |\langle f, 2^{-js} {\psi}_{j,k,m,\iota} \rangle_{H^{s}(\mathbb{R}^{2})} |^{2}, $$

for all$f\in H^{s}(\mathbb {R}^{2})$.

Proof

Let $\bar {c} \geq c>0$. We define

$$ \hat{\mu}^{c}:=\mathcal{X}_{\left[-\frac{1}{2c_{1}},\frac{1}{2c_{1}}\right]^{2}}\hat{\mu}, \qquad \hat{\theta}^{c}:=\mathcal{X}_{\left[-\frac{1}{2c_{1}},\frac{1}{2c_{1}}\right]^{2}}\hat{\theta} \qquad, \hat{\widetilde{\theta}}^{c}:=\mathcal{X}_{\left[-\frac{1}{2c_{1}},\frac{1}{2c_{1}}\right]^{2}}\hat{\widetilde{\theta}}. $$

By using the fact, that $\mathcal {SH}(\mu ,\theta ,\tilde {\theta },c)$ is a frame for $L^{2}(\mathbb {R}^{2}),$ we deduce

$$ \begin{array}{@{}rcl@{}} \|f\|_{H^{s}(\mathbb{R}^{2})}^{2}\ &\sim& \| (1+|\cdot|^{2})^{\frac{s}{2}} \hat{f}\|_{L^{2}(\mathbb{R}^{2})}^{2}\\ & \lesssim& \sum\limits_{(j,k,m,\iota)\in{\Lambda}} |\langle (1+|\cdot|^{2})^{\frac{s}{2}} \hat{f}, \widehat{\theta}_{j,k,m,\iota} \rangle_{L^{2}(\mathbb{R}^{2})} |^{2} \\ &\leq& \sum\limits_{(j,k,m,\iota)\in{\Lambda}} |\langle (1+|\cdot|^{2})^{\frac{s}{2}} \hat{f}, \widehat{\theta}^{c}_{j,k,m} \rangle_{L^{2}(\mathbb{R}^{2})}|^{2} \\ &&+ \sum\limits_{(j,k,m,\iota)\in{\Lambda}} |\langle (1+|\cdot|^{2})^{\frac{s}{2}} \hat{f}, \widehat{\theta}_{j,k,m}-\widehat{\theta}^{c}_{j,k,m} \rangle_{L^{2}(\mathbb{R}^{2})}|^{2} = \mathrm{I}_{1} + \mathrm{I}_{2}. \end{array} $$

(17)

Since β_sh > 4, there exists 𝜖 > 0 such that β_sh − 1 − 𝜖 > 3 and for all $\xi \in \mathbb {R}^{2}$

$$ \begin{array}{@{}rcl@{}} \min \{1, |\xi|^{{\alpha}_{\text{sh}}}\} \min \{ 1, |\xi|^{-{\beta}_{\text{sh}}}\} \leq \min \{1, |{\xi}_{1}|^{{\alpha}_{\text{sh}}}\} \min \{ 1, |{\xi}_{1}|^{-{\beta}_{\text{sh}}+1+\epsilon}\} (\max \{ 1, |{\xi}_{1}|\})^{-1-\epsilon}. \end{array} $$

Thus, we have that $(\theta _{j,k,m,\iota }- \theta _{j,k,m,\iota }^{c})_{(j,k,m,\iota ) \in {\Lambda }}$ satisfies the assumptions of Lemma A.2 with $\sqrt {\tilde {C}} = 2c_{1}^{1+\epsilon }$. Hence, I₂ can be estimated by $c_{1}^{2+2\epsilon }/|\det {(M_{c})}| \|f\|_{H^{s}(\mathbb {R}^{2})}^{2}$ and since $\det {(M_{c}}) = {c_{1}^{2}}$ this term is negligible for c₁ small enough. Therefore we obtain

$$ \begin{array}{@{}rcl@{}} \|f\|_{H^{s}(\mathbb{R}^{2})}^{2} & \lesssim& \sum\limits_{(j,k,m,\iota)\in{\Lambda}} |\langle (1+|\cdot|^{2})^{\frac{s}{2}} \hat{f}, \widehat{\theta}^{c}_{j,k,m} \rangle_{L^{2}(\mathbb{R}^{2})}|^{2}\\ &=& \sum\limits_{(j,k,m)\in{\Lambda}^{\prime}} |\langle (1+|\cdot|^{2})^{\frac{s}{2}} \hat{f}, \widehat{\theta}^{c}_{j,k,m} \rangle_{L^{2}(\mathbb{R}^{2})} |^{2}\\ &&+ \sum\limits_{(j,k,m)\in{\Lambda}^{\prime}} |\langle (1+|\cdot|^{2})^{\frac{s}{2}} \hat{f},\widehat{\widetilde{\theta}}^{c}_{j,k,m} \rangle_{L^{2}(\mathbb{R}^{2})} |^{2} \\ &&+ \sum\limits_{m\in\mathbb{Z}^{2}} |\langle (1+|\cdot|^{2})^{\frac{s}{2}} \hat{f}, \mathcal{F}(\mu^{c}(\cdot-c_{1}m)) \rangle_{L^{2}(\mathbb{R}^{2})} |^{2} \end{array} $$

$$ \begin{array}{@{}rcl@{}} &=& \sum\limits_{(j,k,m)\in{\Lambda}^{\prime}} 2^{\frac{3j}{2}}\left\vert {\int}_{\mathbb{R}^{2}} \hat{f}(\xi)(1+|\xi|^{2})^{\frac{s}{2}} \overline{\mathcal{F}(\theta^{c}(S_{k}A_{j}\cdot-M_{c}m))(\xi)}d\xi \right\vert^{2} \\ &&+\sum\limits_{(j,k,m)\in{\Lambda}^{\prime}} 2^{\frac{3j}{2}}\left\vert {\int}_{\mathbb{R}^{2}} \hat{f}(\xi)(1+|\xi|^{2})^{\frac{s}{2}} \overline{\mathcal{F}(\widetilde{\theta}^{c}({S_{k}^{T}}\tilde{A}_{j}\cdot-M_{\tilde{c}}m))(\xi)}d\xi \right\vert^{2} \\ &&+\sum\limits_{m\in\mathbb{Z}^{2}} \left\vert{\int}_{\mathbb{R}^{2}}\hat{f}(\xi)(1+|\xi|^{2})^{\frac{s}{2}}\overline{\mathcal{F}(\mu^{c}(\cdot-c_{1}m))(\xi)} d\xi\right\vert^{2} \\ &=& \sum\limits_{(j,k,m)\in{\Lambda}^{\prime}} 2^{-\frac{3j}{2}}\left\vert {\int}_{\mathbb{R}^{2}} \hat{f}(\xi)(1+|\xi|^{2})^{\frac{s}{2}} \overline{\hat{\theta}^{c}(S_{k}^{-T}A_{j}^{-1}\xi)}e^{2\pi i\langle M_{c}S_{k}^{-T}A_{j}^{-1}\xi, m \rangle}d\xi \right\vert^{2} \\ &&+\sum\limits_{(j,k,m)\in{\Lambda}^{\prime}} 2^{-\frac{3j}{2}}\left\vert {\int}_{\mathbb{R}^{2}} \hat{f}(\xi)(1+|\xi|^{2})^{\frac{s}{2}} \overline{\hat{\widetilde{\theta}}^{c}(S_{k}^{-1}\tilde{A}_{j}^{-1}\xi)} e^{2\pi i\langle M_{\tilde{c}}S_{k}^{-1}\tilde{A}_{j}^{-1}\xi,m\rangle}d\xi \right\vert^{2} \\ &&+\sum\limits_{m\in\mathbb{Z}^{2}} \left\vert{\int}_{\mathbb{R}^{2}}\hat{f}(\xi)(1+|\xi|^{2})^{\frac{s}{2}}\overline{\hat{\mu}^{c}(\xi)}e^{2\pi i\langle c_{1}\xi,m \rangle} d\xi\right\vert^{2} = \text{II}. \end{array} $$

Now we substitute

$$ \begin{array}{@{}rcl@{}} \xi\leadsto A_{j}{S_{k}^{T}}M_{c}^{-1}\xi, \qquad \xi\leadsto \tilde{A}_{j}S_{k}M_{\tilde{c}}^{-1}\xi, \qquad \xi\leadsto \frac{\xi}{c_{1}}. \end{array} $$

(18)

Furthermore, we use that μ^c, 𝜃^c, and $\tilde {\theta }^{c}$ are all supported in $[-\frac {1}{2c_{1}},\frac {1}{2c_{1}}]^{2}$.

$$ \begin{array}{@{}rcl@{}} \text{II}&=& \sum\limits_{(j,k,m)\in{\Lambda}^{\prime}} \frac{2^{\frac{3j}{2}}}{|\det(M_{c})|^{2}}\cdot \left\vert {\int}_{[-\frac{1}{2},\frac{1}{2}]^{2}} \hat{f}(A_{j}{S_{k}^{T}}M_{c}^{-1}\xi)(1+|A_{j}{S_{k}^{T}}M_{c}^{-1}\xi|^{2})^{\frac{s}{2}}\overline{\hat{\theta}^{c}(M_{c}^{-1}\xi)}e^{2\pi i\langle \xi,m \rangle} d\xi\right\vert^{2} \\ &&+ \sum\limits_{(j,k,m)\in{\Lambda}^{\prime}} \frac{2^{\frac{3j}{2}}}{|\det(M_{c})|^{2}}\cdot \left\vert {\int}_{[-\frac{1}{2},\frac{1}{2}]^{2}} \hat{f}(\tilde{A}_{j}S_{k}M_{\tilde{c}}^{-1}\xi)(1+|\tilde{A}_{j}S_{k} M_{\tilde{c}}^{-1}\xi|^{2})^{\frac{s}{2}}\overline{\hat{\widetilde{\theta}}^{c}(M_{\tilde{c}}^{-1}\xi)}e^{2\pi i\langle \xi,m \rangle} d\xi\right\vert^{2} \\ &&+ \sum\limits_{m\in\mathbb{Z}^{2}} \frac{1}{{c_{1}^{2}}}\left\vert {\int}_{[-\frac{1}{2},\frac{1}{2}]^{2}} \hat{f}\left( \frac{\xi}{c_{1}} \right) \left( 1+\left\vert \frac{\xi}{c_{1}} \right\vert^{2} \right)^{\frac{s}{2}} \overline{\hat{\mu}^{c}\left( \frac{\xi}{c_{1}}\right)}e^{2\pi i\langle\xi,m\rangle}d\xi\right\vert^{2}. \end{array} $$

By using the Parseval identity we obtain

$$ \begin{array}{@{}rcl@{}} \text{II}&=& \sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}} \frac{2^{\frac{3j}{2}}}{|\det(M_{c})|^{2}}\cdot \left\| \hat{f}(A_{j}{S_{k}^{T}}M_{c}^{-1}\cdot)(1+|A_{j}{S_{k}^{T}}M_{c}^{-1}\cdot|^{2})^{\frac{s}{2}}\hat{\theta}^{c}(M_{c}^{-1}\cdot)\right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &&+ \sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}} \frac{2^{\frac{3j}{2}}}{|\det(M_{c})|^{2}}\cdot \left\| \hat{f}(\tilde{A}_{j}S_{k}M_{\tilde{c}}^{-1}\cdot)(1+|\tilde{A}_{j}S_{k} M_{\tilde{c}}^{-1}\cdot|^{2})^{\frac{s}{2}}\hat{\widetilde{\theta}}^{c}(M_{\tilde{c}}^{-1}\cdot)\right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &&+ \frac{1}{{c_{1}^{2}}}\left\| \hat{f}\left( \frac{\cdot}{c_{1}} \right) \left( 1+\left\vert \frac{\cdot}{c_{1}} \right\vert^{2} \right)^{\frac{s}{2}} \hat{\mu}^{c}\left( \frac{\cdot}{c_{1}}\right)\right\|_{L^{2}(\mathbb{R}^{2})}^{2}. \end{array} $$

Now we substitute

$$ \begin{array}{@{}rcl@{}} \xi\leadsto M_{c}\xi, \qquad \xi\leadsto M_{\tilde{c}}\xi, \qquad \xi\leadsto c_{1}\xi \end{array} $$

(19)

to arrive at

$$ \begin{array}{@{}rcl@{}} \text{II}&=& \sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}} 2^{\frac{3j}{2}} \left\| (1+|A_{j}{S_{k}^{T}}\cdot|^{2})^{\frac{s}{2}}\hat{f}(A_{j}{S_{k}^{T}}\cdot) \hat{\theta}^{c}(\cdot) \right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &&+ \sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}}2^{\frac{3j}{2}} \left\| (1+|\tilde{A}_{j}S_{k}\cdot|^{2})^{\frac{s}{2}} \hat{f}(\tilde{A}_{j}S_{k}\cdot) \hat{\widetilde{\theta}}^{c}(\cdot) \right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &&+ \left\| (1+|\cdot|^{2})^{\frac{s}{2}}\cdot\hat{f}\cdot \hat{\phi} \right\|_{L^{2}(\mathbb{R}^{2})}^{2}. \end{array} $$

Now set

$$ \hat{\phi}^{c}:=\mathcal{X}_{[-\frac{1}{2c_{1}},\frac{1}{2c_{1}}]^{2}}\hat{\phi}, \qquad \hat{\psi}^{c}:=\mathcal{X}_{[-\frac{1}{2c_{1}},\frac{1}{2c_{1}}]^{2}}\hat{\psi} \qquad, \hat{\widetilde{\psi}}^{c}:=\mathcal{X}_{[-\frac{1}{2c_{1}},\frac{1}{2c_{1}}]^{2}}\hat{\widetilde{\psi}}. $$

Using the definition of $\theta ^{c},\widetilde {\theta }^{c},\mu ^{c}$ then yields

$$ \begin{array}{@{}rcl@{}} \text{II}&=&\sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}} 2^{\frac{3j}{2}} \left\| (1+|A_{j}{S_{k}^{T}}\cdot|^{2})^{\frac{s}{2}}\hat{f}(A_{j}{S_{k}^{T}}\cdot) |(\cdot)_{1}|^{s} \hat{\psi}^{c}(\cdot) \right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &&+ \sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}}2^{\frac{3j}{2}} \left\| (1+|\tilde{A}_{j}S_{k}\cdot|^{2})^{\frac{s}{2}} \hat{f}(\tilde{A}_{j}S_{k}\cdot) |(\cdot)_{2}|^{s} \hat{\widetilde{\psi}}^{c}(\cdot) \right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &&+ \left\| (1+|\cdot|^{2})^{\frac{s}{2}}\cdot\hat{f}\cdot \hat{\phi} \right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &=& \sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}} 2^{\frac{3j}{2}} \left\| (1+|A_{j}{S_{k}^{T}}\cdot|^{2})^{\frac{s}{2}}\hat{f}(A_{j}{S_{k}^{T}}\cdot) 2^{-js} |(A_{j}{S_{k}^{T}}\cdot)_{1}|^{s} \hat{\psi}^{c}(\cdot) \right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &&+ \sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}}2^{\frac{3j}{2}} \left\| (1+|\tilde{A}_{j}S_{k}\cdot|^{2})^{\frac{s}{2}} \hat{f}(\tilde{A}_{j}S_{k}\cdot) 2^{-js} |(\tilde{A}_{j}S_{k}\cdot)_{2}|^{s} \hat{\psi}^{c}(\cdot) \right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &&+ \left\| (1+|\cdot|^{2})^{\frac{s}{2}}\cdot\hat{f}\cdot \hat{\phi}^{c} \right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &\leq& \sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}} 2^{\frac{3j}{2}}\left\| (1+|A_{j}{S_{k}^{T}}\cdot|^{2})^{s}\hat{f}(A_{j}{S_{k}^{T}}\cdot) 2^{-js} \hat{\psi}^{c}(\cdot) \right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &&+ \sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}} 2^{\frac{3j}{2}}\left\| (1+|\tilde{A}_{j}S_{k}\cdot|^{2})^{s} \hat{f}(\tilde{A}_{j}S_{k}\cdot) 2^{-js} \hat{\widetilde{\psi}}^{c}(\cdot) \right\|_{L^{2}(\mathbb{R}^{2})}^{2} \\ &&+ \left\| (1+|\cdot|^{2})^{s}\cdot\hat{f}\cdot \hat{\phi}^{c} \right\|_{L^{2}(\mathbb{R}^{2})}^{2} = \text{III}. \end{array} $$

We observe that

$$ (1+|\xi|^{2})^{s} \lesssim \sum\limits_{|\mathbf{a}|\leq s} (2\pi)^{2|\mathbf{a}|} \xi^{2\mathbf{a}} \text{ for all } \xi \in \mathbb{R}^{2}. $$

Thus we can estimate

$$ \begin{array}{@{}rcl@{}} \text{III} \lesssim &\sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}} 2^{\frac{3j}{2}}\cdot\left\|\sum\limits_{|\mathbf{a}|\leq s} (2\pi)^{2|\mathbf{a}|} (A_{j}{S_{k}^{T}}\cdot)^{2\mathbf{a}} \hat{f}(A_{j}{S_{k}^{T}}\cdot)2^{-js} \hat{\psi}^{c}(\cdot)\right\|_{L^{2}(\mathbb{R}^{2})}^{2}\\ &\quad + \sum\limits_{(j,k)\in{\Lambda}^{\prime\prime}} 2^{\frac{3j}{2}}\cdot\left\|\sum\limits_{|\mathbf{a}|\leq s} (2\pi)^{2|\mathbf{a}|} (\tilde{A}_{j}S_{k}\cdot)^{2\mathbf{a}}\hat{f}(\tilde{A}_{j}S_{k}\cdot) 2^{-js} \hat{\widetilde{\psi}}^{c}(\cdot)\right\|_{L^{2}(\mathbb{R}^{2})}^{2}\\ &\qquad + \left\|\sum\limits_{|\mathbf{a}|\leq s} (2\pi)^{2|\mathbf{a}|} (\cdot)^{2\mathbf{a}} \hat{f}\hat{\phi}^{c} \right\|_{L^{2}(\mathbb{R}^{2})}^{2} = \text{IV}. \end{array} $$

Invoking Parseval’s identity again and reversing the transformations (19) as well as (18) from earlier shows that

$$ \begin{array}{@{}rcl@{}} \text{IV} &= &\sum\limits_{(j,k,m,\iota)\in {\Lambda}} |\langle \hat{f}, \sum\limits_{|\mathbf{a}|\leq s} (2\pi)^{2|\mathbf{a}|} (\cdot)^{2\mathbf{a}} 2^{-js}\widehat{\psi}^{c}_{j,k,m,\iota} \rangle_{L^{2}(\mathbb{R}^{2})} |^{2} \\ &= &\sum\limits_{(j,k,m,\iota)\in{\Lambda}} |\sum\limits_{|\mathbf{a}|\leq s} \langle \hat{f}, (-1)^{|\mathbf{a}|} (2\pi i)^{2|\mathbf{a}|} (\cdot)^{2\mathbf{a}} 2^{-js}\widehat{\psi}^{c}_{j,k,m,\iota} \rangle_{L^{2}(\mathbb{R}^{2})} |^{2}. \end{array} $$

By standard results on derivatives and the Fourier transform we have that

$$ (2\pi i \cdot)^{2\mathbf{a}} 2^{-js}\widehat{\psi}^{c}_{j,k,m,\iota} = \mathcal{F}(2^{-js} D^{2\mathbf{a}} {\psi}^{c}_{j,k,m,\iota}). $$

Thus we can invoke the Plancherel identity and partial integration to obtain

$$ \begin{array}{@{}rcl@{}} \text{IV}&= &\sum\limits_{(j,k,m,\iota)\in{\Lambda}} |\sum\limits_{|\mathbf{a}|\leq s} \langle f, (-1)^{|\mathbf{a}|} 2^{-js}D^{2\mathbf{a}} {\psi}^{c}_{j,k,m,\iota} \rangle_{L^{2}(\mathbb{R}^{2})} |^{2}\\ &= &\sum\limits_{(j,k,m,\iota)\in{\Lambda}} |\sum\limits_{|\mathbf{a}|\leq s} \langle D^{\mathbf{a}} f, 2^{-js}D^{\mathbf{a}} {\psi}^{c}_{j,k,m,\iota} \rangle_{L^{2}(\mathbb{R}^{2})} |^{2}\\ &= &\sum\limits_{(j,k,m,\iota)\in{\Lambda}} | \langle f, 2^{-js} {\psi}^{c}_{j,k,m,\iota} \rangle_{H^{s}(\mathbb{R}^{2})} |^{2}. \end{array} $$

We still need to transition back from ${\psi }^{c}_{j,k,m,\iota }$ to ψ_{j, k, m, ι}. We proceed by invoking a triangle inequality

$$ \begin{array}{@{}rcl@{}} \text{IV} &\lesssim & \ \sum\limits_{(j,k,m,\iota)\in{\Lambda}} | \langle f, 2^{-js} {\psi}_{j,k,m,\iota} \rangle_{H^{s}(\mathbb{R}^{2})} |^{2} \\ &&+ \sum\limits_{(j,k,m,\iota)\in{\Lambda}} | \langle f, 2^{-js} ({\psi}_{j,k,m,\iota} - {\psi}^{c}_{j,k,m,\iota}) \rangle_{H^{s}(\mathbb{R}^{2})} |^{2}\\ &= & \ \text{IV}_{1} + \text{IV}_{2}, \end{array} $$

Using a similar estimate as for I₂ in (17) we can estimate by invoking Lemma A.2 that IV₂ is negligible for c small enough. This yields the result. □

To conclude this subsection we would like to examine how the auxiliary functions $\theta ,\widetilde {\theta }$ and μ can be chosen such that they fulfill the frame property required in the proof of Theorem 3.3. Therefore assume $\psi , \widetilde {\psi },$ and ϕ fulfill the assumptions of Theorem 3.3 with γ + 4 > α_sh > γ > 4 + s. Then it follows that

$$ \begin{array}{@{}rcl@{}} \theta(x) &=& \frac{1}{(-2\pi i)^{s}} D^{s}({\psi}^{1})(x_{1}) \phi^{1}(x_{2}),\\ \widetilde{\theta}(x) &=& \phi^{1}(x_{1}) \frac{1}{(-2\pi i)^{s}} D^{s}({\psi}^{1})(x_{2}),\\ \mu &=& \phi^{1}(x_{1}) \phi^{1}(x_{2}). \end{array} $$

Furthermore D^s(ψ¹) and ϕ¹ satisfy the assumptions of Theorem 3.3 and therefore there exists a sampling parameter $\bar {c}>0$ such that for $c_{1}=c_{2} \leq \bar c$ and c = (c₁, c₂) the system $\mathcal {SH}(\phi , \psi , \widetilde {\psi }, c)$ constitutes a frame for $L^{2}(\mathbb {R}^{2})$.

Appendix B: Localization of shearlet and wavelet frames

We now turn to the proof of Proposition 3.2. For this, we will require the following technical lemma.

Lemma B.1

[40] Let$\psi \in L^{2}(\mathbb {R}^{2})$be such that there exists C > 0 with

$$ \begin{array}{@{}rcl@{}} |\widehat{\psi}({\xi}_{1},{\xi}_{2})| \leq C \frac{\min\{1,|{\xi}_{1}|^{\alpha }\}}{\max\{1, |{\xi}_{1}|^{\beta} \} \max\{1, |{\xi}_{2}|^{\beta} \}}, \quad \text{for a.e. } ({\xi}_{1}, {\xi}_{2}) \in \mathbb{R}^{2}, \end{array} $$

where β/2 > α > 1. Then, for ι = − 1, 1,

$$ \begin{array}{@{}rcl@{}} \sum\limits_{|k| \leq 2^{j/2}}|({\psi}_{j,k,m,\iota})^{\wedge}({\xi}_{1},{\xi}_{2})| \leq 2^{-3/4j} C^{\prime} \frac{1}{\max\{1,|2^{-j}{\xi}_{1}|^{\beta/2}\}}\frac{1}{\max\{1,|2^{-j}{\xi}_{2}|^{\beta/2}\}}, \end{array} $$

for a.e. $({\xi }_{1}, {\xi }_{2}) \in \mathbb {R}^{2}$and a constant$C^{\prime }$.

Proof

See [40, Lemma 3.2.1] for a proof. □

Now we are ready to prove Proposition 3.2.

Proof

of Proposition 3.2 First of all, by the definition of ${\Theta }_{\tau , t}^{c}$ one has that for all $(j_{\text {sh}},k,m_{\text {sh}},\iota ) \in {{\Lambda }_{0}^{c}}$ and $(j_{\mathrm {w}},m_{\mathrm {w}},\upsilon ) \in {\Theta }_{\tau , t}^{c}$ such that $j_{\mathrm {w}} < 1/(2\tau ) j_{\text {sh}} + t$ we have that $supp (2^{-j_{\mathrm {w}}s}\omega _{j_{\mathrm {w}},m_{\mathrm {w}},\upsilon })^{d} \cap supp {\psi }_{j_{\text {sh}},k,m_{\text {sh}},\iota } = \emptyset$. Hence, we can assume in the sequel that $j_{\mathrm {w}} > 1/(2\tau ) j_{\text {sh}} + t$.

Furthermore, the total number of wavelet translates for a fixed level j_w is of order $2^{2j_{\mathrm {w}}}$. W.l.o.g., the following computations can be done for υ = 1. Using this observation and Plancherel’s identity, we obtain

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{(j_{\mathrm{w}},m_{\mathrm{w}},1)\in {\Theta}_{\tau, t }^{c}} \sum\limits_{(j_{\text{sh}},k,m_{\text{sh}},\iota) \in {{\Lambda}_{0}^{c}}} |\langle (2^{-j_{\mathrm{w}}s}\omega_{j_{\mathrm{w}},m_{\mathrm{w}},{1}})^{d}, 2^{-j_{\text{sh}} s}{\psi}_{j_{\text{sh}},k,m_{\text{sh}},\iota} \rangle_{H^{s}({\Omega})}|^{2} \\ &\!\lesssim\! & \ \sum\limits_{j_{\mathrm{w}} = 0}^{\infty}\sum\limits_{j_{\text{sh}} =0}^{(2\tau)(j_{\mathrm{w}}-t)} \sum\limits_{|k| \leq 2^{j_{\text{sh}}/2} } 2^{2j_{\mathrm{w}}}\max\limits_{m_{\text{sh}}, m_{\mathrm{w}}}(\sum\limits_{|\mathbf{a}| \leq s}|\langle (\cdot)^{\mathbf{a}} \widehat{(2^{-j_{\mathrm{w}} s}\omega_{j_{\mathrm{w}},m_{\mathrm{w}},{1}})^{d}}, (\cdot)^{\mathbf{a}} 2^{-j_{\text{sh}} s}\widehat{{\psi}_{j_{\text{sh}},k,m_{\text{sh}},\iota}} \rangle_{L^{2}(\mathbb{R}^{2})}|^{2})\\ &\!\lesssim\! & \ \sum\limits_{|\mathbf{a}| \leq s}\sum\limits_{j_{\mathrm{w}} = 0}^{\infty}\sum\limits_{j_{\text{sh}} =0}^{(2\tau)(j_{\mathrm{w}}-t)} \sum\limits_{|k| \leq 2^{j_{\text{sh}}/2} } 2^{2j_{\mathrm{w}}}\max\limits_{m_{\text{sh}}, m_{\mathrm{w}}}(|\langle (\cdot)^{\mathbf{a}} \widehat{(2^{-j_{\mathrm{w}} s}\omega_{j_{\mathrm{w}},m_{\mathrm{w}},{1}})^{d}}, (\cdot)^{\mathbf{a}} 2^{-j_{\text{sh}} s}\widehat{{\psi}_{j_{\text{sh}},k,m_{\text{sh}},\iota}} \rangle_{L^{2}(\mathbb{R}^{2})}|^{2}). \end{array} $$

(9)

Leveraging on the frequency decay of the corresponding shearlet atoms, applying Lemma B.1, and using (W1) yields for any |a|≤ s

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{j_{\text{sh}} = 0}^{(2\tau)(j_{\mathrm{w}}-t)} \sum\limits_{{|k|}\leq 2^{j_{\text{sh}}/2}} 2^{2j_{\mathrm{w}}}\max\limits_{m_{\mathrm{w}},m_{\text{sh}}} |\langle (\cdot)^{\mathbf{a}} \widehat{(2^{-j_{\mathrm{w}} s}\omega_{j_{\mathrm{w}},m_{\mathrm{w}},{1}})^{d}}, (\cdot)^{\mathbf{a}} 2^{-j_{\text{sh}} s} \widehat{{\psi}_{j_{\text{sh}},k,m_{\text{sh}},\iota}} \rangle_{L^{2}(\mathbb{R}^{2})}|^{2}\\ & \lesssim& \sum\limits_{j_{\text{sh}} = 0}^{(2\tau)(j_{\mathrm{w}}-t)} 2^{- 3/2j_{\text{sh}}} \left( {\int}_{\mathbb{R}^{2}} \frac{2^{-j_{\mathrm{w}} s}\xi^{\mathbf{a}} \min\{1, |2^{-j_{\mathrm{w}}} {\xi}_{1}|^{{\alpha}_{\mathrm{w}}}\}}{\max\{1, |2^{-j_{\mathrm{w}}}{\xi}_{1}|^{{\beta}_{\mathrm{w}}}\}\max\{1, |2^{-j_{\mathrm{w}}}{\xi}_{2}|^{{\beta}_{\mathrm{w}}}\}} \right. \\ && \quad \cdot \left. \frac{ 2^{-j_{\text{sh}} s}\xi^{\mathbf{a}}}{\max\{1, |2^{- j_{\text{sh}}}{\xi}_{1}|^{{\beta}_{\text{sh}}/2}\}\max\{1, |2^{-j_{\text{sh}}}{\xi}_{2}|^{{\beta}_{\text{sh}}/2}\}} d\xi \right)^{2} =: \mathrm{I}. \end{array} $$

By a simple computation we obtain that if β_w ≥ s

$$ \begin{array}{@{}rcl@{}} \frac{2^{-j_{\mathrm{w}} s}\xi^{\mathbf{a}} }{\max\{1, |2^{-j_{\mathrm{w}}}{\xi}_{1}|^{{\beta}_{\mathrm{w}}}\max\{1, |2^{-j_{\mathrm{w}}}{\xi}_{2}|^{{\beta}_{\mathrm{w}}-s}\} }\lesssim \frac{1}{\max\{1, |2^{-j_{\mathrm{w}}}{\xi}_{1}|^{{\beta}_{\mathrm{w}}-s}\} \max\{1, |2^{-j_{\mathrm{w}}}{\xi}_{2}|^{{\beta}_{\mathrm{w}}-s}\} }. \end{array} $$

We plug this estimate into the estimate above and obtain with ${\beta }_{\mathrm {w}}^{\prime }={\beta }_{\mathrm {w}}-s$ and ${\beta }_{\text {sh}}^{\prime }={\beta }_{\text {sh}}/2-s$:

$$ \begin{array}{@{}rcl@{}} \mathrm{I} & \lesssim& \sum\limits_{j_{\text{sh}} = 0}^{(2\tau)(j_{\mathrm{w}}-t)} 2^{- 3/2j_{\text{sh}}} \left( {\int}_{\mathbb{R}^{2}} \frac{ \min\{1, |2^{-j_{\mathrm{w}}} {\xi}_{1}|^{{\alpha}_{\mathrm{w}}}\}}{\max\{1, |2^{-j_{\mathrm{w}}}{\xi}_{1}|^{{\beta}_{\mathrm{w}}^{\prime}}\}\max\{1, |2^{-j_{\mathrm{w}}}{\xi}_{2}|^{{\beta}_{\mathrm{w}}^{\prime}}\}} \right. \\ & &\quad \cdot \left. \frac{1}{\max\{1, |2^{- j_{\text{sh}}}{\xi}_{1}|^{{\beta}_{\text{sh}}^{\prime}}\}\max\{1, |2^{-j_{\text{sh}}}{\xi}_{2}|^{{\beta}_{\text{sh}}^{\prime}}\}} d\xi \right)^{2} = \text{II}. \end{array} $$

We continue by applying the substitution $\xi \mapsto 2^{j_{\text {sh}}} \xi$

$$ \begin{array}{@{}rcl@{}} \text{II} &\lesssim& \sum\limits_{j_{\text{sh}} = 0}^{(2\tau)(j_{\mathrm{w}}-t)} 2^{5/2j_{\text{sh}}} \left( {\int}_{\mathbb{R}^{2}} \frac{\min\{1, | 2^{j_{\text{sh}} - j_{\mathrm{w}}}{\xi}_{1}|^{{\alpha}_{\mathrm{w}}}\}}{\max\{1, |2^{j_{\text{sh}} - j_{\mathrm{w}}} {\xi}_{1}|^{{\beta}_{\mathrm{w}}^{\prime}}\}\max\{1, |2^{j_{\text{sh}} - j_{\mathrm{w}}}{\xi}_{2}|^{{\beta}_{\mathrm{w}}^{\prime}}\}} \right. \\ & &\quad \cdot \left. \frac{1}{\max\{1, |{\xi}_{1}|^{{\beta}_{\text{sh}}^{\prime}}\}\max\{1, |{\xi}_{2}|^{{\beta}_{\text{sh}}^{\prime}}\}} d\xi \right)^{2}. \\ &\lesssim &\sum\limits_{j_{\text{sh}} = 0}^{\infty} 2^{5/2j_{\text{sh}}} \left( {\int}_{\mathbb{R}^{2}} \frac{\min\{1, |2^{j_{\text{sh}} - j_{\mathrm{w}}} {\xi}_{1}|^{{\alpha}_{\mathrm{w}}}\}}{\max\{1, |{\xi}_{1}|^{{\beta}_{\text{sh}}^{\prime}}\}\max\{1, |{\xi}_{2}|^{{\beta}_{\text{sh}}^{\prime}}\}} d\xi \right)^{2} \\ &\lesssim &\sum\limits_{j_{\text{sh}} = 0}^{\infty} 2^{5/2j_{\text{sh}} + 2{\alpha}_{\mathrm{w}}(j_{\text{sh}}-j_{\mathrm{w}})} \left( {\int}_{\mathbb{R}^{2}} \frac{ |{\xi}_{1}|^{{\alpha}_{\mathrm{w}}} }{\max\{1, |{\xi}_{1}|^{{\beta}_{\text{sh}}^{\prime}}\}\max\{1, |{\xi}_{2}|^{{\beta}_{\text{sh}}^{\prime}}\}} d\xi \right)^{2}. \end{array} $$

Since ${\beta }_{\text {sh}}^{\prime }-{\alpha }_{\mathrm {w}}>1$ we obtain that the integral above is finite and hence we conclude

$$ \begin{array}{@{}rcl@{}} \mathrm{I} \lesssim \sum\limits_{j_{\text{sh}} = 0}^{(2\tau)(j_{\mathrm{w}}-t)} 2^{5/2j_{\text{sh}} + 2{\alpha}_{\mathrm{w}}(j_{\text{sh}}-j_{\mathrm{w}})}. \end{array} $$

(10)

We rewrite the last sum above as

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j_{\text{sh}} = 0}^{(2\tau)(j_{\mathrm{w}}-t)} 2^{5/2j_{\text{sh}} + 2{\alpha}_{\mathrm{w}}(j_{\text{sh}}-j_{\mathrm{w}})} = 2^{-2 {\alpha}_{\mathrm{w}} \epsilon j_{\mathrm{w}} }\sum\limits_{j_{\text{sh}} = 0}^{(2\tau)(j_{\mathrm{w}}-t)} 2^{5/2j_{\text{sh}} + 2{\alpha}_{\mathrm{w}}(j_{\text{sh}}-(1-\epsilon)j_{\mathrm{w}})}. \end{array} $$

(11)

Since j_w > 1/(2τ)j_sh + t, we can now estimate

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j_{\text{sh}} = 0}^{\infty} 2^{5/2j_{\text{sh}} + 2{\alpha}_{\mathrm{w}}(j_{\text{sh}}-(1-\epsilon)j_{\mathrm{w}})} \lesssim 2^{-2 {\alpha}_{\mathrm{w}} (1-\epsilon)t} \sum\limits_{j_{\text{sh}}=0}^{\infty} 2^{5/2j_{\text{sh}} + 2{\alpha}_{\mathrm{w}}(j_{\text{sh}}-(1-\epsilon) (1/(2\tau)j_{\text{sh}}) }. \end{array} $$

Since α_w((1 − ε)/τ − 2) > 5/2 by assumption, the latter sum is finite. This leads to the estimate

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j_{\text{sh}} = 0}^{\infty} 2^{5/2j_{\text{sh}} + 2{\alpha}_{\mathrm{w}}(j_{\text{sh}}-(1-\epsilon)j_{\mathrm{w}})} \lesssim 2^{-2 {\alpha}_{\mathrm{w}} (1-\epsilon)t}. \end{array} $$

(12)

Now (23) in combination with (22) and (21) implies together with (20) that

$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{(j_{\text{sh}},k,m,\iota) \in {{\Lambda}_{0}^{c}}} \sum\limits_{(j_{\mathrm{w}},m^{\prime},1)\in {\Theta}_{\tau, t }^{c}} |\langle ({2^{-j_{\mathrm{w}} s}\omega_{j_{\mathrm{w}},m^{\prime},\upsilon}})^{d}, 2^{-j_{\text{sh}} s}{\psi}_{j_{\text{sh}},k,m,\iota} \rangle_{H^{s}({\Omega})}|^{2} \\ &\lesssim& \sum\limits_{j_{\mathrm{w}} = 0}^{\infty} 2^{-2 {\alpha}_{\mathrm{w}} \epsilon j_{\mathrm{w}} } 2^{-2 {\alpha}_{\mathrm{w}} (1-\epsilon)t}\lesssim 2^{-2 {\alpha}_{\mathrm{w}} (1-\epsilon)t}. \end{array} $$

□