Lemma 3.1
Let \(\phi , {\widetilde{\phi }} \in L^2({{\mathbb {R}}^+})\) be given. Then the collection \(\big \{\phi (x\ominus \sigma ):\sigma \in \Lambda ^+\big \}\) is biorthogonal to \(\big \{\widetilde{\phi }(x\ominus \sigma ):\sigma \in \Lambda ^+\big \}\) if and only if
$$\begin{aligned} \sum _{\sigma \in \Lambda ^+_+}{\widehat{\phi }}(\zeta \oplus \sigma )\overline{\widehat{\widetilde{\phi }}(\zeta \oplus \sigma )}=1\quad a.e\; \zeta \in {\mathbb {R}}^+. \end{aligned}$$
Proof
For \(\gamma \in \Lambda ^+\), it follows that \(\left\langle \phi (x\ominus \sigma ), \widetilde{\phi }(x\ominus \gamma )\right\rangle =\delta _{\sigma , \gamma } \Leftrightarrow \left\langle \phi , \widetilde{\phi }(x\ominus \gamma )\right\rangle =\delta _{0, \gamma }\). Moreover, we have
$$\begin{aligned} \begin{array}{rcl} \big \langle \phi , {\widetilde{\phi }}(x\ominus \gamma )\big \rangle &{}=&{}\left\langle {\widehat{\phi }},\widehat{ {\widetilde{\phi }}}(x\ominus \gamma )\right\rangle \\ \ &{}=&{}\displaystyle \int _{{\mathbb {R}}^+}{\widehat{\phi }}(\zeta )\overline{\widehat{{\widetilde{\phi }}}(\zeta )}\overline{\chi (\gamma ,\zeta )}\mathrm{d}\zeta \\ &{}=&{}\displaystyle \int _0^{1/2}\left\{ \sum _{m \in {\mathbb {Z}}}{\widehat{\phi }}\left( \zeta \oplus \frac{m}{2}\right) \overline{\widehat{{\widetilde{\phi }}}\left( \zeta \oplus \frac{m}{2}\right) }\chi (\gamma ,m)\right\} \overline{\chi (\gamma ,\zeta )}\mathrm{d}\zeta . \end{array} \end{aligned}$$
Using the fact that \(\big \{\overline{\chi (\gamma ,\zeta )}:\gamma \in \Lambda ^+\big \}\) is an orthonormal basis of \(L^2\left[ 0, \frac{1}{2}\right)\), we obtain the desired result. \(\square\)
Now we proceed to establish a sufficient condition for the translates of a function to be linearly independent.
Lemma 3.2
Let \(\phi \in L^2({{\mathbb {R}}^+})\). Suppose there exists two constants \(C, D>0\) such that
$$\begin{aligned} C \le \sum _{\sigma \in \Lambda ^+}\left| {\widehat{\phi }}(\zeta \oplus \sigma )\right| ^2\le D\quad for\; a.e \; \zeta \in {{\mathbb {R}}^+}. \end{aligned}$$
(3.1)
Then, the set \(\big \{\phi (x\ominus \sigma ):\sigma \in \Lambda ^+\big \}\) is linearly independent.
Proof
For the proof of the lemma, it is sufficient to find another function say \({\widetilde{\phi }}\) whose translates are biorthogonal to \(\phi\). To do this, we define the function \({\widetilde{\phi }}\) by
$$\begin{aligned} \widehat{{\widetilde{\phi }}}(\zeta )=\dfrac{\widehat{\phi }(\zeta )}{\displaystyle \sum _{\sigma \in \Lambda ^+}\left| {\widehat{\phi }}(\zeta \oplus \sigma )\right| ^2}. \end{aligned}$$
Equation (3.1) implies that \({\widetilde{\phi }}\) is well defined and
$$\begin{aligned} \begin{array}{rcl} \displaystyle \sum _{\gamma \in \Lambda ^+}{\widehat{\phi }}(\zeta \oplus \gamma )\overline{\widehat{{\widetilde{\phi }}}(\zeta \oplus \gamma )}&{}=&{}\displaystyle \sum _{\gamma \in \Lambda ^+}{\widehat{\phi }}(\zeta \oplus \gamma )\dfrac{\overline{{\widehat{\phi }}(\zeta \oplus \gamma )}}{\displaystyle \sum _{\sigma \in \Lambda ^+}\left| {\widehat{\phi }}(\zeta \oplus \sigma \oplus \gamma )\right| ^2}\\ &{}=&{}\dfrac{\displaystyle \sum _{\gamma \in \Lambda ^+}\left| {\widehat{\phi }}(\zeta \oplus \gamma )\right| ^2}{\displaystyle \sum _{\nu \in \sigma }\left| {\widehat{\phi }}(\zeta +\nu )\right| ^2}\\ &{}=&{}1. \end{array} \end{aligned}$$
Applying Lemma 3.1, it follows that the set \(\big \{\phi (x\ominus \sigma ):\sigma \in \Lambda ^+\big \}\) is linearly independent. Thus the proof is completed . \(\square\)
Lemma 3.3
Assume that the scaling function \(\phi\) satisfies inequality (3.1). Let \(g=\sum _{\sigma \in \Lambda ^+}h_\sigma \phi (x\ominus \sigma )\), where \(g \in {\text {span}}\big \{\phi (x\ominus \sigma ):\sigma \in \Lambda ^+\big \}\) and \(\big \{h_\sigma \big \}\) is a finite sequence. Define the Fourier transform of h by \(\widehat{h}(\zeta )=\displaystyle \sum\nolimits _{\sigma \in \Lambda ^+}h_\sigma \overline{\chi (\sigma ,\zeta )}\). Then
$$\begin{aligned} C\int _0^{1/2}\big |{\widehat{h}}(\zeta )\big |^2\mathrm{d}\zeta \le \big \Vert g\big \Vert _2^2\le D\int _0^{1/2}\big |{\widehat{h}}(\zeta )\big |^2\mathrm{d}\zeta . \end{aligned}$$
Proof
By using Placherel’s theorem, we obtain
$$\begin{aligned} \begin{array}{rcl} \displaystyle \int _{{\mathbb {R}}^+}\big |g(x)\big |^2dx&{}=&{}\displaystyle \int _{{\mathbb {R}}^+}\left| \sum _{\sigma \in \Lambda ^+}h_\sigma \phi (x\ominus \sigma )\right| ^2dx\\ \ &{}=&{}\displaystyle \int _{{\mathbb {R}}^+}\left| \sum _{\sigma \in \Lambda ^+}h_\sigma {\widehat{\phi }}(\zeta )\overline{\chi (\sigma ,\zeta )}\right| ^2\mathrm{d}\zeta \\ \ &{}=&{}\displaystyle \int _{{\mathbb {R}}^+}\big |{\widehat{\phi }}(\zeta )\big |^2\left| \sum _{\sigma \in \Lambda ^+}h_\sigma \overline{\chi (\sigma ,\zeta )}\right| ^2\mathrm{d}\zeta \\ \ &{}=&{}\displaystyle \int _{{\mathbb {R}}^+}\big |{\widehat{\phi }}(\zeta )\big |^2\big |{\widehat{h}}(\zeta )\big |^2\mathrm{d}\zeta \\ \ &{}=&{}\displaystyle \int _0^{1/2}\sum _{m \in {\mathbb {Z}}}\left| {\widehat{\phi }}\left( \zeta \oplus \frac{m}{2}\right) \right| ^2 \left| {\widehat{h}}(\zeta )\right| ^2\mathrm{d}\zeta . \end{array} \end{aligned}$$
Using identity (3.1), the result follows. \(\square\)
Theorem 3.4
Let \(\big \{\phi (x\ominus \sigma ):\sigma \in \Lambda ^+\big \}\) be a Riesz basis for its closed linear span. Suppose that there exists a function \(\big \{\widetilde{\phi }(x\ominus \sigma ):\sigma \in \Lambda ^+\big \}\) which is biorthogonal to \(\big \{\phi (x\ominus \sigma ):\sigma \in \Lambda ^+\big \}\). Then, for every \(f \in \overline{{\text {span}}}\left\{ \phi (x\ominus \sigma ):\sigma \in \Lambda ^+\right\}\), we have
$$\begin{aligned} f =\sum _{\sigma \in \Lambda ^+}\left\langle f, \widetilde{\phi }(x\ominus \sigma )\right\rangle \phi (x\ominus \sigma ); \end{aligned}$$
(3.2)
and there exists constants \(C, D>0\) such that
$$\begin{aligned} C\big \Vert f\big \Vert _2^2\le \sum _{\sigma \in \Lambda ^+}\left| \left\langle f, \widehat{\widetilde{\phi }}(\zeta \ominus \sigma )\right\rangle \right| ^2\le D\big \Vert f\big \Vert _2^2. \end{aligned}$$
(3.3)
Proof
We first prove (3.2) and (3.3) for any \(f \in {\text {span}}\big \{\phi (x\ominus \sigma ):\sigma \in \Lambda ^+\big \}\) and then generalize it to \(\overline{{\text {span}}}\big \{\phi (x\ominus \sigma ):\sigma \in \Lambda ^+\big \}\). Let \(f \in {\text {span}}\big \{\phi (x\ominus \sigma ):\sigma \in \Lambda ^+\big \}\), then there exists a finite sequence \(\big \{h_\sigma : \sigma \in \Lambda ^+\big \}\) such that \(f =\sum _{\sigma \in \Lambda ^+}h_\sigma \phi (x\ominus \sigma )\). Also, the biorthogonality condition implies that
$$\begin{aligned} \big \langle f, {\widetilde{\phi }}(x\ominus \gamma )\big \rangle= & {} \left\langle \displaystyle \sum _{\sigma \in \Lambda ^+}h_\sigma \phi (x\ominus \sigma ), {\widetilde{\phi }}(x\ominus \gamma )\right\rangle \\= & {} \displaystyle \sum _{\sigma \in \Lambda ^+}h_\sigma \big \langle \phi (x\ominus \sigma ), {\widetilde{\phi }}(x\ominus \gamma )\big \rangle \\= & {} h_\sigma , \end{aligned}$$
which proves (3.2). In order to prove (3.3), we make the use of Lemma 3.3 to get
$$\begin{aligned} D^{-1}\big \Vert f\big \Vert _2^2\le \int _0^{1/2}\big |\widehat{h}(\zeta )\big |^2\mathrm{d}\zeta \le C^{-1}\big \Vert f\big \Vert _2^2. \end{aligned}$$
Using the Placherel formula for Fourier series and the fact that \(h_\sigma =\left\langle f, \widetilde{\phi }(x\ominus \sigma )\right\rangle\), we obtain
$$\begin{aligned} \int _0^{1/2}\big |{\widehat{h}}(\zeta )\big |^2\mathrm{d}\zeta =\sum _{\sigma \in \Lambda ^+}\big |h_\sigma \big |^2=\sum _{\sigma \in \Lambda ^+}\left| \big \langle f, {\widetilde{\phi }}(x\ominus \sigma )\big \rangle \right| ^2. \end{aligned}$$
This proves (3.3). We now generalize the results to \(\overline{{{{\text {span}}}}}\left\{ \phi (x\ominus \sigma ):\sigma \in \Lambda ^+\right\}\). For \(f \in \overline{{{{\text {span}}}}}\left\{ \widetilde{\phi }(x\ominus \sigma ):\sigma \in \Lambda ^+\right\}\), there exists a sequence \(\{f_m:m \in {\mathbb {Z}}\}\) in \({\text {{{span}}}}\left\{ {\widetilde{\phi }}(x\ominus \sigma ):\sigma \in \Lambda ^+\right\}\) such that \(\Vert f_m-f\Vert _2 \rightarrow 0\) as \(m \rightarrow \infty .\) Thus, for each \(\sigma \in \Lambda ^+\), we have
$$\begin{aligned} \big \langle f_m, {\widetilde{\phi }}(x\ominus \sigma )\big \rangle \rightarrow \big \langle f, {\widetilde{\phi }}(x\ominus \sigma )\big \rangle \quad \text {as}\; m \rightarrow \infty . \end{aligned}$$
Hence, the result holds for each \(f_m\). Thus, we have
$$\begin{aligned} \sum _{\sigma \in \Lambda ^+}\left| \big \langle f, \widetilde{\phi }(x\ominus \sigma )\big \rangle \right| ^2&=\sum _{\sigma \in \Lambda ^+}\lim _{m \rightarrow \infty }\left| \big \langle f_m, \widetilde{\phi }(x\ominus \sigma )\big \rangle \right| ^2\nonumber \\&=\lim _{m \rightarrow \infty }\sum _{\sigma \in \Lambda ^+}\left| \big \langle f_m, {\widetilde{\phi }}(x\ominus \sigma )\big \rangle \right| ^2\nonumber \\&\le D\lim _{m \rightarrow \infty }\big \Vert f_m\big \Vert _2^2\nonumber \\&=D\big \Vert f\big \Vert _2^2. \end{aligned}$$
(3.4)
Moreover, we have
$$\begin{aligned} \left\{ \displaystyle \sum _{\sigma \in \Lambda ^+}\left| \big \langle f_m, \widetilde{\phi }(x\ominus \sigma )\big \rangle \right| ^2\right\} ^{1/2}&\le \left\{ \displaystyle \sum _{\sigma \in \Lambda ^+}\left| \big \langle f_m-f, \widetilde{\phi }(x\ominus \sigma )\big \rangle \right| ^2\right\} ^{1/2}+\left\{ \displaystyle \sum _{\sigma \in \Lambda ^+}\left| \big \langle f, \widetilde{\phi }(x\ominus \sigma )\big \rangle \right| ^2\right\} ^{1/2}. \end{aligned}$$
As the upper bound in (3.3) holds for \(f_m-f\) and lower bound for each \(f_m\), we infer that
$$\begin{aligned} C^{1/2}\big \Vert f\big \Vert _2 \le D^{1/2}\big \Vert f_m-f\big \Vert _2+\left( \sum _{\sigma \in \Lambda ^+}\left| \big \langle f_m, \widetilde{\phi }(x\ominus \sigma )\big \rangle \right| ^2\right) ^{1/2}, \end{aligned}$$
from which we conclude that
$$\begin{aligned} C\big \Vert f\big \Vert _2^2 \le \sum _{\sigma \in \Lambda ^+}\left| \left\langle f, \widetilde{\phi }(x\ominus \sigma )\right\rangle \right| ^2. \end{aligned}$$
(3.5)
Combining (3.4) and (3.5), we obtain (3.3). Similarly, we can prove (3.2) for
$$\begin{aligned} f \in \overline{{{{\text {span}}}}}\left\{ \phi (x\ominus \sigma ):\sigma \in \Lambda ^+\right\} \end{aligned}$$
and the proof is completed. \(\square\)
Now we proceed to establish the properties of Nonuniform Biorthogonal wavelets on positive half line.
Let \(\{{{\mathcal {V}}}_j: j \in {\mathbb {Z}}\}\) and \(\{\widetilde{{\mathcal {V}}}_j: j \in {\mathbb {Z}}\}\) be biorthogonal NUMRA’s with scaling functions \(\phi\) and \({\widetilde{\phi }}\). Then there exists integral periodic functions \(m_0\) and \({\widetilde{m}}_0\) with the property \(\widehat{\phi }(\zeta )=m_0\left( \zeta /N\right) \widehat{\phi }\left( \zeta /N\right)\) and \(\widehat{\widetilde{\phi }}(\zeta )={\widetilde{m}}_0\left( \zeta /N\right) \widehat{\widetilde{\phi }}\left( \zeta /N\right)\). Suppose there exists integral periodic functions \(m_\ell\) and \({\widetilde{m}}_\ell , 1 \le \ell \le N-1\) such that
$$\begin{aligned} {\mathcal {M}}(\zeta ) \overline{\widetilde{{\mathcal {M}}}(\zeta )}=I, \end{aligned}$$
(3.6)
where
$$\begin{aligned} {\mathcal {M}}(\zeta )=\left( \begin{array}{cccc} m_0\left( \dfrac{\zeta }{N}\right) &{}m_0\left( \dfrac{\zeta }{N}\oplus \dfrac{1}{2N}\right) &{}\dots &{}m_0\left( \dfrac{\zeta }{N}\oplus \dfrac{N-1}{2N}\right) \\ m_1\left( \dfrac{\zeta }{N}\right) &{}m_2\left( \dfrac{\zeta }{N}\oplus \dfrac{1}{2N}\right) &{}\dots &{}m_2\left( \dfrac{\zeta }{N}\oplus \dfrac{N-1}{2N}\right) \\ \vdots &{}\vdots &{}\ddots &{}\vdots \\ m_{N-1}\left( \dfrac{\zeta }{N}\right) &{}m_{N-1}\left( \dfrac{\zeta }{N}\oplus \dfrac{1}{2N}\right) &{}\dots &{}m_{N-1}\left( \dfrac{\zeta }{N}\oplus \dfrac{N-1}{2N}\right) \end{array} \right) \end{aligned}$$
and
$$\begin{aligned} \widetilde{{\mathcal {M}}}(\zeta )=\left( \begin{array}{cccc} {\widetilde{m}}_0\left( \dfrac{\zeta }{N}\right) &{}{\widetilde{m}}_0\left( \dfrac{\zeta }{N}\oplus \dfrac{1}{2N}\right) &{}\dots &{}{\widetilde{m}}_0\left( \dfrac{\zeta }{N}\oplus \dfrac{N-1}{2N}\right) \\ {\widetilde{m}}_1\left( \dfrac{\zeta }{N}\right) &{}{\widetilde{m}}_2\left( \dfrac{\zeta }{N}\oplus \dfrac{1}{2N}\right) &{}\dots &{}{\widetilde{m}}_2\left( \dfrac{\zeta }{N}\oplus \dfrac{N-1}{2N}\right) \\ \vdots &{}\vdots &{}\ddots &{}\vdots \\ {\widetilde{m}}_{N-1}\left( \dfrac{\zeta }{N}\right) &{}{\widetilde{m}}_{N-1}\left( \dfrac{\zeta }{N}\oplus \dfrac{1}{2N}\right) &{}\dots &{}{\widetilde{m}}_{N-1}\left( \dfrac{\zeta }{N}\oplus \dfrac{N-1}{2N}\right) \end{array} \right) . \end{aligned}$$
For \(1 \le \ell \le N-1\), define the associated biorthgonal wavelets as \(\psi _\ell\) and \({\widetilde{\psi }}_\ell\) by
$$\begin{aligned} {\widehat{\psi }}_\ell (\zeta )=m_\ell \left( \zeta /N\right) \widehat{\phi }\left( \zeta /N\right) \quad \text {and} \quad \widehat{\widetilde{\psi }}_\ell (\zeta )={\widetilde{m}}_\ell \left( \zeta /N\right) \widehat{{\widetilde{\phi }}}\left( \zeta /N\right) . \end{aligned}$$
Definition 3.5
A pair of NUMRA’s \(\{{{\mathcal {V}}}_j: j \in {\mathbb {Z}}\}\) and \(\{\widetilde{{\mathcal {V}}}_j: j \in {\mathbb {Z}}\}\) with scaling functions \(\phi\) and \({\widetilde{\phi }}\), respectively are said to be biorthogonal to each other if \(\{\phi (\cdot \ominus \sigma ): \sigma \in \Lambda ^+\}\) and \(\{{\widetilde{\phi }}(\cdot \ominus \sigma ): \sigma \in \Lambda ^+\}\) are biorthogonal.
Definition 3.6
Let \(\phi\) and \({\widetilde{\phi }}\) be scaling functions for biorthogonal NUMRA’s. For each \(j \in {\mathbb {Z}}\), define the operators \(P_j\) and \({\widetilde{P}}_j\) on \(L^2({\mathbb {R}}^+)\) by
$$\begin{aligned} P_j f=\sum _{\sigma \in \Lambda ^+}\big \langle f, {\widetilde{\phi }}_{j, \sigma }\big \rangle \phi _{j, \sigma }\quad \text {and} \quad {\widetilde{P}}_j f=\sum _{\sigma \in \Lambda ^+}\big \langle f, \phi _{j, \sigma }\big \rangle {\widetilde{\phi }}_{j, \sigma }, \end{aligned}$$
respectively. It is easy to verify that these operators are uniformly bounded on \(L^2({\mathbb {R}}^+)\) and both the series are convergent in \(L^2({\mathbb {R}}^+)\).
Remark 3.7
The operators \(P_j\) and \({\widetilde{P}}_j\) satisfy the following properties.
-
(a)
\(P_j f=f \iff f \in V_j\) and \({\widetilde{P}}_j f=f \iff f \in {\widetilde{V}}_j\).
-
(b)
\(\displaystyle \lim _{j \rightarrow \infty }\big \Vert P_j f-f\big \Vert _2=0\) and \(\displaystyle \lim _{j \rightarrow -\infty }\big \Vert P_j f\big \Vert _2=0\) for every \(f \in L^2({\mathbb {R}}^+).\)
Lemma 3.8
Let \(\phi\) and \({\widetilde{\phi }}\) be the scaling functions for biorthogonal NUMRA’s and \(\psi _\ell\) and \({\widetilde{\psi }}_\ell , 1 \le \ell \le N-1\) be the associated wavelets satisfying (3.6). Then, we have the following
-
(a)
\(\big \{\psi _{\ell ,0,\sigma }: \sigma \in \Lambda ^+\big \}\) is biorthogonal to \(\big \{{\widetilde{\psi }}_{\ell ,0,\gamma }:\gamma \in \Lambda ^+\big \}\),
-
(b)
\(\big \langle \psi _{\ell ,0,\sigma }, {\widetilde{\phi }}_{0, \gamma }\big \rangle =\big \langle {\widetilde{\psi }}_{\ell ,0,\sigma }, \phi _{0, \gamma }\big \rangle ,\quad {\text {for all}} \;\sigma , \gamma \in \Lambda ^+\).
Proof
we have
$$\begin{aligned}&\sum _{t \in {\mathbb {Z}}}{\widehat{\psi }}_\ell \left( \zeta \oplus \dfrac{t}{2}\right) \overline{\widehat{{\widetilde{\psi }}}_\ell \left( \zeta \oplus \dfrac{t}{2}\right) }\\&\quad =\displaystyle \sum _{t \in {\mathbb {Z}}}\left\{ m_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{t}{2N}\right) {\widehat{\phi }}\left( \dfrac{\zeta }{N} \oplus \dfrac{t}{2N}\right) \overline{{\widetilde{m}}_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{t}{2N}\right) }\,\overline{\widehat{{\widetilde{\phi }}}\left( \dfrac{\zeta }{N} \oplus \dfrac{t}{2N}\right) }\right\} \\&\quad =\sum _{s=0}^{N-1}\sum _{t \in {\mathbb {Z}}}\left\{ m_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{t}{2}\oplus \dfrac{s}{2N}\right) {\widehat{\phi }}\left( \dfrac{\zeta }{N} \oplus \dfrac{t}{2}\oplus \dfrac{s}{2N}\right) \overline{{\widetilde{m}}_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{t}{2}\oplus \dfrac{s}{2N}\right) }\,\overline{\widehat{{\widetilde{\phi }}}\left( \dfrac{\zeta }{N} \oplus \dfrac{t}{2}\oplus \dfrac{s}{2N}\right) }\right\} \\&\quad =\sum _{s=0}^{N-1}\left\{ m_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{s}{2N}\right) \overline{{\widetilde{m}}_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{s}{2N}\right) }\right\} \\&\quad =1. \end{aligned}$$
Hence, by Lemma 3.1, \(\big \{\psi _{\ell ,0,\sigma }: \sigma \in \Lambda ^+\big \}\) is biorthogonal to \(\big \{{\widetilde{\psi }}_{\ell ,0,\sigma }:\sigma \in \Lambda ^+\big \}\). This proves part (a). To prove part (b), we have for, \(\sigma , \gamma \in \Lambda ^+\)
$$\begin{aligned} \displaystyle \big \langle \psi _{\ell ,0,\sigma }, {\widetilde{\phi }}_{0, \gamma }\big \rangle= & {} \big \langle \psi _\ell (x\ominus \sigma ), {\widetilde{\phi }}(x\ominus \gamma )\big \rangle \\= & {} \displaystyle \left\langle {\widehat{\psi }}_\ell \,\overline{\chi (\sigma . \zeta )}, \widehat{{\widetilde{\phi }}}\,\overline{\chi (\gamma ,\zeta )}\right\rangle \\= & {} \displaystyle \int _{{\mathbb {R}}^+} m_\ell \left( \dfrac{\zeta }{N}\right) \widehat{\phi }\left( \dfrac{\zeta }{N}\right) \overline{\chi (\sigma ,\zeta )} \, \overline{\widetilde{m}_0\left( \dfrac{\zeta }{N}\right) }\,\overline{\widehat{\widetilde{\phi }}\left( \dfrac{\zeta }{N}\right) }\overline{\chi (\gamma ,\zeta )} \mathrm{d}\zeta \\= & {} \displaystyle \int _0^{1/2}\sum _{p \in {\mathbb {Z}}}\left\{ m_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{p}{2N}\right) {\widehat{\phi }}\left( \dfrac{\zeta }{N} \oplus \dfrac{p}{2N}\right) \right. \\&\left. \qquad \times \overline{{\widetilde{m}}_0\left( \dfrac{\zeta }{N} \oplus \dfrac{p}{2N}\right) }\,\overline{\widehat{{\widetilde{\phi }}}\left( \dfrac{\zeta }{N} \oplus \dfrac{p}{2N}\right) }\right\} \chi (\gamma \ominus \sigma ,\zeta )\mathrm{d}\zeta \\= & {} \displaystyle \int _0^{1/2}\sum _{s=0}^{N-1}\sum _{p \in {\mathbb {Z}}}\left\{ m_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{p}{2}\oplus \dfrac{s}{2N}\right) {\widehat{\phi }}\left( \dfrac{\zeta }{N} \oplus \dfrac{p}{2}\oplus \dfrac{s}{2N}\right) \right. \\&\left. \times \overline{{\widetilde{m}}_0\left( \dfrac{\zeta }{N} \oplus \dfrac{p}{2}\oplus \dfrac{s}{2N}\right) }\,\overline{\widehat{{\widetilde{\phi }}}\left( \dfrac{\zeta }{N} \oplus \dfrac{p}{2}\oplus \dfrac{s}{2N}\right) }\right\} \chi (\gamma \ominus \sigma ,\zeta ) \mathrm{d}\zeta \\= & {} \displaystyle \int _0^{1/2}\sum _{s=0}^{N-1}\left\{ m_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{s}{2N}\right) \overline{{\widetilde{m}}_0\left( \dfrac{\zeta }{N} \oplus \dfrac{s}{2N}\right) }\right\} \chi (\gamma \ominus \sigma ,\zeta )\mathrm{d}\zeta \\= & {} 0. \end{aligned}$$
The dual one can also be shown equal to zero in a similar manner. This proves part (b) and hence the proof is completed. \(\square\)
Theorem 3.9
Let \(\phi ,{\widetilde{\phi }},\psi _\ell\) and \({\widetilde{\psi }}_\ell , 1 \le \ell \le N-1\) be as in Theorem 3.8. Let \(\psi _0=\phi\) and \({\widetilde{\psi }}_0={\widetilde{\phi }}\). Then, for every \(f \in L^2({{\mathbb {R}}})\), we have
$$\begin{aligned} P_1f=P_0f+\sum _{\ell =1}^{N-1}\sum _{\sigma \in \Lambda ^+}\big \langle f, {\widetilde{\psi }}_{\ell , 0, \sigma }\big \rangle \psi _{\ell , 0, \sigma } \end{aligned}$$
(3.7)
and
$$\begin{aligned} {\widetilde{P}}_1f={\widetilde{P}}_0f+\sum _{\ell =1}^{N-1}\sum _{\sigma \in \Lambda ^+}\big \langle f, \psi _{\ell , 0, \sigma }\big \rangle \widetilde{\psi }_{\ell , 0, \sigma }. \end{aligned}$$
(3.8)
where the series in (3.7) and (3.8) converges in \(L^2({{\mathbb {R}}^+})\).
Proof
For the sake of convenience, we will only prove (3.7), as (3.8) is an easy consequence. In particular, we will prove it in the weak sense only. For this, let \(f, g \in L^2({{\mathbb {R}}})\). Then, we have
$$\begin{aligned}&\sum _{\ell =0}^{N-1}\sum _{\sigma \in \Lambda ^+}\left\langle f, {\widetilde{\psi }}_{\ell , 0, \sigma }\right\rangle \overline{\big \langle g, \psi _{\ell , 0, \sigma }\big \rangle }\nonumber \\&\quad =\displaystyle \sum _{\ell =0}^{N-1}\sum _{\sigma \in \Lambda ^+}\left\{ \int _{{\mathbb {R}}^+} {\widehat{f}}(\zeta )\overline{\widehat{{\widetilde{\psi }}}_\ell (\zeta ) }\chi ( \sigma ,\zeta )\mathrm{d}\zeta \right\} \left\{ \int _{{\mathbb {R}}^+} \overline{{\widehat{g}}(\zeta )}{\widehat{\psi }}_\ell (\zeta ) \overline{\chi ( \sigma ,\zeta )}\mathrm{d}\zeta \right\} \nonumber \\&\quad =\displaystyle \sum _{\ell =0}^{N-1}\sum _{\sigma \in \Lambda ^+}\left\{ \int _0^{1/2}\sum _{p \in {\mathbb {Z}}} {\widehat{f}}\left( \zeta \oplus \dfrac{p}{2}\right) \overline{\widehat{{\widetilde{\psi }}}_\ell \left( \zeta \oplus \dfrac{p}{2}\right) }\chi ( \sigma ,\zeta )\mathrm{d}\zeta \right\} \nonumber \\&\qquad \times \left\{ \displaystyle \int _0^{1/2}\sum _{q \in {\mathbb {Z}}} \overline{{\widehat{g}}\left( \zeta \oplus \dfrac{q}{2}\right) }{\widehat{\psi }}_\ell \left( \zeta \oplus \dfrac{q}{2}\right) \overline{\chi ( \sigma ,\zeta )}\mathrm{d}\zeta \right\} \nonumber \\&\quad =\displaystyle \sum _{\ell =0}^{N-1}\int _0^{1/2}\left\{ \sum _{p \in {\mathbb {Z}}} {\widehat{f}}\left( \zeta \oplus \dfrac{p}{2}\right) \overline{\widehat{{\widetilde{\psi }}}_\ell \left( \zeta \oplus \dfrac{p}{2}\right) }\right\} \left\{ \sum _{q \in {\mathbb {Z}}}\overline{{\widehat{g}}\left( \zeta \oplus \dfrac{q}{2}\right) }{\widehat{\psi }}_\ell \left( \zeta \oplus \dfrac{q}{2}\right) \right\} \mathrm{d}\zeta \nonumber \\ \end{aligned}$$
$$\begin{aligned}&=\displaystyle \int _0^{1/2}\sum _{\ell =0}^{N-1}\left\{ \sum _{p \in {\mathbb {Z}}} {\widehat{f}}\left( \zeta \oplus \dfrac{p}{2}\right) \overline{{\widetilde{m}}_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{p}{2N}\right) }\,\overline{\widehat{{\widetilde{\phi }}}\left( \dfrac{\zeta }{N} \oplus \dfrac{p}{2N}\right) }\right. \nonumber \\&\qquad \times \left. \displaystyle \sum _{q \in {\mathbb {Z}}}\overline{{\widehat{g}}\left( \zeta \oplus \dfrac{q}{2}\right) }m_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{q}{2N}\right) {\widehat{\phi }}\left( \dfrac{\zeta }{N} \oplus \dfrac{q}{2N}\right) \right\} \mathrm{d}\zeta \nonumber \\&\quad =\displaystyle \int _0^{1/2}\sum _{\ell =0}^{N-1}\left\{ \sum _{r=0}^{N-1}\sum _{p^\prime \in {\mathbb {Z}}} {\widehat{f}}\left( \zeta \oplus \dfrac{p^\prime }{2}N\oplus \dfrac{r}{2}\right) \overline{{\widetilde{m}}_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{r}{2N}\oplus \dfrac{p^\prime }{2}\right) }\,\overline{\widehat{{\widetilde{\phi }}}\left( \dfrac{\zeta }{N} \oplus \dfrac{r}{2N}\oplus \dfrac{p^\prime }{2}\right) }\right. \nonumber \\&\qquad \times \displaystyle \sum _{s=0}^{N-1}\sum _{q^\prime \in {\mathbb {N}}_0}\overline{{\widehat{g}}\left( \zeta \oplus \dfrac{q^\prime }{2}N\oplus \dfrac{s}{2}\right) }m_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{s}{2N}\oplus \dfrac{q^\prime }{2}\right) \left. {\widehat{\phi }}\left( \dfrac{\zeta }{N} \oplus \dfrac{s}{2N}\oplus \dfrac{q^\prime }{2}\right) \right\} \mathrm{d}\zeta \nonumber \\&\quad =\displaystyle \int _0^{1/2}\sum _{r=0}^{N-1}\sum _{p^\prime \in {\mathbb {N}}_0}\sum _{s=0}^{N-1}\sum _{q^\prime \in {\mathbb {N}}_0}\left\{ \sum _{\ell =0}^{N-1} \overline{{\widetilde{m}}_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{r}{2N}\right) }m_\ell \left( \dfrac{\zeta }{N} \oplus \dfrac{s}{2N}\right) \right\} \nonumber \\&\qquad \times {\widehat{f}}\left( \zeta \oplus \dfrac{p^\prime }{2}N\oplus \dfrac{r}{2}\right) \overline{\widehat{{\widetilde{\phi }}}\left( \dfrac{\zeta }{N} \oplus \dfrac{r}{2N}+\dfrac{p^\prime }{2}\right) }\,\overline{{\widehat{g}}\left( \zeta \oplus \dfrac{q^\prime }{2}N\oplus \dfrac{s}{2}\right) }{\widehat{\phi }}\left( \dfrac{\zeta }{N} \oplus \dfrac{s}{2N}\oplus \dfrac{q^\prime }{2}\right) \mathrm{d}\zeta \nonumber \\&\quad =\displaystyle \int _0^{1/2}\sum _{p^\prime \in {\mathbb {N}}_0}\sum _{q^\prime \in {\mathbb {N}}_0}\sum _{s=0}^{N-1}{\widehat{f}}\left( \zeta \oplus \dfrac{p^\prime }{2}N\oplus \dfrac{s}{2}\right) \overline{\widehat{{\widetilde{\phi }}}\left( \dfrac{\zeta }{N} \oplus \dfrac{s}{2N}\oplus \dfrac{p^\prime }{2}\right) }\nonumber \\&\qquad \times \overline{{\widehat{g}}\left( \zeta \oplus \dfrac{q^\prime }{2}N\oplus \dfrac{s}{2}\right) }{\widehat{\phi }}\left( \dfrac{\zeta }{N} \oplus \dfrac{s}{2N}\oplus \dfrac{p^\prime }{2}\right) \mathrm{d}\zeta \nonumber \\&\quad =\displaystyle \sum _{s=0}^{N-1}\int _0^{{s+1}/2}\sum _{p^\prime \in {\mathbb {N}}_0}\sum _{q^\prime \in {\mathbb {N}}_0}{\widehat{f}}\left( \zeta \oplus \dfrac{p^\prime }{2}N\right) \overline{\widehat{{\widetilde{\phi }}}\left( \dfrac{\zeta }{N} \oplus \dfrac{p^\prime }{2}\right) }\,\overline{{\widehat{g}}\left( \zeta \oplus \dfrac{q^\prime }{2}N\right) }{\widehat{\phi }}\left( \dfrac{\zeta }{N} \oplus \dfrac{p^\prime }{2}\right) \mathrm{d}\zeta . \, \end{aligned}$$
(3.9)
Furthermore, we have
$$\begin{aligned}&\displaystyle \sum _{\sigma \in \Lambda ^+}\left\langle f, \widetilde{\phi }_{1, \sigma }\right\rangle \overline{\left\langle g, \phi _{1, \sigma }\right\rangle }\nonumber \\&\quad =\displaystyle \sum _{\sigma \in \Lambda ^+}\left\{ \int _{{\mathbb {R}}} {\widehat{f}}(\zeta )\overline{\widehat{{\widetilde{\phi }}}\left( \dfrac{\zeta }{N}\right) }e^{2\pi i \zeta /N}\mathrm{d}\zeta \right\} \left\{ \int _{{\mathbb {R}}} \overline{{\widehat{g}}(\zeta )}{\widehat{\phi }}\left( \dfrac{\zeta }{N}\right) e^{-2\pi i \zeta /N}\mathrm{d}\zeta \right\} \nonumber \\&\quad =\displaystyle \int _0^{1/2}\sum _{p\in {\mathbb {Z}}} \widehat{f}\left( \zeta \oplus \dfrac{p}{2}N\right) \overline{\widehat{\widetilde{\phi }}\left( \dfrac{\zeta }{N}\oplus \dfrac{p}{2}\right) }\mathrm{d}\zeta \int _0^{1/2}\sum _{q\in {\mathbb {Z}}} \overline{\widehat{g}\left( \zeta \oplus \dfrac{q}{2}N\right) }\widehat{\phi }\left( \dfrac{\zeta }{N}\oplus \dfrac{q}{2}\right) \mathrm{d}\zeta \nonumber \\&\quad =\displaystyle \int _0^{1/2}\sum _{p\in {\mathbb {Z}}} \widehat{f}\left( \zeta \oplus \dfrac{p}{2}N\right) \overline{\widehat{\widetilde{\phi }}\left( \dfrac{\zeta }{N}\oplus \dfrac{p}{2}\right) }\mathrm{d}\zeta \int _0^{1/2}\sum _{q\in {\mathbb {Z}}} \overline{\widehat{g}\left( \zeta \oplus \dfrac{q}{2}N\right) }\widehat{\phi }\left( \dfrac{\zeta }{N}\oplus \dfrac{q}{2}\right) \mathrm{d}\zeta \nonumber \\&\quad =\displaystyle \int _0^{1/2}\sum _{p\in {\mathbb {Z}}}\sum _{q\in {\mathbb {Z}}} {\widehat{f}}\left( \zeta \oplus \dfrac{p}{2}N\right) \overline{\widehat{\widetilde{\phi }}\left( \dfrac{\zeta }{N}\oplus \dfrac{p}{2}\right) }\overline{{\widehat{g}}\left( \zeta \oplus \dfrac{q}{2}N\right) }\widehat{\phi }\left( \dfrac{\zeta }{N}\oplus \dfrac{q}{2}\right) \mathrm{d}\zeta . \end{aligned}$$
(3.10)
Combing (3.9) and (3.10), we get the desired result. \(\square\)
Theorem 3.10
Let \(\phi\) and \({\widetilde{\phi }}\) be the scaling functions for biorthogonal NUMRA’s and \(\psi _\ell\) and \({\widetilde{\psi }}_\ell , 1 \le \ell \le N-1\) be the associated wavelets satisfying the matrix condition (3.6). Then, the collection \(\big \{\psi _{\ell , j, \sigma }: 1 \le \ell \le N-1, j \in {\mathbb {Z}}, \sigma \in \Lambda ^+ \big \}\) and \(\big \{{\widetilde{\psi }}_{\ell , j, \sigma }: 1 \le \ell \le N-1, j \in {\mathbb {Z}}, \sigma \in \Lambda ^+ \big \}\) are biorthogonal. Moreover, if
$$\begin{aligned} \big |{\widehat{\phi }}(\zeta )\big |\le K(1+|\zeta |)^{-\frac{1}{2}-\epsilon },\,\big |\widehat{{\widetilde{\phi }}}(\zeta )\big |\le K(1+|\zeta |)^{-\frac{1}{2}-\epsilon },\,\big |{\widehat{\psi }}_\ell (\zeta )\big |\le K|\zeta |\; and \;\big |\widehat{{\widetilde{\psi }}}(\zeta )\big |\le K|\zeta |, \end{aligned}$$
(3.11)
for some constant \(K>0,\,\epsilon >0\) and for a.e. \(\zeta \in {\mathbb {R}}\), then \(\big \{\psi _{\ell , j, \sigma }: 1 \le \ell \le N-1, j \in {\mathbb {Z}}, \sigma \in \Lambda ^+ \big \}\) and \(\big \{\widetilde{\psi }_{\ell , j, \sigma }: 1 \le \ell \le N-1, j \in {\mathbb {Z}}, \sigma \in \Lambda ^+ \big \}\) form Riesz bases for \(L^2({{\mathbb {R}}})\).
Proof
First we show that \(\big \{\psi _{\ell , j, \sigma }: 1 \le \ell \le N-1, j \in {\mathbb {Z}}, \sigma \in \Lambda ^+ \big \}\) and \(\big \{{\widetilde{\psi }}_{\ell , j, \sigma }: 1 \le \ell \le N-1, j \in {\mathbb {Z}}, \sigma \in \Lambda ^+ \big \}\) are biorthogonal to each other. For this, we will show that for each \(\ell ,\, 1\le \ell \le N-1\) and \(j \in {\mathbb {Z}}\),
$$\begin{aligned} \big \langle \psi _{\ell , j, \sigma }, {\widetilde{\psi }}_{\ell , j, \gamma }\big \rangle =\delta _{\sigma , \gamma }. \end{aligned}$$
(3.12)
In fact, we have already proved (3.12) for \(j=0\). For \(j\ne 0\), we have
$$\begin{aligned} \big \langle \psi _{\ell , j, \sigma }, {\widetilde{\psi }}_{\ell , j, \gamma }\big \rangle =\big \langle D_{-j} \psi _{\ell , 0, \sigma }, D_{-j}{\widetilde{\psi }}_{\ell , 0, \gamma }\big \rangle =\big \langle \psi _{\ell , 0, \sigma }, \widetilde{\psi }_{\ell , 0, \gamma }\big \rangle =\delta _{\sigma , \gamma }. \end{aligned}$$
Also, for fixed \(\sigma , \gamma \in \Lambda ^+\) and \(j, j^\prime \in {\mathbb {Z}}\) with \(j<j^\prime\), we claim that
$$\begin{aligned} \big \langle \psi _{\ell , j, \sigma }, {\widetilde{\psi }}_{\ell ^\prime , j^\prime , \gamma }\big \rangle =0. \end{aligned}$$
(3.13)
As \(\psi _{\ell , 0, \sigma } \in {{\mathcal {V}}}_1\), hence \(\psi _{\ell , j, \sigma }=D_{-j} \psi _{\ell , 0, \sigma }\in {{\mathcal {V}}}_{j+1}\subseteq {{\mathcal {V}}}_{j^\prime }\). Therefore, it is enough to show that \({\widetilde{\psi }}_{\ell ^\prime , j^\prime , \gamma }\) is orthogonal to every element of \({{\mathcal {V}}}_{j^\prime }\). Let \(g \in \mathcal{V}_{j^\prime }\). Since \(\big \{\phi _{j^\prime , \sigma }: \sigma \in \Lambda ^+\big \}\) is a Riesz basis for \({{\mathcal {V}}}_{j^\prime }\), hence there exists an \(l^2\)-sequence \(\big \{d_\sigma : \sigma \in \Lambda ^+\big \}\) such that \(g=\sum _{\sigma \in \Lambda ^+}d_\sigma \phi _{j^\prime , \sigma }\) in \(L^2({\mathbb {R}}^+)\). Using part (b) of Lemma 3.6, we have
$$\begin{aligned} \big \langle {\widetilde{\psi }}_{\ell ^\prime , j^\prime , \gamma }, \phi _{j^\prime , \sigma }\big \rangle =\big \langle D_{-j^\prime } {\widetilde{\psi }}_{\ell ^\prime , 0, \gamma }, D_{-j^\prime }\phi _{0, \sigma }\big \rangle =0. \end{aligned}$$
Therefore,
$$\begin{aligned} \big \langle {\widetilde{\psi }}_{\ell ^\prime , j^\prime , \gamma }, g\big \rangle =\Big \langle {\widetilde{\psi }}_{\ell ^\prime , j^\prime , \gamma }, \displaystyle \sum _{\sigma \in \Lambda ^+}d_\sigma \phi _{j^\prime , \sigma }\Big \rangle =\displaystyle \sum _{\sigma \in \Lambda ^+}d_\sigma \big \langle {\widetilde{\psi }}_{\ell ^\prime , j^\prime , \gamma }, \phi _{j^\prime , \sigma }\big \rangle =0. \end{aligned}$$
We now show that these two collections form Riesz bases for \(L^2({\mathbb {R}}^+)\). The linear independence is clear from the fact that these collections are biorthogonal to each other. So, we have to check the frame inequalities only i.e., there exists constants \(C, {\widetilde{C}}, D, {\widetilde{D}}>0\) such that
$$\begin{aligned} C\big \Vert g\big \Vert _2^2\le \sum _{\ell =1}^{N-1}\sum _{j \in \mathbb {Z}}\sum _{\sigma \in \Lambda ^+}\left| \big \langle g, \psi _{\ell , j, \sigma }\big \rangle \right| ^2\le D\big \Vert g\big \Vert _2^2,\quad \forall \; f \in L^2({\mathbb {R}}^+), \end{aligned}$$
(3.14)
and
$$\begin{aligned} {\widetilde{C}}\big \Vert g\big \Vert _2^2\le \sum _{\ell =1}^{N-1}\sum _{j \in {\mathbb {Z}}}\sum _{\sigma \in \Lambda ^+}\left| \big \langle g, \widetilde{\psi }_{\ell , j, \sigma }\big \rangle \right| ^2\le \widetilde{D}\big \Vert g\big \Vert _2^2,\quad \forall \; f \in L^2({\mathbb {R}}^+). \end{aligned}$$
(3.15)
Let us first check the existence of the upper bounds in (3.14) and (3.15 ). For this, we have
$$\begin{aligned} \sum _{\sigma \in \Lambda ^+}\left| \big \langle g, {\widetilde{\psi }}_{\ell , j, \sigma }\big \rangle \right| ^2= & {} \displaystyle \sum _{\sigma \in \Lambda ^+}\left| \int _{{\mathbb {R}}^+} {\widehat{g}}(\zeta )(N)^{-j/2}\,\overline{{\widehat{\psi }}_\ell \big (N^{-j}\zeta \big )}\chi ( \sigma ,N^{-j}\zeta ) \mathrm{d}\zeta \right| ^2\\= & {} N^{-j}\displaystyle \sum _{\sigma \in \Lambda ^+}\left| \int _0^{1/2} \sum _{p \in {\mathbb {Z}}}{\widehat{g}}\left( \zeta \oplus (N)^j\dfrac{p}{2}\right) \overline{{\widehat{\psi }}_\ell \left( (N)^{-j}\zeta \oplus \dfrac{p}{2}\right) }\chi ( \sigma ,N^{-j}\zeta )\mathrm{d}\zeta \right| ^2\\= & {} \displaystyle \int _0^{1/2}\left| \sum _{p \in {\mathbb {Z}}}{\widehat{g}}\left( \zeta \oplus (N)^j\dfrac{p}{2}\right) \overline{{\widehat{\psi }}_\ell \left( (N)^{-j}\zeta \oplus \dfrac{p}{2}\right) }\right| ^2\mathrm{d}\zeta \\= & {} \displaystyle \int _0^{1/2}\left\{ \sum _{p \in {\mathbb {Z}}}\left| {\widehat{g}}\left( \zeta \oplus (N)^j\dfrac{p}{2}\right) \right| ^2\left| {\widehat{\psi }}_\ell \left( (N)^{-j}\zeta \oplus \dfrac{p}{2}\right) \right| ^{2\delta }\right\} \\&\qquad \times \left\{ \displaystyle \sum _{q \in {\mathbb {Z}}}\left| {\widehat{\psi }}_\ell \left( (N)^{-j}\zeta \oplus \dfrac{q}{2}\right) \right| ^{2(1-\delta )}\right\} \mathrm{d}\zeta \\= & {} \displaystyle \int _{{\mathbb {R}}^+}\left| {\widehat{g}}(\zeta )\right| ^2\big |{\widehat{\psi }}_\ell \left( (N)^{-j}\zeta \right) \big |^{2\delta }\sum _{q \in {\mathbb {Z}}}\left| {\widehat{\psi }}_\ell \left( (N)^{-j}\zeta \oplus \dfrac{q}{2}\right) \right| ^{2(1-\delta )}\mathrm{d}\zeta . \end{aligned}$$
By our assumption (3.11), we have \(|{\widehat{\psi }}_\ell (\zeta )|\le K\left( 1+|(N)^{-1}\zeta |\right) ^{-1/2-\epsilon }\) and therefore, it follows that \(\sum _{q \in {\mathbb {Z}}}\big |\widehat{\psi }_\ell \left( (N)^{-j}\zeta +q/2\right) \big |^{2(1-\delta )}\) is uniformly bounded if \(\delta <2\epsilon (1+2\epsilon )^{-1}\). Thus, there exists a constant \(K>0\) such that
$$\begin{aligned} \displaystyle \sum _{\sigma \in \Lambda ^+}\left| \big \langle g, {\widetilde{\psi }}_{\ell , j, \sigma }\big \rangle \right| ^2\le & {} K\displaystyle \int _{{\mathbb {R}}^+}\big | \widehat{g}(\zeta )\big |^2\sum _{j \in {\mathbb {Z}}}\big |\widehat{\psi }_\ell \left( (N)^{-j}\zeta \right) \big |^{2\delta }\mathrm{d}\zeta \\\le & {} K\sup \left\{ \displaystyle \sum _{j \in {\mathbb {Z}}}\big |{\widehat{\psi }}_\ell \left( (N)^{-j}\zeta \right) \big |^{2\delta }: 1 \le \zeta \le N\right\} \big \Vert g\big \Vert _2^2. \end{aligned}$$
Also for \(\zeta \in [1, N]\), we have
$$\begin{aligned} \displaystyle \sum _{j=-\infty }^0\big |{\widehat{\psi }}_\ell \left( (N)^{-j}\zeta \right) \big |^{2\delta }\le & {} \displaystyle \sum _{j=-\infty }^0\dfrac{K^{2\delta }}{\big (1+|(N)^{j-1}\zeta |\big )^{\delta (1+2\epsilon )}}\\\le & {} \displaystyle \sum _{j=-\infty }^0\dfrac{K^{2\delta }}{(N)^{(j-1)\delta (1+2\epsilon )}}\\\le & {} \displaystyle K^{2\delta }\dfrac{q^{\delta (1+2\epsilon )}}{1-(N)^{-\delta (1+2\epsilon )}}. \end{aligned}$$
Furthermore, we have
$$\begin{aligned} \sum _{j=1}^\infty \big |\widehat{\psi }_\ell \left( (N)^{-j}\zeta \right) \big |^{2\delta }\le \sum _{j=1}^\infty \big (K(N)^{-j}|\zeta |\big )^{2\delta }\le K^{2\delta }\sum _{j=1}^\infty (N)^{(-j+1)2\delta }=K^{2\delta }\dfrac{1}{1-(N)^{-2\delta }}, \end{aligned}$$
and hence, it follows that \(\sup \big \{\sum _{j \in \mathbb {Z}}\big |{\widehat{\psi }}_\ell \left( (N)^{-j}\zeta \right) \big |^{2\delta }: 1 \le \zeta \le N\big \}\) is finite. Therefore, there exist \(D>0\) such that of (3.15) holds. Similarly, we can show for dual one also. The existence of lower bounds for both the cases can be shown in similar fashion. Thus, we have
$$\begin{aligned} \big \Vert g\big \Vert _2^2= & {} \big \langle g, g \big \rangle \\= & {} \left\langle \displaystyle \sum _{\ell =1}^{N-1}\sum _{j \in {\mathbb {Z}}}\sum _{\sigma \in \Lambda ^+}\big \langle g, {\widetilde{\psi }}_{\ell , j, \sigma }\big \rangle \psi _{\ell , j, \sigma }, g \right\rangle \\= & {} \displaystyle \sum _{\ell =1}^{N-1}\sum _{j \in {\mathbb {Z}}}\sum _{\sigma \in \Lambda ^+} \big \langle g, {\widetilde{\psi }}_{\ell , j, \sigma }\big \rangle \big \langle \psi _{\ell , j, \sigma }, g \big \rangle \\\le & {} \left( \displaystyle \sum _{\ell =1}^{N-1}\sum _{j \in {\mathbb {Z}}}\sum _{\sigma \in \Lambda ^+} \left| \big \langle g, {\widetilde{\psi }}_{\ell , j, \sigma }\big \rangle \right| ^2\right) ^{1/2}\left( \displaystyle \sum _{\ell =1}^{N-1}\sum _{j \in {\mathbb {Z}}}\sum _{\sigma \in \Lambda ^+} \left| \big \langle g, \psi _{\ell , j, \sigma }\big \rangle \right| ^2\right) ^{1/2}\\\le & {} ({\widetilde{D}})^{1/2}\big \Vert g\big \Vert _2\left( \displaystyle \sum _{\ell =1}^{N-1}\sum _{j \in {\mathbb {Z}}}\sum _{\sigma \in \Lambda ^+} \left| \big \langle g, \psi _{\ell , j, \sigma }\big \rangle \right| ^2\right) ^{1/2}. \end{aligned}$$
Hence,
$$\begin{aligned} \dfrac{1}{{\widetilde{D}}}\,\big \Vert g\big \Vert _2^2\le \sum _{\ell =1}^{N-1}\sum _{j \in {\mathbb {Z}}}\sum _{\sigma \in \Lambda ^+} \left| \big \langle g, \psi _{\ell , j, \sigma }\big \rangle \right| ^2. \end{aligned}$$
The dual case can be proved in similar lines. This completes the proof. \(\square\)