$\Lambda _s$-NONUNIFORM MULTIRESOLUTION ANALYSIS

Abstract

Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal. 158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.

1 Introduction

A multiresolution analysis, with dilation factor 2 [2, 3, 7], is an increasing sequence of closed subspaces $\{V_j\}_{j\in \mathbb Z}$ of $L^2(\mathbb R)$ along with a $\ \phi \in V_0$ satisfying: $\{\phi (\cdot -k):k\in \mathbb Z\}$ is an orthonormal basis for $V_0$ , $\cap _{j\in \mathbb Z}V_j=\{0\}$ , $\overline {\cup _{j\in \mathbb Z}V_j}=L^2(\mathbb R)$ and $f\in V_j$ if and only if $f(2\cdot ) \in V_{j+1}$ for all $j\in \mathbb Z$ . The function $\phi $ is called a scaling function. Multiresolution analysis (MRA) is an important tool, which was introduced by Mallat and Meyer, in constructing a wavelet $\psi \in L^2(\mathbb R)$ such that the collection $\{2^{\kern1.5pt j/2}\psi (2^{\kern1.5pt j}x-k):j,k\in \mathbb Z\}$ is an orthonormal basis for $L^2(\mathbb R)$ . The concept of MRA has been extended in several ways in past years, like generalizing to $L^2(\mathbb R^n)$ , allowing the subspaces of MRA to be generated by a Riesz basis instead of an orthonormal basis, admitting a finite number of scaling functions and replacing the dilation factor of two by an integer $N\geq 2$ . All these concepts were developed such that the translation set is always a subgroup of $\mathbb R$ . Gabardo and Nashed [4] considered a generalization of Mallat’s classical theory of multiresolution analysis to nonuniform multiresolution analysis (NUMRA) based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,({r}/{N})\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime.

Definition 1.1. Let $N\in \mathbb N$ and r be an odd integer relatively prime to N such that $1\leq r\leq 2N-1$ and $\Lambda =\{0,({r}/{N})\}+2\mathbb Z$ . An associated nonuniform multiresolution analysis (NUMRA) is a collection $\{V_j\}_{j\in \mathbb Z}$ of closed subspaces of $L^2(\mathbb R)$ satisfying the following conditions:

1. (i) there exists a $\phi \in V_0$ , called a scaling function, such that $\{\phi (\cdot -\lambda ):\lambda \in \Lambda \}$ is an orthonormal basis for $V_0$ ;

2. (ii) $V_j\subseteq V_{j+1}$ for all $j\in \mathbb Z$ ;

3. (iii) $f\in V_j$ if and only if $f(2N\cdot )\in V_{j+1}$ for all $j\in \mathbb {Z}$ ;

4. (iv) $\overline {\cup _{j\in \mathbb Z}V_j}=L^2(\mathbb R)$ ;

5. (v) $\cap _{j\in \mathbb Z}V_j=\{0\}$ .

Note that if $N=1$ , one obtains the standard definition of multiresolution analysis with dyadic dilation 2. In [5], Gabardo and Yu have constructed wavelets associated with nonuniform multiresolution analysis; these wavelets are called nonuniform wavelets. The main activity in signal processing is that of recovering signals from the samples, but the traditional methods assume that the samples are of uniform spacing; recently, much research has been dedicated to the case of nonuniform spacing of the data samples. The problem of signal reconstruction [4, 9] from nonuniformly sampled data arises in many applications, including sampling systems with sampling jitter, the design of irregularly spaced antenna arrays, reconstruction of signals from noisy samples and the processing of geophysical data. Some fundamental results for nonuniform wavelets and wavelet sets related to spectral pairs can be found in [9]. The nonuniform wavelet construction from the NUMRA could be used in signal processing, sampling theory, speech recognition and various other areas, where nonuniform translations are needed. This motivates us to consider the nonuniform translation set ${\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_{q}}/{N}\}+s\mathbb Z}$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that ${1\leq r_i\leq sN-1}$ and $N\geq 2$ instead of $\Lambda =\{0,({r}/{N})\}+2\mathbb Z$ . Now one can ask, is it possible to define a NUMRA with $\widetilde {\Lambda }$ as a translation set? In this paper we prove that the concept of nonuniform multiresolution analysis with the translation set $\widetilde {\Lambda }$ can be defined only if $\widetilde {\Lambda }$ is of the form $\{0,({r}/{N})\}+s\mathbb Z$ . Hence, we replace the translation set $\Lambda =\{0,({r}/{N})\}+2\mathbb Z$ by $\Lambda _s=\{0,({r}/{N})\}+s\mathbb Z$ and the dilation factor $2N$ by $sN$ , where s is a positive even integer and r is an odd integer, to introduce $\Lambda _s$ -nonuniform multiresolution analysis with dilation factor $sN$ . For a given $\phi \in L^2(\mathbb R)$ , we define

(1-1) $$ \begin{align} V_j = \left\{ \begin{array}{@{}ll} \overline{\text{span}}\{\phi(\cdot-\lambda):\lambda\in \Lambda \} & \text { if } j = 0,\\[3pt] \{f : f((2N)^{-j}~\cdot)\in V_0 \} & \text{ if } j\in \mathbb Z\setminus\{0\}. \end{array} \right. \end{align} $$

We say that a function $\phi \in L^2(\mathbb R)$ generates a NUMRA if the spaces in (1-1) together with $\phi $ form a NUMRA. Recently, in [7], the authors have characterized the scaling functions that generate a NUMRA. This also motivates us to look for a characterization for scaling functions associated with $\Lambda _s$ -nonuniform multiresolution analysis. This characterization is useful for the construction of wavelets associated with a $\Lambda _s$ -NUMRA.

2 NUMRA with translation set ${\Lambda }_s$

In this section, we consider a set $ \widetilde {\Lambda }=\{{r_0}/{N},{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where $r_0=0$ , $q, r_i, N\in \mathbb N, (r_i,N)=1, i=1,2,\ldots ,q$ and $s\in 2\mathbb N$ . We prove that the nonuniform multiresolution analysis with the translation set $\widetilde {\Lambda }$ can be defined only if $\widetilde {\Lambda }$ is of the form $\{0,({r}/{N})\}+s\mathbb Z$ , where $1\leq r\leq sN-1$ is an odd integer such that r and N are relatively prime. We denote the dilation and translation operators on $L^2(\mathbb R)$ by

$$ \begin{align*}(D^{\kern1.5pt j}f)(x)=(sN)^{\kern1.5pt j/2}f((sN)^{\kern1.5pt j}x) \quad\text{and}\quad (T_\lambda f)(x)=f(x-\lambda),\end{align*} $$

respectively. For $\phi \in L^2(\mathbb R)$ , the following theorem gives a characterization for the translation system $\{T_\lambda \phi : \lambda \in \widetilde {\Lambda }\}$ to be an orthonormal system in $L^2(\mathbb R)$ .

Theorem 2.1. For $\phi \in L^2(\mathbb R), \{T_\lambda \phi : \lambda \in \widetilde {\Lambda }\}$ is an orthonormal system if and only if

$$ \begin{align*}\sum_{p=0}^{sN-1}e^{2\pi i({p}/{s})(({r_i-r_j})/{N})}w\bigg(\xi+\dfrac{p}{s}\bigg)=s\delta_{ij}\quad\text{a.e. } \xi\in\mathbb R,\end{align*} $$

where $w(\xi )=\sum _{k\in \mathbb Z}|\widehat {\phi }(\xi +kN)|^2$ , $i,j\in \{0,1,2,\ldots ,q\}.$

Proof. Suppose that $\{T_\lambda \phi : \lambda \in \widetilde {\Lambda }\}$ is an orthonormal system. Then

$$ \begin{align*} \delta_{\lambda\sigma} &= \langle T_\lambda\phi , T_\sigma\phi \rangle \\ &= \langle \widehat{T_\lambda\phi }, \widehat{T_\sigma\phi} \rangle \\ &= \int_{\mathbb R}|\widehat{\phi}(\xi)|^2 e^{-2\pi i\xi(\lambda-\sigma)} \,d\xi. \end{align*} $$

Suppose that $\lambda =({r_j}/{N})+sm$ and $\sigma =({r_i}/{N})+sn$ , $i,j\in \{0,1,\ldots ,q\},$ $m,n\in \mathbb Z$ . Then

$$ \begin{align*} \delta_{ij}\delta_{mn} &= \int_{\mathbb R}|\widehat{\phi}(\xi)|^2 e^{-2\pi i\xi(sm+({r_j}/{N}))} e^{2\pi i\xi(sn+({r_i}/{N}))} \,d\xi \\ &= \int_{\mathbb R}|\widehat{\phi}(\xi)|^2 e^{-2\pi i\xi s(m-n)} e^{-2\pi i\xi(({r_j-r_i})/{N})} \,d\xi \\[-12pt] \\ &= \sum_{k\in\mathbb Z}\int_{kN}^{(k+1)N}|\widehat{\phi}(\xi)|^2 e^{-2\pi i\xi s(m-n)}e^{-2\pi i\xi(({r_j-r_i})/{N})} \,d\xi \\[2pt] &= \int_0^N \sum_{k\in\mathbb Z}|\widehat{\phi}(\eta+kN)|^2 e^{-2\pi i\eta s(m-n)}e^{-2\pi i\eta(({r_j-r_i})/{N})} \,d\eta\\[2pt] &= \sum_{p=0}^{sN-1}\int_{{p}/{s}}^{({p}/{s})+({1}/{s})} w(\eta) e^{-2\pi i\eta s(m-n)}e^{-2\pi i\eta(({r_j-r_i})/{N})} \,d\eta\\[2pt] &= \int_{0}^{{1}/{s}} \sum_{p=0}^{sN-1} w\bigg(\xi+\frac{p}{s}\bigg) e^{-2\pi i\xi s(m-n)} e^{-2\pi i(\xi+({p}/{s}))(({r_j-r_i})/{N})} \,d\xi\\[2pt] &= \int_{0}^{{1}/{s}} e^{-2\pi i\xi(({r_j-r_i})/{N})} \sum_{p=0}^{sN-1} e^{-2\pi i({p}/{s})(({r_j-r_i})/{N})}w\bigg(\xi+\frac{p}{s}\bigg) e^{-2\pi i\xi s(m-n)}\,d\xi. \end{align*} $$

Since $\{\!\!\sqrt {s}e^{-2\pi i\xi sl} : l\in \mathbb Z\}$ is an orthonormal basis for $L^2([0,{1}/{s}))$ ,

$$ \begin{align*}e^{-2\pi i\xi(({r_j-r_i})/{N})} \sum_{p=0}^{sN-1} e^{-2\pi i({p}/{s})(({r_j-r_i})/{N})}w\bigg(\xi+\frac{p}{s}\bigg)=s\delta_{ij}\quad\text{a.e. } \xi\in\mathbb R.\end{align*} $$

Thus,

$$ \begin{align*}\sum_{p=0}^{sN-1} e^{-2\pi i({p}/{s})(({r_j-r_i})/{N})}w\bigg(\xi+\frac{p}{s}\bigg)=s\delta_{ij}\quad\text{a.e. } \xi\in\mathbb R\end{align*} $$

and so

$$ \begin{align*}\sum_{p=0}^{sN-1}e^{2\pi i({p}/{s})(({r_i-r_j})/{N})}w\bigg(\xi+\frac{p}{s}\bigg)=s\delta_{ij}\quad\text{a.e. } \xi\in\mathbb R.\end{align*} $$

The converse of the proof follows by just retracing the steps.

Theorem 2.2. If $\{T_\lambda \phi : \lambda \in \widetilde {\Lambda }\}$ is an orthonormal system in $L^2(\mathbb R)$ , then $\widetilde {\Lambda }=\{0,({r}/{N})\}+s\mathbb Z$ and $1\leq r\leq sN-1$ is an odd integer.

Proof. Suppose that $\{T_\lambda \phi : \lambda \in \widetilde {\Lambda }\}$ is an orthonormal system. Then, from Theorem 2.1,

(2-1) $$ \begin{align} \sum_{p=0}^{sN-1}\sum_{k\in\mathbb{Z}}e^{2\pi i ({p}/{s})(({r_i-r_j})/{N})}\bigg|\widehat{\phi}\bigg(\xi+\frac{p}{s}+kN\bigg)\bigg|^2=s\delta_{ij}. \end{align} $$

Now let $C_p(\xi )=\sum _{k\in \mathbb {Z}}|\widehat {\phi }(\xi +({p}/{s})+kN)|^2, 0\leq p\leq sN-1$ . Then, by (2-1), for $i=j$ ,

(2-2) $$ \begin{align} \nonumber s &= \sum_{p=0}^{sN-1} C_p(\xi) \\[-2pt] \nonumber &= \sum_{p=0}^{({sN}/{2})-1}C_p(\xi)+\sum_{p=({sN}/{2})}^{sN-1}C_p(\xi) \\[-2pt] &= \sum_{p=0}^{({sN}/{2})-1}(C_p(\xi)+C_{({sN}/{2})+p}(\xi)). \end{align} $$

Again from (2-1), for $i\neq j$ ,

$$ \begin{align*} 0 &= \sum_{p=0}^{sN-1}e^{2\pi i ({p}/{s})(({r_i-r_j})/{N})} C_p(\xi) \\[-2pt] &= \sum_{p=0}^{({sN}/{2})-1}e^{2\pi i ({p}/{s})(({r_i-r_j})/{N})}C_p(\xi)+\sum_{p=({sN}/{2})}^{sN-1}e^{2\pi i ({p}/{s})(({r_i-r_j})/{N})}C_p(\xi) \\[-2pt] &= \sum_{p=0}^{({sN}/{2})-1}e^{2\pi i ({p}/{s})(({r_i-r_j})/{N})}(C_p(\xi)+e^{\pi i(r_i-r_j)}C_{({sN}/{2})+p}(\xi)). \end{align*} $$

If $r_i-r_j$ is even, then

$$ \begin{align*}\sum_{p=0}^{({sN}/{2})-1}e^{2\pi i ({p}/{s})(({r_i-r_j})/{N})}(C_p(\xi)+C_{({sN}/{2})+p}(\xi))=0.\end{align*} $$

Since $\{ e^{2\pi i ({p}/{s})(({r_i-r_j})/{N})} : p\in \{0,1,\ldots ,({sN}/{2})-1\} \}$ is linearly independent in $L^2(\mathbb Z_{{sN}/{2}})$ , $C_p(\xi )+C_{({sN}/{2})+p}(\xi )=0$ for almost every (a.e.) $\xi \in \mathbb R$ and for all ${p\in \{0,1,\ldots ,({sN}/{2})-1\}}$ , that is a contradiction to (2-2). Thus, $r_i-r_j$ is an odd integer for all $i\neq j$ and hence $r_i=r_i-r_0$ is an odd integer for all $i=1,2,\ldots ,q$ . Using the fact that $r_i$ and $r_i-r_j, i\neq j, i=1,2,\ldots , q, j=0,1,\ldots , q$ are odd integers, we can observe that the cardinality of $\{r_1,r_2,\ldots , r_q\}$ is 1. Suppose not. Then there exist $r_m$ and $r_n, m\neq n$ , such that $r_m, r_n$ and $r_m-r_n$ are odd integers, which is not possible. Thus, the cardinality of $\{r_1,r_2,\ldots , r_q\}$ is 1. Hence, $\widetilde {\Lambda }=\{0,({r}/{N})\}+s\mathbb Z$ .

Definition 2.3. Let $N\in \mathbb {N}$ , s be an even positive integer and $1\leq r\leq sN-1$ be an odd integer relatively prime to N. An associated $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) with dilation factor $sN$ is a collection $\{V_j\}_{j\in \mathbb Z}$ of closed subspaces of $L^2(\mathbb R)$ satisfying the following conditions:

1. (i) there exists a $\phi \in V_0$ , called a scaling function, such that $\{T_\lambda \phi :\lambda \in \Lambda _s\}$ is an orthonormal basis for $V_0$ , where $\Lambda _s=\{0,({r}/{N})\}+s\mathbb Z$ ;

2. (ii) $V_j\subseteq V_{j+1}$ for all $j\in \mathbb Z$ ;

3. (iii) $f\in V_j$ if and only if $f(sN\cdot )\in V_{j+1}$ for all $j\in \mathbb Z$ ;

4. (iv) $\cap _{j\in \mathbb Z}V_j=\{0\}$ ;

5. (v) $\overline {\cup _{j\in \mathbb Z}V_j}=L^2(\mathbb R)$ .

When $s=2$ , it gives NUMRA and for $N=1$ and $s=2$ it gives classical MRA. The following theorem gives an equivalent condition for a system $\{T_\lambda \phi :\lambda \in \Lambda _s\}$ to be an orthonormal system in $L^2(\mathbb R)$ .

Theorem 2.4. Let $\phi \in L^2(\mathbb R)$ , $s\in 2\mathbb{N}$ and $1\leq r\leq sN-1$ be an odd integer. Then $\{T_\lambda \phi :\lambda \in \Lambda _s\}$ is an orthonormal system in $L^2(\mathbb R)$ if and only if

(2-3) $$ \begin{align} \sum_{p=0}^{({sN}/{2})-1}\sum_{k\in\mathbb Z}\bigg|\widehat{\phi}\bigg(\xi+\frac{p}{s}+kN\bigg)\bigg|^2=\frac{s}{2} \quad\text{a.e. }\xi\in\mathbb R. \end{align} $$

Proof. Suppose that $\{T_\lambda \phi :\lambda \in \Lambda _s\}$ is an orthonormal system. Then, from Theorem 2.1,

(2-4) $$ \begin{align} \sum_{p=0}^{sN-1}\sum_{k\in\mathbb Z}\bigg|\widehat{\phi}\bigg(\xi+\frac{p}{s}+kN\bigg)\bigg|^2=s \quad\text{a.e. }\xi\in\mathbb R \end{align} $$

and

(2-5) $$ \begin{align} \sum_{p=0}^{sN-1}\sum_{k\in\mathbb Z}e^{2\pi i({p}/{s})({r}/{N})}\bigg|\widehat{\phi}\bigg(\xi+\frac{p}{s}+kN\bigg)\bigg|^2=0 \quad\text{a.e. }\xi\in\mathbb R. \end{align} $$

Let $C_j(\xi )=\sum _{k\in \mathbb Z}|\widehat {\phi }(\xi +({j}/{s})+kN)|^2, 0\leq j\leq sN-1$ . Then $C_{j}(\xi )=C_{sN+j}(\xi )$ for $0\leq j\leq sN-1$ . From (2-4),

(2-6) $$ \begin{align} \nonumber s = \sum_{p=0}^{sN-1}C_p(\xi) &= \sum_{p=0}^{({sN}/{2})-1}C_p(\xi)+ \sum_{p=({sN}/{2})}^{sN-1}C_p(\xi) \\[-2pt] &= \sum_{p=0}^{({sN}/{2})-1}[C_p(\xi)+C_{({sN}/{2})+p}(\xi)]. \end{align} $$

From (2-5),

$$ \begin{align*} \nonumber 0 &= \sum_{p=0}^{sN-1}e^{2\pi i({p}/{s})({r}/{N})}C_p(\xi)\\[-2pt] \nonumber &= \sum_{p=0}^{({sN}/{2})-1}e^{2\pi i({p}/{s})({r}/{N})}C_p(\xi)+ \sum_{p=({sN}/{2})}^{sN-1}e^{2\pi i({p}/{s})({r}/{N})}C_p(\xi) \\[-2pt] \nonumber &= \sum_{p=0}^{({sN}/{2})-1}e^{2\pi i({p}/{s})({r}/{N})}C_p(\xi)+ e^{2\pi i({r}/{sN})(({sN}/{2})+p)}C_{({sN}/{2})+p}(\xi) \\[-2pt] \nonumber &= \sum_{p=0}^{({sN}/{2})-1}e^{2\pi i({p}/{s})({r}/{N})}[C_p(\xi)+ e^{\pi ir}C_{({sN}/{2})+p}(\xi)]. \end{align*} $$

Since r is an odd integer, say $r=2l+1$ ,

(2-7) $$ \begin{align} \sum_{p=0}^{({sN}/{2})-1}e^{2\pi i({p}/{s})2l}[C_p(\xi)-C_{({sN}/{2})+p}(\xi)]e^{2\pi i({p}/{sN})}=0. \end{align} $$

Observe that the left-hand side of (2-7) is a discrete Fourier series on the group ${\mathbb Z}/({({sN}/{2})\mathbb Z})$ and so

$$ \begin{align*}C_p(\xi)=C_{({sN}/{2})+p}(\xi).\end{align*} $$

Hence, from (2-6),

$$ \begin{align*}\sum_{p=0}^{({sN}/{2})-1}\sum_{k\in\mathbb Z}\bigg|\widehat{\phi}\bigg(\xi+\frac{p}{s}+kN\bigg)\bigg|^2=\frac{s}{2} \quad\text{a.e. }\xi\in\mathbb R.\end{align*} $$

By retracing the steps, we obtain the converse part and hence the proof is complete.

3 Characterization of scaling functions for $\Lambda _s$ -NUMRA

In this section, we characterize the functions $\phi \in L^2(\mathbb R)$ that generate $\Lambda _s$ -nonuniform multiresolution analysis. For $\phi \in L^2(\mathbb R)$ , define

(3-1) $$ \begin{align} V_j = \left\{ \begin{array}{@{}ll} \overline{\text{span}}\{\phi(\cdot-\lambda):\lambda\in \Lambda_s\} & \text { if } j = 0,\\[3pt] \{f : f((sN)^{-j}~\cdot)\in V_0 \} & \text{ if } j\neq 0. \end{array} \right. \end{align} $$

Then it is clear that $f\in V_j$ if and only if $f(sN\cdot )\in V_{j+1}$ for all $j\in \mathbb Z$ . One can observe that, if $\{T_\lambda \phi : \lambda \in \Lambda _s\}$ is an orthonormal basis for $V_0$ , then $\{D^{\kern1.5pt j}T_\lambda \phi : \lambda \in \Lambda _s\}$ is an orthonormal basis for $V_j, j\in \mathbb Z$ .

Definition 3.1. A function $\phi \in L^2(\mathbb R)$ is said to generate a $\Lambda _s$ -NUMRA if the spaces $V_j$ defined in (3-1) together with $\phi $ form a $\Lambda _s$ -NUMRA.

Theorem 3.2. Let $\phi \in L^2(\mathbb R)$ be such that $\{T_\lambda \phi : \lambda \in \Lambda _s\}$ is an orthonormal system and the spaces $V_j$ be as defined in (3-1). Then $V_j\subseteq V_{j+1}$ for all $j\in \mathbb Z$ if and only if there exist $({1}/{s})$ -periodic functions $m_0^1$ and $m_0^2$ such that

$$ \begin{align*}\widehat{\phi}(sN\xi)=(m_0^1(\xi)+ m_0^2(\xi)e^{-2\pi i\xi({r}/{N})})\widehat{\phi}(\xi) \quad\text{a.e. }\xi\in\mathbb R.\end{align*} $$

Proof. Suppose that $V_j\subseteq V_{j+1}$ for all $j\in \mathbb Z$ . Then, by (3-1), it is clear that $f\in V_j$ if and only if $f(sN\cdot )\in V_{j+1}$ for all $j\in \mathbb Z$ . Since $\phi \in V_0\subseteq V_1$ , $\phi ({\cdot }/{sN})\in V_0$ . As $\{T_\lambda \phi : \lambda \in \Lambda _s\}$ is an orthonormal basis for $V_0$ ,

$$ \begin{align*}\phi\bigg(\frac{x}{sN}\bigg)=\sum_{\lambda\in\Lambda_s}a_\lambda\phi(x-\lambda).\end{align*} $$

Taking the Fourier transform on both sides,

$$ \begin{align*} \widehat{\phi}(sN\xi) &= \frac{1}{sN}\sum_{\lambda\in\Lambda_s}a_\lambda\widehat{\phi}(\xi)e^{-2\pi i\xi\lambda} \\ &= \bigg(\frac{1}{sN}\sum_{\lambda\in s\mathbb Z}a_\lambda e^{-2\pi i\xi\lambda}+\frac{1}{sN}\sum_{\lambda\in({r}/{N})+s\mathbb Z}a_\lambda e^{-2\pi i\xi\lambda}\bigg)\widehat{\phi}(\xi) \\ &= \bigg(\frac{1}{sN}\sum_{l\in \mathbb Z}a_{sl}e^{-2\pi i\xi sl}+\frac{1}{sN}\sum_{l\in\mathbb Z}a_{({r}/{N})+sl}e^{-2\pi i\xi(({r}/{N})+sl)}\bigg)\widehat{\phi}(\xi). \end{align*} $$

Thus,

$$ \begin{align*}\widehat{\phi}(sN\xi)=(m_0^1(\xi)+ m_0^2(\xi)e^{-2\pi i\xi({r}/{N})})\widehat{\phi}(\xi),\end{align*} $$

where $m_0^1(\xi )=({1}/{sN})\sum _{l\in \mathbb Z}a_{sl}e^{-2\pi i\xi sl}$ and $m_0^2(\xi )=({1}/{sN})\sum _{l\in \mathbb Z}a_{({r}/{N})+sl}e^{-2\pi i\xi sl}$ are $({1}/{s})$ -periodic functions in $L^2([0,{1}/{s}))$ . By reversing the steps above, we get the converse part.

Theorem 3.3. Let $\phi \in L^2(\mathbb R)$ be such that $\{\phi (\cdot -\lambda ): \lambda \in \Lambda _s\}$ is an orthonormal system and the $V_j$ are the spaces as defined in (3-1). Then $\overline {\cup _{j\in \mathbb Z}V_j}=L^2(\mathbb R)$ if and only if $\lim _{j\to \infty }|\widehat {\phi }((sN)^{-j}\xi )|^2=c, ~ c>0,$ a.e. $\xi \in [-({1}/{2s}),({1}/{2s}))$ .

Proof. Assume that $\lim _{j\to \infty }|\widehat {\phi }((sN)^{-j}\xi )|^2=c,~~ c> 0$ . Let $f\in (\overline {\cup _{j\in \mathbb Z}V_j})^{\perp }$ and $P_j$ be the orthogonal projection onto $V_j, j\in \mathbb Z$ . For $\epsilon>0$ , there exists a $g\in L^2(\mathbb R)$ with $\widehat {g}\in C_c(\mathbb R)$ such that

(3-2) $$ \begin{align} \|f-g\|\leq \epsilon. \end{align} $$

Also note that $\|P_jg\|=\|P_j(g-f)\|\leq \epsilon $ for all $j\in \mathbb Z$ .

Let us take $\Lambda _s=\{{r_0}/{N},{r_1}/{N}\}+s\mathbb Z$ , where $r_0=0$ and $r_1=r$ , for simplicity. For $j\in \mathbb Z$ ,

$$ \begin{align*} \|P_jg\|^2 &= \displaystyle\sum_{\lambda\in\Lambda_s}|\langle g, D^{\kern1.5pt j}T_\lambda\phi \rangle|^2 \\ &= \displaystyle\sum_{\lambda\in\Lambda_s}|\langle \widehat{g}, \widehat{D^{\kern1.5pt j}T_\lambda\phi \rangle}|^2 \\ &= \displaystyle\sum_{\lambda\in\Lambda_s}\bigg|\int_{\mathbb R}(sN)^{-({j}/{2})}\widehat{g}(\xi)e^{2\pi i\xi\lambda(sN)^{-j}} \overline{\widehat{\phi}\bigg(\frac{\xi}{(sN)^{\kern1.5pt j}}\bigg)}~ d\xi \bigg|^2\\ &= \displaystyle\sum_{m=0}^{1}\sum_{l\in\mathbb Z}\bigg|\int_{\mathbb R}(sN)^{-({j}/{2})}\widehat{g}(\xi)e^{2\pi i\xi(({r_m}/{N})+sl)(sN)^{-j}} \overline{\widehat{\phi}\bigg(\frac{\xi}{(sN)^{\kern1.5pt j}}\bigg)}~ d\xi \bigg|^2. \end{align*} $$

Next, by changing the variable $\xi =(sN)^{\kern1.5pt j}\eta $ ,

$$ \begin{align*} \|P_jg\|^2 &= (sN)^{\kern1.5pt j}\displaystyle\sum_{m=0}^{1}\sum_{l\in\mathbb Z}\bigg|\int_{\mathbb R}\widehat{g}((sN)^{\kern1.5pt j}\eta) e^{2\pi i\eta({r_m}/{N})}e^{2\pi i\eta sl} \overline{\widehat{\phi}(\eta)}~ d\eta \bigg|^2. \end{align*} $$

Since $\widehat {g}$ has compact support, we can choose j large enough such that supp $(\widehat {g}((sN)^{\kern1.5pt j}\cdot ))\subset [({-1}/{2s}),({1}/{2s})]$ . As $\{\!\!\sqrt {s}e^{2\pi i\xi sl}:l\in \mathbb Z\}$ is an orthonormal basis for $L^2([-{1}/{2s},{1}/{2s}))$ ,

$$ \begin{align*} \|P_jg\|^2 &= \frac{(sN)^{\kern1.5pt j}}{s}\displaystyle\sum_{m=0}^{1}\sum_{l\in\mathbb Z}\bigg|\int_{-{1}/{2s}}^{{1}/{2s}} \widehat{g}((sN)^{\kern1.5pt j}\xi) e^{2\pi i\xi({r_m}/{N})}\sqrt{s}e^{2\pi i\xi sl} \overline{\widehat{\phi}(\xi)}~ d\xi \bigg|^2 \\ &= \frac{(sN)^{\kern1.5pt j}}{s}\sum_{m=0}^{1} \int_{-{1}/{2s}}^{{1}/{2s}}|\kern1.2pt\widehat{g}((sN)^{\kern1.5pt j}\xi)\widehat{\phi}(\xi)|^2\,d\xi\\ &= \frac{2(sN)^{\kern1.5pt j}}{s} \int_{\mathbb R}|\kern1.2pt\widehat{g}((sN)^{\kern1.5pt j}\xi)\widehat{\phi}(\xi)|^2\,d\xi. \end{align*} $$

By change of variable,

$$ \begin{align*} \|P_jg\|^2 &= \frac{2}{s} \int_{\mathbb R}|\kern1.2pt\widehat{g}(\xi)\widehat{\phi}((sN)^{-j}\xi)|^2\,d\xi\\ & = \frac{2}{s} \int_{-{1}/{2s}}^{{1}/{2s}}|\kern1.2pt\widehat{g}(\xi)\widehat{\phi}((sN)^{-j}\xi)|^2\,d\xi. \end{align*} $$

Since $\lim _{j\to \infty }|\widehat {\phi }((sN)^{-j}\xi )|^2=c$ a.e. $\xi \in [-({1}/{2s}),({1}/{2s}))$ ,

$$ \begin{align*} \lim_{j\to\infty} \|P_jg\|^2 &= \lim_{j\to\infty}\frac{2}{s} \int_{-{1}/{2s}}^{{1}/{2s}} |\kern1.2pt\widehat{g}(\xi)\widehat{\phi}((sN)^{-j}\xi)|^2\,d\xi\\ &= \frac{2c}{s}\int_{\mathbb R}|\kern1.2pt\widehat{g}(\xi)|^2 \,d\xi. \end{align*} $$

Note that $\|P_jg\|^2\leq \epsilon ^2$ for all j and hence

$$ \begin{align*}\|\kern1.2pt\widehat{g}\|^2=\|g\|^2\leq \frac{s\epsilon^2}{2c}.\end{align*} $$

Since $\|f\|\leq \epsilon +\|g\|$ , we have $\|f\|\leq \epsilon +\epsilon \sqrt {{s}/{2c}}$ , which proves that $f=0$ and hence $\overline {\bigcup _{j\in \mathbb Z}V_j}=L^2(\mathbb R)$ .

Conversely, suppose that $\overline {\bigcup _{j\in \mathbb Z}V_j}=L^2(\mathbb R)$ . Then $\|f-P_jf\|^2\to 0$ as $j\to \infty $ for all $f\in L^2(\mathbb R)$ . If $f\in L^2(\mathbb R)$ is such that $\widehat {f}(\xi )=\chi _{_{[-({1}/{2s}),{1}/{2s})}}(\xi )$ , then $\|f\|^2=\|\widehat {f}\|^2={1}/{s}$ . Thus $\|P_jf\|^2\to \|f\|^2={1}/{s}$ as $j\to \infty $ . Now

$$ \begin{align*} \|P_jf\|^2 &= \displaystyle\sum_{\lambda\in\Lambda_s}|\langle f, D^{\kern1.5pt j}T_\lambda\phi \rangle|^2 \\ &= \displaystyle\sum_{\lambda\in\Lambda_s}|\langle \widehat{f}, \widehat{D^{\kern1.5pt j}T_\lambda\phi \rangle}|^2 \\ &= \displaystyle\sum_{\lambda\in\Lambda_s}\bigg|\int_{\mathbb R}\widehat{f}(\xi)(sN)^{-({j}/{2})}e^{2\pi i\xi\lambda(sN)^{-j}} \overline{\widehat{\phi}\bigg(\frac{\xi}{(sN)^{\kern1.5pt j}}\bigg)}~ d\xi \bigg|^2\\[-12pt] \\ &= \displaystyle\sum_{\lambda\in\Lambda_s}\bigg|\int_{-{1}/{2s}}^{{1}/{2s}}(sN)^{-({j}/{2})}e^{2\pi i\xi\lambda(sN)^{-j}} \overline{\widehat{\phi}\bigg(\frac{\xi}{(sN)^{\kern1.5pt j}}\bigg)}~ d\xi \bigg|^2\\ &= \displaystyle\sum_{m=0}^{1}\sum_{l\in\mathbb Z}\bigg|\int_{-{1}/{2s}}^{{1}/{2s}}(sN)^{-({j}/{2})}e^{2\pi i\xi(({r_m}/{N})+sl)(sN)^{-j}} \overline{\widehat{\phi}\bigg(\frac{\xi}{(sN)^{\kern1.5pt j}}\bigg)}~ d\xi \bigg|^2. \end{align*} $$

By changing the variable $\xi =(sN)^{\kern1.5pt j}\eta $ ,

$$ \begin{align*} \|P_jf\|^2 &= (sN)^{\kern1.5pt j}\displaystyle\sum_{m=0}^{1}\sum_{l\in\mathbb Z}\bigg|\int_{-({1}/{2s(sN)^{\kern1.5pt j}})}^{{1}/{2s(sN)^{\kern1.5pt j}}} e^{2\pi i\eta({r_m}/{N})}e^{2\pi i\eta sl} \overline{\widehat{\phi}(\eta)}~ d\eta \bigg|^2. \end{align*} $$

For $j\geq 0$ , $\Omega _j:=[-({1}/({2s(sN)^{\kern1.5pt j}})),({1}/({2s(sN)^{\kern1.5pt j}}))]\subseteq [-({1}/{2s}),({1}/{2s})]$ . Hence,

$$ \begin{align*}\|P_jf\|^2=(sN)^{\kern1.5pt j}\displaystyle\sum_{m=0}^{1}\sum_{l\in\mathbb Z}\bigg|\int_{-{1}/{2s}}^{{1}/{2s}}\chi_{_{\Omega_j}}(\xi) e^{2\pi i\xi({r_m}/{N})}e^{2\pi i\xi sl} \overline{\widehat{\phi}(\xi)}~ d\xi \bigg|^2.\end{align*} $$

Since $\{\!\!\sqrt {s}e^{2\pi i\xi sl}:l\in \mathbb Z\}$ is an orthonormal basis for $L^2([-({1}/{2s}),({1}/{2s})))$ ,

$$ \begin{align*} \|P_jf\|^2 & = \frac{(sN)^{\kern1.5pt j}}{s}\displaystyle\sum_{m=0}^{1}\int_{-{1}/{2s}}^{{1}/{2s}}\bigg|\chi_{_{\Omega_j}}(\xi) \overline{\widehat{\phi}(\xi)} \bigg|^2 \,d\xi\\ &= \frac{2}{s}\int_{-{1}/{2s}}^{{1}/{2s}}|{\widehat{\phi}((sN)^{-j}\xi)} |^2 \,d\xi. \end{align*} $$

Taking the limit as $j\to \infty $ ,

$$ \begin{align*}\frac{1}{s}=\lim_{j\to\infty}\frac{2}{s}\int_{-{1}/{2s}}^{{1}/{2s}}|{\widehat{\phi}((sN)^{-j}\xi)} |^2 \,d\xi\end{align*} $$

and hence

$$ \begin{align*}\lim_{j\to\infty}\int_{-{1}/{2s}}^{{1}/{2s}}|{\widehat{\phi}((sN)^{-j}\xi)} |^2 \,d\xi=\frac{1}{2}.\end{align*} $$

From (2-3), and since $\int _{-{1}/{2s}}^{{1}/{2s}}(({s}/{2})- \lim _{j\to \infty }|{\widehat {\phi }((sN)^{-j}\xi )} |^2) \,d\xi =0$ ,

$$ \begin{align*}\displaystyle\lim_{j\to\infty}|{\widehat{\phi}((sN)^{-j}\xi)}|^2= \frac{s}{2} \quad\text{a.e. }\xi\in\bigg[-\frac{1}{2s},\frac{1}{2s}\bigg).\\[-42pt]\end{align*} $$

Theorem 3.4. Let $\phi \in L^2(\mathbb R)$ and the $V_j$ be as defined in (3-1). Then $\phi $ generates $\Lambda _s$ -nonuniform multiresolution analysis with dilation factor $sN$ if and only if:

1. (1) $\sum _{p=0}^{({sN}/{2})-1}\sum _{k\in \mathbb Z}|\widehat {\phi }(\xi +({p}/{s})+kN)|^2={s}/{2} \text { a.e. }\xi \in \mathbb R$ ;

2. (2) $\lim _{j\to \infty }|{\widehat {\phi }((sN)^{-j}\xi )}|^2=c,~~c> 0, \text { a.e. }\xi \in [-({1}/{2s}),{1}/{2s})$ ;

3. (3) there exist $({1}/{s})$ -periodic functions $m_0^1$ and $m_0^2$ such that

   $$ \begin{align*}\widehat{\phi}(sN\xi)=(m_0^1(\xi)+ m_0^2(\xi)e^{-2\pi i\xi({r}/{N})})\widehat{\phi}(\xi).\end{align*} $$

The proof follows from Theorems 2.4, 3.2 and 3.3.

Next we construct an example of a $\Lambda _s$ -NUMRA with dilation factor $sN$ using some results of spectral pairs [4].

Definition 3.5. Let A be a measurable subset of $\mathbb R$ and $|A|$ the Lebesgue measure of A such that $0<|A|<\infty $ and $\Lambda \subset \mathbb R$ is a discrete set. Then the pair $(A,\Lambda )$ is said to be a spectral pair if the collection $\{|A|^{-1/2}e^{2\pi i\xi \lambda }\chi _{_{A}}(\xi ): \lambda \in \Lambda \}$ is a complete orthonormal system for $L_A^2$ , where $L^2_A$ is the subspace of $L^2(\mathbb R)$ that consists of functions vanishing outside of A almost everywhere.

Proposition 3.6 [4]

Let $V_0$ be a closed subspace of $L^2(\mathbb R)$ and suppose that there exist $\phi \in V_0$ and a discrete set $\Lambda \subset \mathbb R$ such that $\{T_\lambda \phi :\lambda \in \Lambda \}$ is an orthonormal basis for $V_0$ . Then, given a measurable set $A\subset \mathbb R$ with $0<|A|<\infty $ , the mapping $F:V_0\rightarrow L^2_A$ defined by $F(T_\lambda \phi )=|A|^{-1/2}e^{2\pi i\xi \lambda }\chi _{_A}(\xi )$ is unitary if and only if the collection $\{|A|^{-1/2}e^{2\pi i\xi \lambda }\chi _{_A}(\xi ):\lambda \in \Lambda \}$ is an orthonormal basis for $L^2_A$ .

The above proposition says that $(A,\Lambda )$ is a spectral pair if and only if the mapping $F:V_0\rightarrow L^2_A$ defined by $F(T_\lambda \phi )=|A|^{-1/2}e^{2\pi i\xi \lambda }\chi _{_A}(\xi )$ is unitary, where $V_0$ is the space defined as in Proposition 3.6. The following result gives an equivalent condition for a spectral pair.

Theorem 3.7. Let $\Lambda \subset \mathbb R$ be discrete and $A\subseteq \mathbb R$ be measurable such that $0<|A|<\infty $ . Then $(A, \Lambda )$ is a spectral pair if and only if

$$ \begin{align*}\sum_{\lambda\in \Lambda}|\widehat{\chi}_{_{A}}(\xi-\lambda)|^2=|A|^2\quad\forall\xi\in\mathbb R.\end{align*} $$

For the proof, we refer to [4].

Let r be a positive odd integer and $s=2r$ . Let $\Lambda =\Lambda _s=\{0,({r}/{N})\}+s\mathbb Z$ and $A=[0,{1}/{s})\cup [{N}/{s}, ({N+1})/{s})$ . To show that $(A,\Lambda )$ is a spectral pair, we have to prove that

$$ \begin{align*} \sum_{\lambda\in \Lambda}|\widehat{\chi}_{_{A}}(\xi-\lambda)|^2=|A|^2\quad\forall \xi\in\mathbb R. \end{align*} $$

That is, we have to prove that

(3-3) $$ \begin{align} \sum_{n\in \mathbb Z}|\widehat{\chi}_{_{A}}(\xi-sn)|^2+ \sum_{n\in \mathbb Z}\bigg|\widehat{\chi}_{_{A}}\bigg(\xi-\frac{r}{N}-sn\bigg)\bigg|^2=|A|^2 \quad\forall \xi\in\mathbb R. \end{align} $$

If $g=\chi _{_{A}}\ast \widetilde {\chi }_{_{A}}$ , where $\widetilde {\chi }_{_{A}}(x)=\chi _{_{A}}(-x),$ then $\widehat {g}=|\widehat {\chi }_{_{A}}|^2\in L^1(\mathbb R)$ .

Hence, by the Fourier inversion theorem, if $\widehat {g}\in L^1(\mathbb R)$ and $b>0$ , then

$$ \begin{align*}\mathcal{F}^{-1}\bigg(\sum_{n\in\mathbb Z}\widehat{g}(\xi-bn)\bigg)=\frac{1}{b}\sum_{n\in\mathbb Z}g\bigg(\frac{n}{b}\bigg)\delta_{{n}/{b}}(\xi).\end{align*} $$

Using the above fact, we see that (3-3) is equivalent to

$$ \begin{align*}\frac{1}{s}\sum_{n\in\mathbb Z}(1+e^{2\pi i({r}/{N})({n}/{s})})(\kern1.2pt \chi_{_A}\ast \widetilde{\chi}_{_{A}})\bigg(\frac{n}{s}\bigg)\delta_{{n}/{s}}=|A|^2\delta_0.\end{align*} $$

Since $s=2r$ ,

(3-4) $$ \begin{align} \frac{1}{s}\sum_{n\in\mathbb Z}(1+e^{\pi i({n}/{N})})(\kern1.2pt \chi_{_A}\ast \widetilde{\chi}_{_{A}})\bigg(\frac{n}{s}\bigg)\delta_{{n}/{s}}=|A|^2\delta_0. \end{align} $$

Thus, we have to prove (3-4) for $A=[0,{1}/{s})\cup [{N}/{s}, ({N+1})/{s})$ . Note that

$$\begin{align*}(\kern1.2pt \chi _{_A}\ast \widetilde {\chi }_{_{A}})({n}/{s})=|A\cap (A+({n}/{s}))|=0\end{align*}$$

for all $n\neq 0,N,-N$ . Next, for $n=N,-N$ , we have $1+e^{\pi i({n}/{N})}=0$ . Thus,

$$ \begin{align*}\frac{1}{s}\displaystyle\sum_{n\in\mathbb Z}(1+e^{\pi i({n}/{N})})(\kern1.2pt \chi_{_A}\ast \widetilde{\chi}_{_{A}})\bigg(\frac{n}{s}\bigg)\delta_{{n}/{s}} = \frac{2}{s}|A|\delta_0=|A|^2\delta_0.\end{align*} $$

Hence, $(A,\Lambda _s)$ is a spectral pair.

Example 1. Let $N\in \mathbb {N}$ and $1\leq r\leq sN-1$ be an odd integer such that r and N are relatively prime, and $s=2r$ . Now define $\phi \in L^2(\mathbb R)$ by $\widehat {\phi }(\xi )=\chi _{_B}(\xi )$ , where $B=[-({1}/{2s}),{1}/{2s})\cup [({2N-1})/{2s}, ({2N+1})/{2s})$ . Since $B=A-({1}/{2s})$ , $(B,\Lambda _s)$ is a spectral pair. Let

$$ \begin{align*}V_0=\overline{\text{span}}\{\phi(\cdot-\lambda): \lambda\in\Lambda_s\},\end{align*} $$

where $\Lambda _s=\{0,({r}/{N})\}+s\mathbb Z$ , and define the spaces $V_j$ as in (3-1). Since the Fourier transform of $\phi (\cdot -\lambda )$ is $e^{-2\pi i\xi \lambda }\chi _{_B}(\xi )$ and $(B,\Lambda _s)$ is a spectral pair, $\{\phi (x-\lambda ): \lambda \in \Lambda _s\}$ is an orthonormal system in $L^2(\mathbb R)$ . Then, from Theorem 2.4,

$$ \begin{align*}\sum_{p=0}^{({sN}/{2})-1}\sum_{k\in\mathbb Z}\bigg|\widehat{\phi}\bigg(\xi+\frac{p}{s}+kN\bigg)\bigg|^2=\frac{s}{2} \quad\text{a.e. }\xi\in\mathbb R.\end{align*} $$

Now $\lim _{j\to \infty }|\widehat {\phi }((sN)^{-j}\xi )|^2=\lim _{j\to \infty }|\chi _{_B}((sN)^{-j}\xi )|^2 =\lim _{j\to \infty }|\chi _{_{(sN)^{\kern1.5pt j}B}}(\xi )|^2=1$ for all $\xi \in [-({1}/{2s}),({1}/{2s}))$ . Next we find the $({1}/{s})$ -periodic functions $m_0^1$ and $m_0^2$ such that

$$ \begin{align*}\widehat{\phi}(sN\xi)=(m_0^1(\xi)+ m_0^2(\xi)e^{-2\pi i\xi({r}/{N})})\widehat{\phi}(\xi).\end{align*} $$

To compute $m_0^1$ and $m_0^2$ , we use the relation

$$ \begin{align*} \widehat{\phi}(sN\xi) &= \chi_{_B}(sN\xi)=\chi_{_C}(\xi) \\ &= (m_0^1(\xi)+ m_0^2(\xi)e^{-2\pi i\xi({r}/{N})} )\chi_{_B}(\xi), \end{align*} $$

where $C=[-({1}/{2s^2N}),{1}/{2s^2N})\cup [({2N-1})/{2s^2N},({2N+1})/{2s^2N})$ . Fixing $\xi \in [-({1}/{2s^2N}),{1}/{2s^2N})$ ,

$$ \begin{align*} m_0^1(\xi)+ m_0^2(\xi)e^{-2\pi i\xi({r}/{N})}=\chi_{_C}(\xi) \end{align*} $$

on the interval $[-({1}/{2s^2N}),{1}/{2s^2N})$ . Note that

$$ \begin{align*}\chi_{_C}\bigg(\xi+\frac{1}{s^2}\bigg)=\chi_{_C}(\xi),~ \xi\in\bigg[-\frac{1}{2s^2N},\frac{1}{2s^2N}\bigg).\end{align*} $$

Hence, one can choose $m_0^1(\xi )=\chi _{_C}(\xi )$ and $m_0^2(\xi )=0$ so that $m_0^1$ and $m_0^2$ are $({1}/{s})$ -periodic functions such that $m_0^1(\xi )+ m_0^2(\xi )e^{-2\pi i\xi ({r}/{N})}=\chi _{_C}(\xi )$ . Hence, from Theorem 3.4, $\{V_j,\phi \}_{j\in \mathbb {Z}}$ is a $\Lambda _s$ -NUMRA.

4 Condition for $\Lambda _s$ -NUMRA wavelet filters

Suppose that the closed subspaces $\{V_j\}_{j\in \mathbb Z}$ of $L^2(\mathbb R)$ form a $\Lambda _s$ -NUMRA with scaling function $\phi \in L^2(\mathbb R)$ . Then there is a function $m_0(\xi )$ of the form

$$ \begin{align*}m_0(\xi)=m_0^1(\xi)+e^{-2\pi i\xi r/N}m_0^2(\xi),\end{align*} $$

where $m_0^1, m_0^2$ are $({1}/{s})$ -periodic functions in $L^2([0,{1}/{s}))$ such that

$$ \begin{align*}\widehat{\phi}(sN\xi)=m_0(\xi)\widehat{\phi}(\xi).\end{align*} $$

Now define the functions $\psi _k\in L^2(\mathbb R), k=1,2,\ldots ,sN-1,$ by

(4-1) $$ \begin{align} \widehat{\psi_k}(sN\xi)=m_k(\xi)\widehat{\phi}(\xi), \end{align} $$

where

(4-2) $$ \begin{align} m_k(\xi)=m_k^1(\xi)+e^{-2\pi i\xi r/N} m_k^2(\xi),\quad k=1,2,\ldots, sN-1. \end{align} $$

The functions $m_k^1$ and $m_k^2, k=1,2,\ldots , sN-1$ , are called $\Lambda _s$ -NUMRA wavelet filters. In this section, we prove the conditions on $m_k,k=1,\ldots ,sN-1$ such that $\{(sN)^{\kern1.5pt j/2}\psi _k((sN)^{\kern1.5pt j}\cdot - \lambda ): k=1,2,\ldots ,sN-1, j\in \mathbb Z, \lambda \in \Lambda _s\}$ is an orthonormal basis for $L^2(\mathbb R)$ ; we call the set of functions $\{\psi _k:k=1,2,\ldots , sN-1\}\ \text{as}\ \Lambda _s$ -NUMRA wavelets. Let us take $\psi _0=\phi $ for notational simplicity. The following proposition gives the equivalent conditions for the system $\{\psi _k(\cdot -\lambda ):k=0,1,\ldots , sN-1, \lambda \in \Lambda _s\}$ to be an orthonormal set in $L^2(\mathbb R)$ . When $s=2$ , these equivalent conditions coincide with the conditions in [4, 5, 9]. Now we prove the following results using similar arguments as in [4].

Proposition 4.1. Let $\psi _k\in L^2(\mathbb R), k=0,1,\ldots ,sN-1$ . Then the system

$$\begin{align*}\{\psi _k(\cdot -\lambda ):k=0,1,\ldots , sN-1, \lambda \in \Lambda _s\}\end{align*}$$

is orthonormal if and only if

(4-3) $$ \begin{align} \sum_{p=0}^{sN-1}w_{kl}\bigg(\xi+\frac{p}{s}\bigg)=s\delta_{kl} \end{align} $$

and

(4-4) $$ \begin{align} \sum_{p=0}^{sN-1}\alpha^p w_{kl}\bigg(\xi+\frac{p}{s}\bigg)=0, \end{align} $$

where $\alpha =e^{-2\pi ir/sN}$ and

(4-5) $$ \begin{align} w_{kl}(\xi)=\displaystyle\sum_{j\in\mathbb Z}\widehat{\psi}_k(\xi+jN)\overline{\widehat{\psi}_l(\xi+jN)},\quad \xi\in\mathbb R. \end{align} $$

Proof. Suppose that $\{\psi _k(\cdot -\lambda ):k=0,1,\ldots , sN-1, \lambda \in \Lambda _s\}$ is an orthonormal system. Then

$$ \begin{align*} \delta_{kl}\delta_{\lambda\sigma} &= \int_{\mathbb R} \psi_k(x-\lambda)\overline{\psi_l(x-\sigma)}\, dx \\ &= \int_{\mathbb R} \widehat{\psi}_k(\xi)\overline{\widehat{\psi}_l(\xi)} e^{-2\pi i\xi(\lambda-\sigma)} \,d\xi \\ &= \sum_{j\in\mathbb Z}\int_{0}^{N} \widehat{\psi}_k(\xi-jN)\overline{\widehat{\psi}_l(\xi-jN)} e^{-2\pi i(\xi-jN)(\lambda-\sigma)} \,d\xi \\ &= \int_{0}^{N}\sum_{j\in\mathbb Z} \widehat{\psi}_k(\xi-jN)\overline{\widehat{\psi}_l(\xi-jN)} e^{-2\pi i\xi(\lambda-\sigma)} \,d\xi \\ &= \int_{0}^{N} w_{kl}(\xi) e^{-2\pi i\xi(\lambda-\sigma)}\, d\xi \\ &= \int_{0}^{{1}/{s}}\sum_{p=0}^{sN-1}w_{kl}\bigg(\xi+\frac{p}{s}\bigg) e^{-2\pi i\xi(\lambda-\sigma)}e^{-2\pi i({p}/{s})(\lambda-\sigma)} \,d\xi. \end{align*} $$

Letting $\lambda =sm$ and $\sigma =sn, m,n\in \mathbb Z$ ,

$$ \begin{align*}\delta_{mn}\delta_{kl}=\int_{0}^{{1}/{s}}\sum_{p=0}^{sN-1}w_{kl}\bigg(\xi+\frac{p}{s}\bigg)e^{-2\pi i\xi s(m-n)} \,d\xi.\end{align*} $$

Since $\{\!\!\sqrt {s}e^{-2\pi i\xi sl}: l\in \mathbb Z\}$ is an orthonormal basis for $L^2([0,{1}/{s}))$ ,

$$ \begin{align*} \sum_{p=0}^{sN-1}w_{kl}\bigg(\xi+\frac{p}{s}\bigg)=s\delta_{kl}. \end{align*} $$

Letting $\lambda =({r}/{N})+sm$ and $\sigma =sn, m,n\in \mathbb Z$ ,

$$ \begin{align*}0=\int_{0}^{{1}/{s}}e^{-2\pi i\xi s(m-n)} e^{-2\pi i\xi r/N}\sum_{p=0}^{sN-1}w_{kl}\bigg(\xi+\frac{p}{s}\bigg)e^{-2\pi ip({r}/{sN})} \,d\xi.\end{align*} $$

Since $\{\!\!\sqrt {s}e^{-2\pi i\xi sl}: l\in \mathbb Z\}$ is an orthonormal basis for $L^2([0,{1}/{s}))$ ,

$$ \begin{align*} \sum_{p=0}^{sN-1}\alpha^p w_{kl}\bigg(\xi+\frac{p}{s}\bigg)=0, \end{align*} $$

where $\alpha =e^{-2\pi ir/sN}$ . By reversing the steps above, we get the converse part.

Now define an $2sN\times 2sN$ matrix $U(\xi )$ with entries $U_{pq}(\xi ), 0\leq p,q\leq 2sN-1,$ defined by

$$ \begin{align*}U_{pq}(\xi)=\left\{\begin{array}{ll} m_q^1\bigg(\xi+\dfrac{p}{s^2N}\bigg) & 0\leq p\leq sN-1,~~ 0\leq q\leq sN-1,\\[12pt] m_q^2\bigg(\xi+\dfrac{p-sN}{s^2N}\bigg) & sN\leq p\leq 2sN-1,~~ 0\leq q\leq sN-1,\\[12pt] \alpha^p m_{q-sN}^1\bigg(\xi+\dfrac{p}{s^2N}\bigg) & 0\leq p\leq sN-1,~~ sN\leq q\leq 2sN-1,\\[12pt] \alpha^pm_{q-sN}^2\bigg(\xi+\dfrac{p-sN}{s^2N}\bigg) & sN\leq p\leq 2sN-1, ~~ sN\leq q\leq 2sN-1.\\ \end{array}\right.\end{align*} $$

The following proposition characterizes when $\{T_\lambda \psi _k :k=0,1,\ldots , sN-1, \lambda \in \Lambda _s\}$ is an orthonormal system in terms of the matrix $U(\xi )$ .

Proposition 4.2. Let $\psi _k\in L^2(\mathbb R), k=0,1,\ldots ,sN-1$ . Then the system

$$\begin{align*}\{T_\lambda \psi _k :k=0,1,\ldots , sN-1, \lambda \in \Lambda _s\}\end{align*}$$

is orthonormal if and only if the matrix $U(\xi )$ is unitary a.e. $\xi \in \mathbb R$ .

Proof. Using (4-1) and (4-5),

$$ \begin{align*}w_{kl}(sN\xi)=\sum_{j\in\mathbb Z}m_k\bigg(\xi+\frac{j}{s}\bigg)\overline{m_l\bigg(\xi+\frac{j}{s}\bigg)}\bigg|\widehat{\phi}\bigg(\xi+\frac{j}{s}\bigg)\bigg|^2.\end{align*} $$

By the $({1}/{s})$ -periodicity of $m_k^1,m_k^2,m_l^1$ and $m_l^2$ ,

$$ \begin{align*} w_{kl}(sN\xi) &= (m_k^1(\xi)\overline{m_l^1(\xi)}+m_k^2(\xi)\overline{m_l^2(\xi)})\displaystyle\sum_{j\in\mathbb Z} \bigg|\widehat{\phi}\bigg(\xi+\frac{j}{s}\bigg)\bigg|^2 \\ &\quad+m_k^1(\xi)\overline{m_l^2(\xi)}\displaystyle\sum_{j\in\mathbb Z}e^{2\pi i(\xi+({j}/{s})){r}/{N}} \bigg|\widehat{\phi}\bigg(\xi+\frac{j}{s}\bigg)\bigg|^2 \\ &\quad+ m_k^2(\xi)\overline{m_l^1(\xi)}\displaystyle\sum_{j\in\mathbb Z}e^{-2\pi i(\xi+({j}/{s})){r}/{N}} \bigg|\widehat{\phi}\bigg(\xi+\frac{j}{s}\bigg)\bigg|^2. \end{align*} $$

By writing $j=p+(sN)q, p\in \{0,1,\ldots ,sN-1\}, q\in \mathbb Z$ ,

$$ \begin{align*} w_{kl}(sN\xi) &= (m_k^1(\xi)\overline{m_l^1(\xi)}+m_k^2(\xi)\overline{m_l^2(\xi)})\displaystyle\sum_{p=0}^{sN-1}w_{00}\bigg(\xi+\frac{p}{s}\bigg) \\ &\quad+m_k^1(\xi)\overline{m_l^2(\xi)}\displaystyle\sum_{p=0}^{sN-1}\alpha^{-p} w_{00}\bigg(\xi+\frac{p}{s}\bigg)\\ &\quad+ m_k^2(\xi)\overline{m_l^1(\xi)}\displaystyle\sum_{p=0}^{sN-1}\alpha^p w_{00}\bigg(\xi+\frac{p}{s}\bigg). \end{align*} $$

Next, using (4-3) and (4-4),

(4-6) $$ \begin{align} w_{kl}(sN\xi)=s(m_k^1(\xi)\overline{m_l^1(\xi)}+m_k^2(\xi)\overline{m_l^2(\xi)}). \end{align} $$

From (4-3) and (4-6),

(4-7) $$ \begin{align} \sum_{p=0}^{sN-1}\bigg(m_k^1\bigg(\xi+\frac{p}{s^2N}\bigg)\overline{m_l^1\bigg(\xi+\frac{p}{s^2N}\bigg)}+m_k^2\bigg(\xi+\frac{p}{s^2N}\bigg) \overline{m_l^2\bigg(\xi+\frac{p}{s^2N}\bigg)}\bigg)=\delta_{kl}, \end{align} $$

$0\leq k,l\leq sN-1.$ From (4-4) and (4-6),

(4-8) $$ \begin{align} \sum_{p=0}^{sN-1}\alpha^p\bigg(m_k^1\bigg(\xi+\frac{p}{s^2N}\bigg)\overline{m_l^1\bigg(\xi+\frac{p}{s^2N}\bigg)}+m_k^2\bigg(\xi+\frac{p}{s^2N}\bigg) \overline{m_l^2\bigg(\xi+\frac{p}{s^2N}\bigg)}\bigg)=0, \end{align} $$

$0\leq k,l\leq sN-1.$

Conditions (4-7) and (4-8) together are equivalent to saying that the matrix $U(\xi )$ is unitary a.e. $\xi \in \mathbb R$ . It is easy to observe that $\{T_\lambda \psi _k : k=0,1,\ldots ,sN-1, \lambda \in \Lambda _s\}$ is an orthonormal system if and only if $U(\xi )$ is unitary a.e. $\xi \in \mathbb R$ .

The solvability of the system (4-7) and (4-8), given in [4] for the case $s=2$ , can be extended for any even number s. The following theorem generalizes the result of Gabardo and Nashed [4, Lemma 3.2].

Theorem 4.3. If $s=2r$ and $\{T_\lambda \psi _k : k=0,1,\ldots ,sN-1, \lambda \in \Lambda _s\}$ is an orthonormal system, then $\{T_\lambda \psi _k : k=0,1,\ldots ,sN-1, \lambda \in \Lambda _s\}$ is complete in $V_1$ .

Proof. Suppose that $\{T_\lambda \psi _k : k=0,1,\ldots ,sN-1, \lambda \in \Lambda _s\}$ is an orthonormal system. Then, from Proposition 4.2, $U(\xi )$ is unitary for almost every $\xi \in \mathbb R$ . From condition (iii) of Definition 2.3, it is enough to prove that

$$ \begin{align*}\bigg\{\frac{1}{sN}\psi_k\bigg(\frac{x}{sN}-\lambda\bigg):\lambda\in\Lambda_s, k=0,1,\ldots, sN-1\bigg\}\end{align*} $$

is complete in $V_0$ . For any $f\in V_0$ , there exists $m_f(\xi )=\sum _{\lambda \in \Lambda _s}b_\lambda e^{-2\pi i\lambda \xi },$ where $\sum _{\lambda \in \Lambda _s}|b_\lambda |^2<\infty $ , such that $\widehat {f}(\xi )=m_f(\xi )\widehat {\phi }(\xi ).$ Let us define a map $V_0 \longrightarrow L^2_A$ , where $A=[0,{1}/{s})\cup [{N}/{s},({N+1})/{s})$ , by $f\mapsto m_f\chi _{_A}$ . Then, by Proposition 3.6, the above map is unitary. Hence, it is enough to prove that

$$ \begin{align*}S:=\{e^{-2\pi i(sN)\xi\lambda} m_k(\xi)\chi_{_A}(\xi):\lambda\in\Lambda_s,k=0,1,\ldots, sN-1\}\end{align*} $$

is complete in $L^2_A$ . Let $g\in L^2_A$ be such that $g\perp S$ . Since $\{e^{2\pi i\lambda \xi }\chi _{_A}(\xi ): \lambda \in \Lambda _s\}$ is an orthonormal basis for $L^2_A$ ,

(4-9) $$ \begin{align} \nonumber g(\xi)&=\sum_{\lambda\in\Lambda_s}a_\lambda e^{-2\pi i\lambda\xi}\chi_{_{A}}(\xi)\\ &= (g_1(\xi)+g_2(\xi)e^{-2\pi i\xi\frac{r}{N}})\chi_{_A}(\xi), \end{align} $$

where $g_1(\xi )=\sum _{l\in \mathbb Z} a_{sl}e^{-2\pi isl\xi }$ and $g_2(\xi )=\sum _{l\in \mathbb Z}a_{_{({r}/{N})+sl}}e^{-2\pi isl\xi }$ are $({1}/{s})$ -periodic functions. As $g\perp S$ , for any $\lambda \in \Lambda _s$ and any $k\in \{0,1,\ldots ,sN-1\}$ ,

$$ \begin{align*} 0 &= \int_{0}^{{1}/{s}} e^{-2\pi isN\xi\lambda} m_k(\xi)\overline{g(\xi)} \,d\xi + \int_{{N}/{s}}^{({N+1})/{s}} e^{-2\pi isN\xi\lambda} m_k(\xi)\overline{g(\xi)} \,d\xi \\ &= \int_{0}^{{1}/{s}} e^{-2\pi isN\xi\lambda} \bigg\{m_k(\xi)\overline{g(\xi)}+m_k\bigg(\xi+\frac{N}{s}\bigg)\overline{g\bigg(\xi+\frac{N}{s}\bigg)}\bigg\}\, d\xi. \end{align*} $$

Using (4-2) and (4-9),

$$ \begin{align*} \displaystyle\int_{0}^{{1}{s}} e^{-2\pi i\xi sN\lambda}&[ 2\{m_k^1(\xi)\overline{g_1(\xi)}+m_k^2(\xi)\overline{g_2(\xi)}\} + e^{2\pi i\xi({r}/{N})}m_k^1(\xi)\overline{g_2(\xi)}(1+e^{2\pi i({r}/{s})})\\ &\,+ e^{-2\pi i\xi({r}/{N})}m_k^2(\xi)\overline{g_1(\xi)}(1+e^{-2\pi i({r}/{s})})] \,d\xi =0. \end{align*} $$

Since $s=2r$ and r is an odd integer,

(4-10) $$ \begin{align} \displaystyle\int_{0}^{({1}/{s})} e^{-2\pi i\xi sN\lambda} \{m_k^1(\xi)\overline{g_1(\xi)}+m_k^2(\xi)\overline{g_2(\xi)}\} \,d\xi=0. \end{align} $$

Now taking $h_k(\xi )=m_k^1(\xi )\overline {g_1(\xi )}+m_k^2(\xi )\overline {g_2(\xi )}$ and letting $\lambda =sm, m\in \mathbb Z$ ,

$$ \begin{align*}\sum_{j=0}^{sN-1}\int_{({j}/{s^2N})}^{(({j+1})/{s^2N})}h_k(\xi)e^{-2\pi i\xi s^2Nm} \,d\xi=0.\end{align*} $$

Next, by changing the variable,

$$ \begin{align*}\int_{0}^{{1}/{s^2N}}\sum_{j=0}^{sN-1}h_k(\xi+({j}/{s^2N}))e^{-2\pi i\xi s^2Nm} \,d\xi=0.\end{align*} $$

Since $\{e^{-2\pi i\xi s^2Nm}:m\in \mathbb Z\}$ is an orthonormal basis for $L^2([0,{1}/{s^2N}))$ ,

(4-11) $$ \begin{align} \sum_{j=0}^{sN-1}h_k\bigg(\xi+\frac{j}{s^2N}\bigg)=0\quad \textrm{a.e.}~ \xi\in\mathbb R. \end{align} $$

Similarly, by letting $\lambda =sm+({r}/{N})$ and using (4-10),

(4-12) $$ \begin{align} \sum_{j=0}^{sN-1}\alpha^{\kern1.5pt j} h_k\bigg(\xi+\frac{j}{s^2N}\bigg)=0\quad \textrm{a.e.}~\xi\in\mathbb R. \end{align} $$

Consider the vector $u(\xi )\in \mathbb C^{2sN}$ defined by

$$ \begin{align*}u_k(\xi)=\left\{ \begin{array}{ll} g_1\bigg(\xi+\dfrac{k}{sN}\bigg) & k=0,1,\ldots, sN-1, \\[12pt] g_1\bigg(\xi+\dfrac{k-sN}{2sN}\bigg)& {k=sN,sN+1, \ldots, 2sN-1.} \end{array} \right. \end{align*} $$

Thus, from (4-11) and (4-12),

$$ \begin{align*}U^\ast(\xi)u(\xi)=0\quad \textrm{a.e.}~\xi\in\mathbb R.\end{align*} $$

As $U(\xi )$ is unitary for almost every $\xi \in \mathbb R$ , we get $u(\xi )=0$ and hence $g=0$ . This proves completeness.

Assume that $s=2r$ and let $W_0=V_0^\perp $ in $V_1$ . Then $V_1=V_0\oplus W_0$ . Writing $V_{j+1}=V_j\oplus W_j,$ where $W_j$ is the orthogonal complement of $V_j$ in $V_{j+1}$ , we obtain from (ii), (iv) and (v) of Definition 2.3 that

$$ \begin{align*}L^2(\mathbb R)=\bigoplus_{j\in\mathbb Z}W_j.\end{align*} $$

Choose $m_k^1,m_k^2, k=1,2,\ldots , sN-1,$ that satisfy (4-7) and (4-8) and define

$$\begin{align*}\psi _k, k=1,2,\ldots , sN-1,\end{align*}$$

as in (4-1). Then, by Proposition 4.2 and Theorem 4.3,

$$\begin{align*}\{T_\lambda \psi _k : k=0,1,\ldots ,sN-1, \lambda \in \Lambda _s\}\end{align*}$$

is an orthonormal basis for $V_1$ . Since $\{T_\lambda \psi _0 :\lambda \in \Lambda _s\}$ is an orthonormal basis for $V_0, \{T_\lambda \psi _k : k=1,\ldots ,sN-1, \lambda \in \Lambda _s\}$ is an orthonormal basis for $W_0$ . Also, it is easy to observe that, for every $j\in \mathbb Z,\ \{(sN)^{\kern1.5pt j/2}\psi _k((sN)^{\kern1.5pt j}\cdot -\lambda ): k=1,\ldots ,sN-1, \lambda \in \Lambda _s\}$ is an orthonormal basis for $W_j$ . Hence,

$$\begin{align*}\{(sN)^{\kern1.5pt j/2}\psi _k((sN)^{\kern1.5pt j}\cdot -\lambda ): k=1,\ldots ,sN-1, j\in \mathbb Z, \lambda \in \Lambda _s\}\end{align*}$$

is an orthonormal basis for $L^2(\mathbb R)$ .