On Compact Wavelet Matrices of Rank $m$ and of Order and Degree $N$

Abstract

A new parametrization (one-to-one onto map) of compact wavelet matrices of rank $m$ and of order and degree $N$ is proposed in terms of coordinates in the Euclidian space $\mathbb {C}^{(m-1)N}$. The developed method depends on Wiener–Hopf factorization of corresponding unitary matrix functions and allows to construct compact wavelet matrices efficiently. Some applications of the proposed method are discussed.

1 Introduction

Let $\mathbb {C}$ be the set of complex numbers, and let $\mathbb {F}\subset \mathbb {C}$ be a subfield invariant under the complex conjugation. A formally infinite matrix $\mathcal {A}$ with $m$ rows

$$\begin{aligned} {\mathcal A}= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \cdots &{} a^0_{-1}&{}a^0_{0}&{}a^0_{1}&{}a^0_{2}&{}\cdots \\ \cdots &{} a^1_{-1}&{}a^1_{0}&{}a^1_{1}&{}a^1_{2}&{}\cdots \\ &{}\vdots &{}\vdots \\ \cdots &{} a^{m-1}_{-1}&{}a^{m-1}_{0}&{}a^{m-1}_{1}&{}a^{m-1}_{2}&{}\cdots \\ \end{array} \right) ,\quad a^i_j\in \mathbb {F}, \end{aligned}$$

(1)

is called a wavelet matrix (of rank $m$) if its rows satisfy the so called shifted orthogonality condition [9]:

$$\begin{aligned} \sum _{k=-\infty }^\infty a^i_{k+mj}\,\overline{a^r_{k+ms}}=\delta _{ir}\delta _{js} \end{aligned}$$

(2)

($\delta$ stands for the Kronecker delta). Such matrices are a generalization of ordinary $m\times m$ unitary matrices and they play the crucial role in the theory of wavelets [12] and multirate filter banks [14]. The reason for us to work with $\mathbb {F}$ instead of just $\mathbb {C}$ is that it will allow more flexibility in applications, since the proposed proofs and discussions apply to a whole range of fields including the set of real numbers, rational numbers, as well as algebraic extensions of rational numbers. A reader not concerned with general fields may assume that $\mathbb {F}=\mathbb {C}$.

In the polyphase representation [15] of matrix $\mathcal {A}$,

$$\begin{aligned} {\mathbf{A}}(z)=\sum _{k=-\infty }^\infty A_kz^k\,, \end{aligned}$$

(3)

where ${\mathcal A}=(\ldots \; A_{-1}\; A_0\;A_1\;A_2\;\ldots )$ is the partition of $\mathcal {A}$ into block $m\times m$ matrices $A_k=(a^i_{km+j}), 0\le i,j\le m-1$, the condition (2) is equivalent to

$$\begin{aligned} {\mathbf{A}}(z)\widetilde{{\mathbf{A}}}(z)=I_m, \end{aligned}$$

(4)

where $\widetilde{{\mathbf{A}}}(z)=\sum \nolimits _{k=-\infty }^\infty A_k^*z^{-k}$ is the adjoint to ${\mathbf{A}}(z)$ ($A^*:=\overline{A}^T$ is the Hermitian conjugate and $I_m$ stands for the $m\times m$ unit matrix). This is easy to see as (2) can be written in the block matrix form $\sum \nolimits _{k=-\infty }^\infty A_k A^*_{l+k}=\delta _{l0}I_m$.

Our notion of a wavelet matrix is weaker than usual. So, as the orthogonal basis of $L^2(\mathbb {R})$ can be developed (see [12, Ch-s 4, 5]), also the linear condition ${\mathbf{A}}(1)\,{\mathbf e}=\sqrt{m}\,{\mathbf e}_1$, where ${\mathbf e}=(1,1,\ldots ,1)^T$ and ${\mathbf e}_1=(1,0,\ldots ,0)^T$, must be satisfied. In our consideration, the linear condition is irrelevant. Instead, we require the condition

$$\begin{aligned} {\mathbf{A}}(1)=I_m \end{aligned}$$

(5)

which much simplifies the whole exposition. For any wavelet matrix $\mathcal {A}, {\mathbf{A}}(1)=\sum \nolimits _{k=-\infty }^\infty A_k$ is unitary [see (4)], so that the conditions (2) and (5) will be satisfied for $\mathcal {A}_0=\big ({\mathbf{A}}(1)\big )^{-1}\mathcal {A}$. Note also that for any wavelet matrix $\mathcal {A}_0$ satisfying (5) and any unitary matrix $U$, the matrix $\mathcal {A}=U\mathcal {A}_0$ is also a wavelet matrix satisfying ${\mathbf{A}}(1)=U$. Thus the additional restriction (5) does not lose generality in the description of wavelet matrices. Observe that the polyphase representation of $U\mathcal {A}$ is $\sum \nolimits _{k=-\infty }^\infty UA_kz^k$ (obviously, there is one-to-one correspondence between the matrices (1) and their polyphase representations (3) and they are naturally identified).

We consider the compact wavelet matrices, which means that only a finite number of coefficients in (1) are non-zero (the corresponding wavelet functions in $L^2(\mathbb {R})$ have then a compact support, and the corresponding filters in signal processing applications are physically realizable). Namely, compact wavelet matrices with polyphase representation

$$\begin{aligned} \mathbf {A}(z)=\sum _{k=0}^N A_kz^k\,, \end{aligned}$$

(6)

where $A_N\not =0$, are called of (rank $m$ and) order $N:=\mathrm{ord}(\mathcal {A})$ (in some works they are called of genus $N+1$; see, e.g., [12]); we write $\mathcal {A}\in \mathcal {W}\mathcal {M}(m,N,\mathbb {F})$. The property (4) for the matrix polynomial (6) means that ${\mathbf{A}}(z)$ is a paraunitary matrix function [14]. Note that, in this case, ${\mathbf{A}}(z)$ is usual unitary matrix for each $z\in \mathbb {T}:=\{z\in \mathbb {C}:|z|=1\}$.

As the determinant of ${\mathbf{A}}(z)\in \mathcal {W}\mathcal {M}(m,N,\mathbb {F})$ is always monomial, i.e., $\det {\mathbf{A}}(z)=c\,z^d$ [since $\det {\mathbf{A}}(z)\det \widetilde{{\mathbf{A}}}(z)=1$], the positive integer $d$ is defined to be the degree of $\mathcal {A}, d=\deg (\mathcal {A})$; see, e.g.,[12, p. 57]. The set of wavelet matrices of rank $m$, order $N$ and degree $d$ will be denoted by $\mathcal {W}\mathcal {M}(m,N,d,\mathbb {F})$. It can be proved that $\deg (\mathcal {A})\ge \mathrm{ord}(\mathcal {A})$ for any $\mathcal {A}\in \mathcal {W}\mathcal {M}(m,N,\mathbb {F})$ and $\deg (\mathcal {A})=\mathrm{ord}(\mathcal {A})$ if and only if $\mathrm{rank}(A_0)=m-1$ (see Lemma 1 below). As $A_0A_N^*=\mathbf {0}$ for each $\mathcal {A}\in \mathcal {W}\mathcal {M}(m,N,\mathbb {F})$ (since it is the $N$th matrix coefficient of the product in (4)) and $A_N\not =\mathbf {0}$, we have $\mathrm{rank}(A_0)<m$. Thus $\deg (\mathcal {A})=N$ whenever $A_0$ has a maximal possible rank and it is assumed to be the nonsingular case (see [12, p. 58]). Throughout the paper, we will intensively study $\mathcal {W}\mathcal {M}(m,N,N,\mathbb {F})$.

As $\mathcal {A}$ and $U\mathcal {A}$, where $U$ is nonsingular, have the same rank, order and degree, we will additionally assume that $\mathcal {A}\in \mathcal {W}\mathcal {M}(m,N,N,\mathbb {F})$ satisfies (5) and the subset of such compact wavelet matrices will be denoted by $\mathcal {W}\mathcal {M}_0(m,N,N,\mathbb {F})$. As has been mentioned above, the linear condition (5) does not lose generality in the description of $\mathcal {W}\mathcal {M}(m,N,N,\mathbb {F})$.

Matrix polynomials $\mathbf {V}(z)\in \mathcal {W}\mathcal {M}_0(m,1,1,\mathbb {F})$ are called primitive wavelet matrices and they always have the form $\mathbf {V}(z)=Q+Pz$, where $P$ and $Q$ are complementary (orthogonal) projections on $\mathbb {F}^m$ and $\mathrm{rank}(P)=1\Leftrightarrow \mathrm{rank}(Q)=m-1$ (see [9, Th. 3.1] or Lemma 2).

Every ${\mathbf{A}}(z)\in \mathcal {W}\mathcal {M}_0(m,N,N,\mathbb {F})$ can be uniquely factorized as (see [7], [12, Th. 4.4.15], [14] or Theorem 2)

$$\begin{aligned} {\mathbf{A}}(z)=\prod _{j=1}^N \mathbf {V}_j(z)=\prod _{j=1}^N \big (Q_j+P_jz\big ), \quad \mathbf {V}_j(z)\in \mathcal {W}\mathcal {M}_0(m,1,1,\mathbb {F}), \end{aligned}$$

(7)

where no consecutive operators $P_j$ are orthogonal to each other, $P_jP_{j+1}\not =0$. This factorization gives rise to the map (parametrization)

$$\begin{aligned} \underbrace{\mathbb {F} \mathrm {P}^{m-1}\times \mathbb {F} \mathrm {P}^{m-1}\times \cdots \times \mathbb {F} \mathrm {P}^{m-1}}_N\supset \mathcal {B} \longleftrightarrow \mathcal {W}\mathcal {M}_0(m,N,N,\mathbb {F}), \end{aligned}$$

(8)

where $\mathbb {F}\mathrm {P}^{m-1}$ is the projective space $\mathbb {F}^m/(\mathbb {F}\backslash \{0\})$ (the space of one-dimensional subspaces of $\mathbb {F}^m$) which is one-to-one and onto but defined only on the subset $\mathcal {B}$ where consecutive directions are not orthogonal, i.e., some singular points are excluded from the set of parameters. Until now it has been the only known simple way of constructing $\mathcal {A}\in \mathcal {W}\mathcal {M}_0(m,N,N,\mathbb {F})$ for arbitrary $N$: choosing nonzero column vectors $v_j\in \mathbb {F}^m, j=1,2,\ldots ,N$, such that $v_j\not \perp v_{j+1}$, taking the corresponding projections $P_j=v_j(v_j^*v_j)^{-1}v_j^*$ and primitive wavelet matrices $\mathbf {V}_j=I_m-P_j+P_jz$, and constructing the product (7) that belongs to $\mathcal {W}\mathcal {M}_0(m,N,N,\mathbb {F})$.

One can observe through the factorization (7) that $\mathcal {W}\mathcal {M}_0(m,N,N,\mathbb {C})$ is an open dense subset of $\cup _{k=1}^N\mathcal {W}\mathcal {M}_0(m,k,N,\mathbb {C})$, both these sets are $(m-1)\times N$-dimensional manifolds, and their difference is a finite union of less dimensional “surfaces”.

Remark 1

[12, p. 68] The representation (7) resembles the factorization for ordinary polynomials $p(z)=\prod \nolimits _{k=1}^N(z-z_k)$ by using their roots. However the following remarkable fact should be emphasized: to find the roots of $p(z)$ we have to go, in general, to a larger field than the field of coefficients of $p(z)$. For the factorization in (7), the coefficients of each factor $\mathbf {V}_j$ belong to the same field $\mathbb {F}$.

Now we turn to our contribution in the study of compact wavelet matrices. This approach has been developed during a search for a new matrix spectral factorization algorithm [8], and it makes possible to parameterize $\mathcal {W}\mathcal {M}(m,N,N,\mathbb {F})$ directly in terms of points in $\mathbb {F}^{(m-1)N}$. Moreover, nonsingular compact wavelet matrices can be constructed in a more efficient way than according to formula (7). Namely, $O(m^2N^2)$ operations (counting only multiplications) are required for construction of a wavelet matrix ${\mathbf{A}}(z)$ by taking the products in (7), while the new method reduces the number of operations to $O(m^2N\log N)+O(mN^2)+O(m^3N)$. This gives a significant difference in computation time whenever $N\gg m \gg 1$ as computer simulations report [13]. Furthermore, an algorithm based on the new method can be divided into $m$ parallel tasks, which opportunity does not exist in the old method.

Other advantages and possible applications of the proposed method are mentioned in the course of discussions below.

Recall that we identify $\mathcal {A}\in \mathcal {W}\mathcal {M}(m,N,N,\mathbb {F})$ with its polyphase representation (6). The linear condition we have introduced is (5) and such a subclass is denoted by $\mathcal {W}\mathcal {M}_0(m,N,N,\mathbb {F})$. We further require that the last row of $A_N$ is not the zero vector and denote such a subclass by $\mathcal {W}\mathcal {M}_1(m,N,N,\mathbb {F})$. This is done without loss of generality as $A_N\not =0$ and we can interchange the rows of $\mathcal {A}$, if necessary. In particular, for any $\mathcal {A}\in \mathcal {W}\mathcal {M}(m,N,N,\mathbb {F})$ there exist constant unitary $m\times m$ matrices $U_0$ and $U_1$ such that $U_0\mathcal {A} U_1\in \mathcal {W}\mathcal {M}_1(m,N,N,\mathbb {F})$ (by right multiplication we assume that $\mathcal {A} U_1=(A_0U_1, A_1U_1,\ldots ,A_NU_1)$, i.e., the polyphase matrix of $\mathcal {A} U_1$ is ${\mathbf{A}}(z)U_1$). Indeed, we can interchange the rows of $\mathcal {A}$, if necessary, by multiplication by $U_0$ and then take $U_1=\big (U_0{\mathbf{A}}(1)\big )^{-1}$. Hence, in what follows, we parameterize $\mathcal {W}\mathcal {M}_1(m,N,N,\mathbb {F})$.

Let $\mathcal {P}_N^+[\mathbb {F}]:=\big \{\!\sum \nolimits _{k=0}^N c_k z^k:c_0,c_1,\ldots ,c_N \in \mathbb {F}\big \}$ be the set of polynomials with coefficients from $\mathbb {F}$ and $\mathcal {P}_N^-[\mathbb {F}]:=\big \{\sum \nolimits _{k=1}^N c_k z^{-k}:c_1,c_2,\ldots ,c_N\in \mathbb {F}\big \}$ which can be naturally identified with $\mathbb {F}^{N+1}$ and $\mathbb {F}^N$, respectively (note that $\mathcal {P}_N^+[\mathbb {F}]\cap \mathcal {P}_N^-[\mathbb {F}]=\{0\}$ according to our notation). Sometimes $[\mathbb {F}]$ is omitted because it is clear from the context.

Let $\mathcal {P}\mathcal {U}_1(m,N,\mathbb {F})$ be a set of $m\times m$ paraunitary matrix functions $U(z)$,

$$\begin{aligned} \widetilde{U}(z)U(z)=I_m, \end{aligned}$$

(9)

of the special form

$$\begin{aligned} U(z)=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c}u_{11}(z)&{}u_{12}(z)&{}\cdots &{}u_{1m}(z)\\ u_{21}(z)&{}u_{22}(z)&{}\cdots &{}u_{2m}(z)\\ \vdots &{}\vdots &{}\vdots &{}\vdots \\ u_{m-1,1}(z)&{}u_{m-1,2}(z)&{}\cdots &{}u_{m-1,m}(z)\\ \widetilde{u_{m1}}(z)&{}\widetilde{u_{m2}}(z)&{}\cdots &{}\widetilde{u_{mm}}(z)\\ \end{array}\right) ,\quad u_{ij}(z)\in \mathcal {P}_N^+[\mathbb {F}], \end{aligned}$$

(10)

$1\le i,j\le m$, with determinant 1,

$$\begin{aligned} \det U(z)=1, \end{aligned}$$

(11)

and with the linear restriction

$$\begin{aligned} U(1)=I_m, \end{aligned}$$

(12)

and such that not all polynomials $u_{mj}$ (these are the adjoint polynomials of the entries in the last row) are zero at the origin,

$$\begin{aligned} \sum _{j=1}^m|u_{mj}(0)|>0. \end{aligned}$$

(13)

Then for each ${\mathbf{A}}(z)\in \mathcal {W}\mathcal {M}_1(m,N,N,\mathbb {F})$, we have $U(z)=\mathrm{diag}[1,\ldots ,1,z^{-N}]{\mathbf{A}}(z)\in \mathcal {P}\mathcal {U}_1(m,N,\mathbb {F})$ (the last row is multiplied by $z^{-N}$), and conversely. Thus there is a simple one-to-one correspondence

$$\begin{aligned} \mathcal {W}\mathcal {M}_1(m,N,N,\mathbb {F})\longleftrightarrow \mathcal {P}\mathcal {U}_1(m,N,\mathbb {F}), \end{aligned}$$

(14)

and we parameterize the latter class using the following

Theorem 1

Let $N\ge 1$. For any Laurent matrix polynomial $F(z)$ of the form

$$\begin{aligned} F(z)=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c}1&{}0&{}0&{}\cdots &{}0&{}0\\ 0&{}1&{}0&{}\cdots &{}0&{}0\\ 0&{}0&{}1&{}\cdots &{}0&{}0\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ 0&{}0&{}0&{}\cdots &{}1&{}0\\ \zeta _{1}(z)&{}\zeta _{2}(z)&{}\zeta _{3}(z)&{}\cdots &{}\zeta _{m-1}(z)&{}1 \end{array}\right) ,\quad \zeta _j(z)\in \mathcal {P}_N^-[\mathbb {F}], \end{aligned}$$

(15)

there exists a unique

$$\begin{aligned} U(z)\in \mathcal {P}\mathcal {U}_1(m,N,\mathbb {F}) \end{aligned}$$

(16)

such that

$$\begin{aligned} F(z)U(z)\in \mathcal {P}^+(m\times m). \end{aligned}$$

(17)

Conversely, for each $U(z)$ satisfying (16), there exists a unique matrix function $F(z)$ of the form (15) such that (17) holds.

$\mathcal {P}^+(m\times m)$ stands for the set of $m\times m$ matrix polynomials with entries from $\mathcal {P}^+$.

Remark 2

The relation (17) written in the equivalent form $U(z)=F^{-1}(z)M(z)$, where $M(z)\in \mathcal {P}^+(m\times m)$ and each $\zeta _j$ is replaced by $-\zeta _j$ in (15) to get $F^{-1}(z)$, means that we have the Wiener–Hopf factorization of the unitary matrix function $U(t)=U(z)|_{z=t}$ defined on $\mathbb {T}$, since $M(z)$ and $F^{-1}(z)$ are analytic regular matrix functions inside and outside $($including infinity$) \mathbb {T}$, respectively. Although we heavily exploited this idea in our previous works on the factorization of matrix functions [2, 8], from which this paper has stemmed out, we tried to reduce to a minimum the application of Wiener–Hopf factorization theory in the presented proofs and discussions. This allowed us to transfer the obtained results to arbitrary subfields $\mathbb {F}\subset \mathbb {C}$ which are invariant under conjugate operation, and thus extend the area of their applications.

The proof of Theorem 1 makes it possible to explicitly construct the corresponding $U(z)$ for a given $F(z)$, and vice versa. Namely, let the functions in (15) be

$$\begin{aligned} \zeta _i(z)=\sum _{k=1}^N \gamma _{ik}z^{-k},\quad i=1,2,\ldots ,m-1, \end{aligned}$$

(18)

and let $\Theta _i$ be the $(N+1)\times (N+1)$ upper triangular Hankel matrix

$$\begin{aligned} \Theta _i=\left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} 0&{} \gamma _{i1}&{} \gamma _{i2}&{}\cdots &{} \gamma _{i,N-1}&{} \gamma _{iN}\\ \gamma _{i1}&{} \gamma _{i2}&{} \gamma _{i3}&{}\cdots &{} \gamma _{iN}&{}0\\ \gamma _{i2}&{} \gamma _{i3}&{} \gamma _{i4}&{}\cdots &{}0&{}0\\ \cdot &{}\cdot &{}\cdot &{}\cdots &{}\cdot &{}\cdot \\ \gamma _{iN}&{}0&{}0&{}\cdots &{}0&{}0\end{array}\right) , \quad i=1,2,\ldots ,m-1, \end{aligned}$$

(19)

and

$$\begin{aligned} \Delta =\sum _{i=1}^{m-1}\Theta _i\overline{\Theta _i}+I_{N+1}. \end{aligned}$$

(20)

Assume also that $B_i$ is the first column of $\Theta _i, i=1,2,\ldots ,m-1$, and $B_m=(1,0,\ldots ,0)^T$. Let $X_j=(\alpha _{j0},\alpha _{j1},\ldots ,\alpha _{jN})^T\in \mathbb {F}^{N+1}$ be the solution of the following linear system of algebraic equations

$$\begin{aligned} \Delta \cdot X=B_j,\;\;j=1,2,\ldots ,m \end{aligned}$$

(21)

($\Delta$ is positive definite and has a displacement structure of rank m, therefore O$(mN^2)$ operations are required for its solution instead of traditional O$(N^3)$; see [8, Appendix]), and let

$$\begin{aligned} v_{mj}(z)&= \sum _{k=0}^N \alpha _{jk}z^{-k},\;\;\;j=1,2,\ldots ,m,\end{aligned}$$

(22)

$$\begin{aligned} v_{ij}(z)&= \big [\widetilde{\zeta _i}(z)v_{mj}(z)\big ]^+-\delta _{ij},\quad i=1,2,\ldots ,m-1, \end{aligned}$$

(23)

where $[\;\cdot \;]^+$ stands for the projection operator, $\left[ \sum \nolimits _{k=-N}^N c_k z^k\right] ^+= \sum \nolimits _{k=0}^N c_k z^k$. Then

$$\begin{aligned} U(z)=V(z)\big (V(1)\big )^{-1}, \end{aligned}$$

(24)

where $V(z)$ is the $m\times m$ Laurent polynomial matrix $V(z)=\left( v_{ij}(z)\right) _{i,j=1}^m$, will be the desired matrix polynomial (16) as proved in Sect. 4.

For a given $U(z)$, the corresponding $F(z)$ can be also explicitly computed as follows (see Sect. 4 for the proof): if $u_{mj}(0)\not =0$, then

$$\begin{aligned} \zeta _i(z)=\left[ \frac{\widetilde{u}_{ij}(z)}{u_{mj}(z)}\right] ^-, \quad i=1,2,\ldots ,m-1, \end{aligned}$$

(25)

where $[\;\cdot \;]^-$ stands for the projection operator: $\left[ \sum \nolimits _{k=-N}^\infty c_k z^k\right] ^-= \sum \nolimits _{k=-N}^{-1} c_k z^k$, and under $\frac{1}{u_{mj}(z)}$ its formal series expansion in a neighborhood of 0 is assumed. Note that we need to take only the first $N+1$ coefficients in this expansion in order to compute $\zeta _i$s.

In consequence, as the set of matrix polynomials of the form (15) can be easily identified with $\mathbb {F}^{(m-1)N}$, we have the following diagram of one-to-one and onto maps

$$\begin{aligned} \mathcal {W}\mathcal {M}_1(m,N,N,\mathbb {F})\longleftrightarrow \mathcal {P}\mathcal {U}_1(m,N,\mathbb {F})\longleftrightarrow \underbrace{\mathbb {F}^N\times \mathbb {F}^N \times \cdots \times \mathbb {F}^N}_{m-1}, \end{aligned}$$

(26)

i.e., a complete parametrization of nonsingular compact wavelet matrices, which can be effectively realized.

The paper is organized as follows. In the next section we provide all notation and definitions used throughout the paper. Most of them have already been introduced in the current section but we collect them together for the convenience of reference. In Sect. 3 we reprove the factorization (7) and related facts as it seems that our approach is somewhat simpler than the traditional one. The main result of our paper, Theorem 1, is proved in Sect. 4. The essential part of this theorem, the existence of the paraunitary matrix function (16) [without showing the property (13)], has already been published (see [8, Th. 1]) for the case $\mathbb {F}=\mathbb {C}$.¹ It is sufficient to note that the proof goes through without any change if we replace $\mathbb {C}$ by $\mathbb {F}$. However, for readers’ convenience, we included the simplified version of this proof in the present paper. [A minor inaccuracy of the discussion between the formulas (50) and (51) in [8] has also been corrected].

In the remaining two sections we consider some possible applications of our method. In particular, in Sect. 5, we solve the following important wavelet matrix completion problem [7, 9]: given the first row of a wavelet matrix $\mathcal {A}$ (sometimes called the scaling vector or low-pass filter), how to find the remaining rows of $\mathcal {A}$ (called the wavelet vectors or high-pass filters). Many authors addressed this and similar problems from different points of view and their solutions are well presented, e.g., in [1, 6, 10, 11]. In the case where $\mathbb {F}=\mathbb {C}$, the solution is unique (up to some normalizer) in the set of nonsingular wavelet matrices and there exists an appropriate algorithm for construction of wavelet vectors (see [7, 12]). We propose another algorithm for solution of this problem which uses our approach. Computer simulations [13] prove that this algorithm is faster than the old one.

Section 6 contains some discussions about other possible applications of the proposed parametrization of nonsingular compact wavelet matrices.

2 Notation and Preliminaries

The fields of rational, real, and complex numbers are denoted by $\mathbb {Q}, \mathbb {R}$, and $\mathbb {C}$, respectively, and $\mathbb {F}$ stands for a subfield of $\mathbb {C}$ which is invariant under the complex conjugation, $a\in \mathbb {F}\Rightarrow \overline{a}\in \mathbb {F}$.

$\mathbb {T}:=\{z\in \mathbb {C}:|z|=1\}, \mathbb {T}_+:=\{z\in \mathbb {C}:|z|<1\}$, and $\mathbb {T}_-:=\{z\in \mathbb {C}:|z|>1\}\cup \{\infty \}$.

Let $\mathbf{e}_1,\mathbf{e}_2,\ldots ,\mathbf{e}_m$ be the standard basis of $\mathbb {F}^m$, e.g., $\mathbf{e}_1=(1,0,\ldots ,0)$. The usual scalar product and norm in the space $\mathbb {F}^m$ are denoted by $\langle \,\cdot ,\cdot \rangle _m$ and $\Vert \cdot \Vert _m$, respectively (they are independent of $\mathbb {F}$ as it is a subfield of $\mathbb {C}$).

Let $\mathbb {F}^{m\times n}$ be the set of $m\times n$ matrices with entries from a field $\mathbb {F}$. For $M\in \mathbb {F}^{m\times n}$, let $M^*=\overline{M}^T\in \mathbb {F}^{n\times m}$ be the Hermitian conjugate of $M$. $\mathrm{diag}[a_1,a_2,\ldots ,a_m]$ is an $m\times m$ diagonal matrix with corresponding entries on the diagonal and $I_m=\mathrm{diag}[1,1,\ldots ,1]$ is the $m\times m$ unit matrix. $U\in \mathbb {F}^{m\times m}$ is called unitary, $U\in \mathcal {U}_m[\mathbb {F}]$, if $UU^*=U^*U=I_m$.

If $M\in \mathbb {F}^{m\times m}$ and $a$ is an entry of $M$, then $\mathrm{cof}(a)\in \mathbb {F}$ is the cofactor of $a$ and $\mathrm{Cof}(M)\in \mathbb {F}^{m\times m}$ is the cofactor matrix, so that $M^{-1}=\frac{1}{\det M}\big (\mathrm{Cof}(M)\big )^T$ if $M$ is nonsingular. The same notation will be used for matrix functions.

A matrix $P\in \mathbb {F}^{m\times m}$ (considered as a linear map from $\mathbb {F}^m$ to $\mathbb {F}^m$) is called the (orthogonal) projection if it is self-adjoint, $P=P^*$, and idempotent, $P^2=P$, however we will reduce this definition to a single formula: $P=PP^*$. Note that if $P\in \mathbb {F}^{m\times m}$ is a projection of rank $r\le m$, then there exists $U\in \mathbb {C}^{m\times r}$ with orthonormal columns, $U^*U=I_r$, such that

$$\begin{aligned} P=UU^* \end{aligned}$$

(27)

(indeed, the factorization (27) with $U\in \mathbb {C}^{m\times r}$ of full rank $r$ exists since $P$ is non-negative definite, and $UU^*UU^*=UU^* \,\Rightarrow \,U^*UU^*U U^*U=U^*UU^*U \, \Rightarrow (U^*U)^{-1}(U^*U)(U^*U)(U^*U)(U^*U)^{-1}=(U^*U)^{-1} (U^*U)(U^*U)(U^*U)^{-1}\,\Rightarrow \, U^*U=I_r$).

If $P$ and $Q$ from $\mathbb {F}^{m\times m}$ are projections and $P+Q=I_m$, then they are called complementary to each other. Obviously, $PQ^*=Q^*P={\mathbf {0}}$ for complementary projections (as $P(I_m-P)^*=P-PP^*=\mathbf {0}$), and if $P$ is a projection, then $I_m-P$ is also the projection complementary to $P$.

$\mathcal {P}[\mathbb {F}]$ denotes the set of Laurent polynomials with coefficients from a field $\mathbb {F}$, and $\mathcal {P}_N[\mathbb {F}]:=\{\sum \nolimits _{k=-N}^N c_kz^k:c_k\in \mathbb {F}, k=-N,\ldots ,N\}$. If we write just $\mathcal {P}$, the field of coefficients will be clear from the context. $\mathcal {P}^+\subset \mathcal {P}$ is the set of polynomials (with non-negative powers of $z, \sum \nolimits _{k=0}^N c_kz^k\in \mathcal {P}^+$) and $\mathcal {P}^-\subset \mathcal {P}$ is the set of Laurent polynomials with negative powers of $z, \sum \nolimits _{k=1}^N c_kz^{-k}\in \mathcal {P}^-$. We emphasize that, according to our notation, constant functions belong only to $\mathcal {P}^+$ so that $\mathcal {P}^+\cap \mathcal {P}^-=\{0\}$. Let also $\mathcal {P}^\pm _N=\mathcal {P}^\pm \cap \mathcal {P}_N$.

$\mathcal {P}(m\times n)$ denotes the set of $m\times n$ (polynomial) matrices with entries from $\mathcal {P}$, and the sets $\mathcal {P}^+(m\times n), \mathcal {P}^-_N[\mathbb {F}](m\times n)$, etc. are defined similarly. The elements of these sets, $P(z)=\big (p_{ij}(z)\big )$, are called (polynomial) matrix functions. When $n=1$ we have the vector functions and such classes are denoted by $\mathcal {P}(m)$ instead of $\mathcal {P}(m\times 1)$. When we speak about the continuous maps between these sets, we mean that they are equipped with a usual topology.

For $p(z)=\sum \nolimits _{k=-N}^N c_kz^k \in \mathcal {P}$, let $\widetilde{p}(z)=\sum \nolimits _{k=-N}^N \overline{c}_kz^{-k}$ and for $P(z)=[p_{ij}(z)]\in \mathcal {P}(m\times n)$ let $\widetilde{P}(z)=[\widetilde{p}_{ij}(z)]^T\in \mathcal {P}(n\times m)$. Note that $\widetilde{P}(z)=(P(z))^*$ when $z\in \mathbb {T}$. Thus usual relations for adjoint matrices like $\widetilde{P_1+P_2}(z)=\widetilde{P}_1(z)+\widetilde{P}_2(z), \widetilde{P_1P_2}(z)=\widetilde{P}_2(z)\widetilde{P}_1(z)$, etc. hold.

We employ also the additional notation of the sets $\widetilde{\mathcal {P}_N^+} [\mathbb {F}]:=\{\widetilde{p}(z):p(z)\in \mathcal {P}_N^+[\mathbb {F}]\}$, which is an extension of $\mathcal {P}_N^-[\mathbb {F}]$, and $\mathcal {P}_N^\oplus [\mathbb {F}](m):= \underbrace{{\mathcal {P}_N^+}[\mathbb {F}]\times \cdots \times {\mathcal {P}_N^+}[\mathbb {F}]\times \widetilde{\mathcal {P}_N^+}[\mathbb {F}]}_m$.

A polynomial matrix (6), where $A_k\in \mathbb {F}^{m\times m}, k=1,2,\ldots ,N$, is called paraunitary, we write ${\mathbf{A}}(z)\in \mathcal {P}\mathcal {U}(m,N,\mathbb {F})$, if (4) holds, which is equivalent to $\sum \nolimits _{k=0}^{N-l}A_kA^*_{k+l}=\delta _{l0}I_m, l=0,1,\ldots ,N$. This means that ${\mathcal A}=(A_0\;A_1\;A_2\;\ldots \;A_N)\in \mathbb {F}^{m\times m(N+1)}$ is a wavelet matrix and we do not make distinction between $\mathcal {A}$ and its polyphase representation (6). If $A_N\not =\mathbf {0}$ in the representation (6), then we say that the wavelet matrix $\mathcal {A}\equiv {\mathbf{A}}(z)$ has rank $m$ and order $N$, we write ${\mathbf{A}}(z)\in \mathcal {W}\mathcal {M}(m,N,\mathbb {F})$. The degree of a wavelet matrix ${\mathbf{A}}(z)$ is the order of a monomial $\det {\mathbf{A}}(z)$, i.e., $\det {\mathbf{A}}(z)=c\cdot z^{\deg (\mathcal {A})}$. The set of wavelet matrices of rank $m$, order $N$, and degree $d$ will be denoted by $\mathcal {W}\mathcal {M}(m,N,d,\mathbb {F})$. In addition, $\mathcal {W}\mathcal {M}_0(m,N,d,\mathbb {F})$ is the subset of wavelet matrices which satisfy (5), and ${\mathbf{A}}(z)\in \mathcal {W}\mathcal {M}_1(m,N,d,\mathbb {F})\subset \mathcal {W}\mathcal {M}_0(m,N,d,\mathbb {F})$ if the last row of $A_N$ differs from the zero row vector.

Recall also that $\mathcal {P}\mathcal {U}_1(m,N,\mathbb {F})$ denotes the set of specific paraunitary matrix functions which satisfy (9)–(13).

For power series $f(z)=\sum \nolimits _{k=-\infty }^\infty c_k z^k$ and $N\ge 1$, let $[f(z)]^-, [f(z)]^+, \, [f(z)]^-_N$, and $[f(z)]^+_N$, denote respectively $\sum \nolimits _{k=-\infty }^{-1} c_k z^k, \sum \nolimits _{k=0}^\infty c_k z^k, \, \sum \nolimits _{k=-N}^{-1} c_k z^k$, and $\sum \nolimits _{k=0}^N c_k z^k$ and the corresponding functions are assumed under these expressions if the convergence domains of these power series are clear from the context (the dependence of coefficients $c_k$ on the function $f$ is sometimes expressed by $c_k\{f\}=c_k$). Obviously $[f]^-=f-[f]^+$ under these notation.

A matrix function $S(z)\in \mathcal {P}[\mathbb {C}](m\times m)$ is called positive definite if $S(z)$ is such ($XS(z)X^*>0$ for each $0\not =X\in \mathbb {C}^{1\times m}$) for almost every $z\in \mathbb {T}$. The polynomial matrix spectral factorization theorem (see e.g., [3]) asserts that every positive definite $S(z)\in \mathcal {P}_N(m\times m)$ can be factorized as

$$\begin{aligned} S(z)=S^+(z)\widetilde{S^+}(z),\quad z\in \mathbb {C}\backslash \{0\}, \end{aligned}$$

(28)

where $S^+\in \mathcal {P}^+_N(m\times m)$ and $\det S^+(z)\not =0$ for each $z\in \mathbb {T}_+$ (consequently, $\widetilde{S^+}(z)$ is analytic and invertible in $\mathbb {T}_-$), and the representation (28) is unique in a sense that if $S(z)=S^+_1(z)\widetilde{S^+_1}(z)$ is another spectral factorization of $S(z)$, then $S^+_1(z)=S^+(z)U$ for some unitary $U\in \mathcal {U}_m$.

3 Wavelet Matrix Factorization Theorem

The material of this section is mostly well known, however we provide compact proofs of the given statements making emphasis on arbitrariness of a field $\mathbb {F}$.

Lemma 1

Let

$$\begin{aligned} {\mathbf{A}}(z)=\sum _{k=0}^N A_kz^k,\quad A_k\in \mathbb {F}^{m\times m},\quad A_N\not =0, \end{aligned}$$

(29)

be a wavelet matrix of rank $m$, order $N$ and degree $d, {\mathbf{A}}(z)\in \mathcal {W}\mathcal {M}(m,N,d,\mathbb {F})$. Then

• (a) $d\ge N$;

• (b) $d=N$ if and only if $\mathrm{rank}(A_0)=m-1$.

Proof

We have

$$\begin{aligned} \sum _{k=0}^N A^*_kz^{-k}= \widetilde{{\mathbf{A}}}(z)={\mathbf{A}}^{-1}(z)=\frac{1}{\det {\mathbf{A}}(z)}\big (\mathrm{Cof}{\mathbf{A}}(z)\big )^T= cz^{-d} \big (\mathrm{Cof}{\mathbf{A}}(z)\big )^T, \end{aligned}$$

and since $\mathrm{Cof}{\mathbf{A}}(z)\in \mathcal {P}^+(m\times m)$ and $A_N\not =0$, the equation $\sum \nolimits _{k=0}^N A^*_kz^{-k}\!= \frac{c}{z^{d}} \big (\mathrm{Cof}{\mathbf{A}}(z)\big )^T$ implies (a). Using the same reasoning, we conclude that $\mathrm{rank}(A_0)<m-1\Longleftrightarrow \mathrm{Cof}(A_0)=\mathbf {0}\Longleftrightarrow d>N$ (note that if $\mathrm{Cof}{\mathbf{A}}(z)=\sum \nolimits _{k=0} C_kz^k$, then $C_0=\mathrm{Cof}(A_0)$). Hence (b) follows as well (recall that $\mathrm{rank}(A_0)\not =m$ since $A_0A^*_N=0$).    $\square$

The following lemma will be used in the sequel only for $d=1$.

Lemma 2

(cf. [9, Th. 3.1]) Let $\mathbf {V}(z)=Q+Pz, P, Q\in \mathbb {F}^{m\times m}$, be a matrix polynomial. Then it is a wavelet matrix of rank $m$, order 1 and degree $d$ satisfying $V(1)=I_m, \mathbf {V}(z)\in \mathcal {W}\mathcal {M}_0(m,1,d,\mathbb {F})$, if and only if $P$ and $Q$ are complementary projections on $\mathbb {F}^m$ and $\mathrm{rank}(P)=d$.

Proof

The first part of the lemma can be proved by observation that $(Q+Pz)(Q^*+P^*z)=I_m$ and $\mathbf {V}(1)=I_m \, \Longleftrightarrow PQ^*=0,\;PP^*+QQ^*=I_m$, and $P+Q=I_m \Longleftrightarrow$ $P$ and $Q$ are complementary projections.

Now the property of $P$ to be a projection is equivalent to the existence of $U\in \mathbb {C}^{m\times d}, \mathrm{rank}(U)=\mathrm{rank}(P)=d$, with orthonormal columns, $U^*U=I_d$, such that (27) holds. Let $U_0\in \mathbb {C}^{m\times m}$ be any unitary matrix which completes the columns of $U$. Then

$$\begin{aligned} U_0^*V(z)U_0=U_0^*(I-UU^*+UU^*z)U_0=\mathrm{diag}[\underbrace{z,z,\ldots ,z}_d, \underbrace{1,1,\ldots ,1}_{m-d}] \end{aligned}$$

whose determinant is $z^d$. Hence the lemma is proved. $\square$

Theorem 2

(see [12, Th. 4.4.15]) Let ${\mathbf{A}}(z)$ be a wavelet matrix of rank $m$ and of order and degree $N$ satisfying the linear condition (5), ${\mathbf{A}}(z)\in \mathcal {W}\mathcal {M}_0(m,N,N,\mathbb {F})$. Then there exists a unique factorization (7).

Proof

By virtue of Lemma 1, $\mathrm{rank}(A_0)=m-1$. Let a nonzero column vector $v\in \mathbb {F}^m$ be orthogonal to the columns of $A_0, v^*A_0=0$, and $\mathbf {V}_1(z)=I_m-P+Pz$, where $P=v(v^*v)^{-1}v^*$ is the projection. Then $\mathbf {V}_1(z)\in \mathcal {W}\mathcal {M}_0(m,1,1,\mathbb {F})$ and $\mathbf {V}_1(z)$ divides ${\mathbf{A}}(z)$ on the left (the quotient $\mathbf {B}(z):=\mathbf {V}_1^{-1}(z){\mathbf{A}}(z)=\widetilde{\mathbf {V}_1}(z){\mathbf{A}}(z)\in \mathcal {P}^+(m\times m)$ as $P^*A_0=0$).

As $\deg (\mathbf {B})=\deg ({\mathbf{A}})-\deg (\mathbf {V}_1)=N-1$ and $\deg (\mathbf {B})\ge \mathrm{ord}(\mathbf {B})$ by virtue of Lemma 1, we have $\mathrm{ord}(\mathbf {B})=N-1$ (obviously, it cannot be less than $N-1$ as $\mathbf {V}_1(z)\mathbf {B}(z)={\mathbf{A}}(z)$). Hence $\mathbf {B}(z)\in \mathcal {W}\mathcal {M}_0(m,N-1,N-1,\mathbb {F})$ and if we continue these divisions, we get the factorization (7).

The uniqueness of a divisor $\mathbf {V}_1(z)$ (and hence of other factors) follows from the fact that if $\mathbf {V}_1(z)=Q+Pz\in \mathcal {W}\mathcal {M}_0(m,1,1,\mathbb {F})$, where $P$ and $Q$ are complementary projections and $\mathrm{rank}(P)=1$ (see Lemma 2), then $PA_0=0$ as $\mathbf {V}_1^{-1}(z){\mathbf{A}}(z)=(Pz^{-1}+Q){\mathbf{A}}(z)$ does not have the coefficient at $z^{-1}$, and since $\mathrm{rank}(A_0)=m-1$, such a projection $P$ is unique. $\square$

4 Proof of the Main Result

We introduce the following system of $m$ conditions which plays a key role in the proof of Theorem 1 (where this system comes from is explained in [4, Lemma 5]). Namely, for given Laurent polynomials

$$\begin{aligned} \zeta _j(z)\in \mathcal {P}_N^-[\mathbb {F}],\;\;\; j=1,2,\ldots ,m-1, \end{aligned}$$

(30)

let

$$\begin{aligned} \left\{ \begin{array}{l} \zeta _1(z)x_m(z)-\widetilde{x_1}(z)\in \mathcal {P}^+,\\ \zeta _2(z)x_m(z)-\widetilde{x_2}(z)\in \mathcal {P}^+,\\ \vdots \\ \zeta _{m-1}(z)x_m(z)-\widetilde{x_{m-1}}(z)\in \mathcal {P}^+,\\ \zeta _1(z)x_1(z)+\zeta _2(z)x_2(z)+\cdots +\zeta _{m-1}(z)x_{m-1}(z) +\widetilde{x_m}(z)\in \mathcal {P}^+. \end{array}\right. \end{aligned}$$

(31)

We say that a vector function

$$\begin{aligned} \mathbf {u}(z)=\big (u_1(z),u_2(z),\ldots ,u_{m-1}(z),\widetilde{u_m}(z)\big )^T, \quad u_i(z)\in \mathcal {P}_N^+[\mathbb {F}],\,i=1,2,\ldots ,m,\nonumber \\ \end{aligned}$$

(32)

(we emphasize that the first $m-1$ entries of (32) belong to $\mathcal {P}_N^+[\mathbb {F}]$ and the last entry belongs to $\widetilde{\mathcal {P}_N^+}[\mathbb {F}]$) is a solution of (31) if and only if all the conditions in (31) are satisfied whenever $x_i(z)=u_i(z), i=1,2,\ldots ,m$ (it is assumed that $\widetilde{x_m}(z)=\widetilde{u_m}(z)$). Observe that the set of solutions of (31) is a linear subspace of $\mathcal {P}_N^\oplus [\mathbb {F}](m):= \underbrace{{\mathcal {P}_N^+} [\mathbb {F}]\times \cdots \times {\mathcal {P}_N^+}[\mathbb {F}]\times \widetilde{\mathcal {P}_N^+}[\mathbb {F}]}_m$. We will see that actually this subspace is always $m$ dimensional.

We make essential use of the following

Lemma 3

Let (30) hold, and let

$$\begin{aligned} \mathbf {u}(z)=\big (u_1(z),u_2(z),\ldots ,u_{m-1}(z),\widetilde{u_m}(z)\big )^T \in \mathcal {P}_N^\oplus [\mathbb {F}](m), \end{aligned}$$

(33)

and

$$\begin{aligned} \mathbf {v}(z)=\big (v_1(z), v_2(z),\ldots ,v_{m-1}(z),\widetilde{v_m}(z)\big )^T \in \mathcal {P}_N^\oplus [\mathbb {F}](m), \end{aligned}$$

(34)

be two (possibly the same) solutions of the system (31). Then $\langle {\mathbf {u}}(z),{\mathbf {v}}(z)\rangle _m$ is the same for every $z\in \mathbb {C}\backslash \{0\}$, i.e.,

$$\begin{aligned} \sum _{i=1}^{m-1}u_i(z)\widetilde{v_i}(z)+\widetilde{u_m}(z)v_m(z)=\mathrm{Const}. \end{aligned}$$

(35)

Proof

Substituting $x_i=v_i$ in the first $m-1$ conditions and $x_i=u_i$ in the last condition of (31), and then multiplying the functions in the first $m-1$ conditions by $u_i$ and the function in the last condition by $v_m$, we get

$$\begin{aligned} \left\{ \begin{array}{l} \zeta _1v_mu_1-\widetilde{v_1}u_1\in \mathcal {P}^+,\\ \zeta _2v_mu_2-\widetilde{v_2}u_2\in \mathcal {P}^+,\\ \vdots \\ \zeta _{m-1}v_mu_{m-1}-\widetilde{v_{m-1}}u_{m-1}\in \mathcal {P}^+,\\ \zeta _1u_1v_m+\zeta _2u_2v_m+\cdots +\zeta _{m-1}u_{m-1}v_m +\widetilde{u_m}v_m\in \mathcal {P}^+. \end{array}\right. \end{aligned}$$

Subtracting the first $m-1$ functions from the last function in the latter system, we get

$$\begin{aligned} \sum _{i=1}^{m-1}u_i(z)\widetilde{v_i}(z)+\widetilde{u_m}(z)v_m(z)\in \mathcal {P}^+. \end{aligned}$$

(36)

We can interchange the roles of $u$ and $v$ in the above discussion to get in a similar manner that

$$\begin{aligned} \sum _{i=1}^{m-1}v_i(z)\widetilde{u_i}(z)+\widetilde{v_m}(z)u_m(z)\in \mathcal {P}^+. \end{aligned}$$

(37)

It follows from the relations (36) and (37) that the function in (35) and its adjoint belong to $\mathcal {P}^+$, which implies (35). $\square$

Corollary 1

If (33) and (34) are two solutions of the system (31), and

$$\begin{aligned} \mathbf {u}(1)=\mathbf {v}(1), \end{aligned}$$

(38)

then

$$\begin{aligned} \mathbf {u}(z)=\mathbf {v}(z) \;\text { for each }z\in \mathbb {C}\backslash \{0\}. \end{aligned}$$

(39)

In particular, if $\mathbf {u}(1)=\mathbf {0}\in \mathbb {F}^m$, then $\mathbf {u}(z)=\mathbf {0}$ for each $z\in \mathbb {C}\backslash \{0\}$ since the trivial vector function $\mathbf {v}(z)=\mathbf {0}$ is always a solution of the system (31).

Proof

Since $\mathbf {u}(z)-\mathbf {v}(z)$ is also a solution of (31), it follows from Lemma 3 that $\Vert {\mathbf {u}}(z)-{\mathbf {v}}(z)\Vert _{m}= \Vert {\mathbf {u}}(1)-{\mathbf {v}}(1)\Vert _{m}=0$ for each $z\in \mathbb {C}\backslash \{0\}$. Hence (39) holds. $\square$

Remark 3

One can see that Corollary 1 remains valid if we take any fixed point $z_0\not =0$ in the role of 1 in the Eq. (38).

Lemma 4

Let (30) hold, and let vector functions

$$\begin{aligned} \mathbf {v}_j(z)=\big (v_{1j}(z),v_{2j}(z),\ldots , \widetilde{v_{mj}}(z)\big )^T\in \mathcal {P}_N^\oplus [\mathbb {F}](m),\quad j=1,2,\ldots ,m,\quad \end{aligned}$$

(40)

be any $m$ solutions of the system (31). Then the determinant of the $m\times m$ Laurent polynomial matrix

$$\begin{aligned} V(z)=({\mathbf {v}_1}(z),{\mathbf {v}_2}(z),\ldots , {\mathbf {v}_m}(z)) \end{aligned}$$

(41)

is constant,

$$\begin{aligned} \det V(z)=\mathrm{Const}, \quad z\in \mathbb {C}\backslash \{0\}. \end{aligned}$$

(42)

Proof

Since the vector polynomials (40) satisfy the last condition of the system (31), we have

$$\begin{aligned} F(z)V(z)\in \mathcal {P}^+(m\times m), \end{aligned}$$

(43)

where $F(z)$ is defined by (15). Consequently, as $\det F(z)=1$,

$$\begin{aligned} \det V(z)\in \mathcal {P}^+. \end{aligned}$$

(44)

Since the vector polynomials (40) satisfy the first $m-1$ conditions of the system (31), we have

$$\begin{aligned} \phi _{ij}(z):=\zeta _i(z)v_{mj}(z)-\widetilde{v_{ij}}(z)\in \mathcal {P}^+, \end{aligned}$$

$1\le j\le m, 1\le i< m$. Direct computations show that

$$\begin{aligned} \widetilde{V}(z)F^{-1}(z)=\Phi (z)\in \mathcal {P}^+(m\times m), \end{aligned}$$

where $F^{-1}(z)$ is obtained from $F(z)$ by replacing each $\zeta _i$ by $-\zeta _i$ and

$$\begin{aligned} \Phi (z)= \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} -{\phi _{11}(z)}&{}-{\phi _{21}(z)}&{}\cdots &{}-{\phi _{m-1,1}(z)}&{} {v_{m1}(z)}\\ -{\phi _{12}(z)}&{}-{\phi _{22}(z)}&{}\cdots &{}-{\phi _{m-1,2}(z)}&{} {v_{m2}(z)}\\ \vdots &{} \vdots &{}\vdots &{}\vdots &{}\vdots \\ -{\phi _{1m}(z)}&{}-{\phi _{2m}(z)}&{}\cdots &{}-{\phi _{m-1,m}(z)}&{}{v_{m,m}(z)}\\ \end{array}\right) . \end{aligned}$$

Consequently $\det \widetilde{V}(z)=\det \Phi (z)\in \mathcal {P}^+$ and together with (44) this gives (42). $\square$

Remark 4

Observe that the relation (43) holds under the hypothesis of the lemma.

Now we construct explicitly $m$ independent solutions (40) of the system (31). We seek for a nontrivial solution

$$\begin{aligned} \mathbf {x}(z)=\big (x_1(z),x_2(z),\ldots ,\widetilde{x_m}(z)\big )^T\in \mathcal {P}_N^\oplus [\mathbb {F}](m) \end{aligned}$$

(45)

of (31), where

$$\begin{aligned} x_i(z)=\sum _{k=0}^N \alpha _{ik} z^k,\quad i=1,2,\ldots , m, \end{aligned}$$

(46)

and the coefficients $\alpha _{ik}$ are to be determined.

Equating the coefficients of $z^{-k}, k=0,1,2,\ldots ,N$, of the Laurent polynomials in the system (31) to 0, except the 0th coefficient of the $j$th function which we equate to 1, we get the following system of algebraic equations in the block matrix form which we denote by $\mathbb {S}_j$:

$$\begin{aligned} \mathbb {S}_j:= \left\{ \begin{array}{l} \Theta _1 X_m-\overline{X_1}=\mathbf{0}, \\ \Theta _2 X_m-\overline{X_2}=\mathbf{0}, \\ \vdots \\ \Theta _j X_m-\overline{X_j}=\mathbf{1}, \\ \vdots \\ \Theta _{m-1} X_m-\overline{X_{m-1}}=\mathbf{0}, \\ \Theta _1 X_1+\Theta _2 X_2+\cdots +\Theta _{m-1} X_{m-1}+\overline{X_m}=\mathbf{0}, \end{array}\right. \end{aligned}$$

(47)

where $\Theta _i$ is defined from the Eqs. (18) and (19), $\mathbf{0}=(0,0,\ldots ,0)^T\in \mathbb {F}^{N+1}, \mathbf{1}=(1,0,0,\ldots ,0)^T\in \mathbb {F}^{N+1}$, and the column vectors

$$\begin{aligned} X_i=(\alpha _{i0},\alpha _{i1},\ldots ,\alpha _{iN})^T,\;\;i=1,2,\ldots ,m, \end{aligned}$$

(48)

[see (46)] are unknown.

Remark 5

We emphasize that if $(X_1,X_2,\ldots ,X_m)$ defined by (48) is the solution of the system (47), then the vector function (45) defined by (46) will be the solution of the system (31).

It is easy to show that the system (47) $\mathbb {S}_j$ has a solution for each $j=1,2,\ldots ,m$. Indeed, determining $X_i, i=1,2,\ldots ,m-1$, from the first $m-1$ equations in (47),

$$\begin{aligned} X_i=\overline{\Theta _i}\cdot \overline{X_m}-\delta _{ij}\mathbf{1},\quad i=1,2,\ldots ,m-1, \end{aligned}$$

(49)

and then substituting into the last equation of (47), we get

$$\begin{aligned} \Theta _1\cdot \overline{\Theta _1}\cdot \overline{X_m}+\Theta _2\cdot \overline{\Theta _2}\cdot \overline{X_m} +\cdots +\Theta _{m-1}\cdot \overline{\Theta _{m-1}}\cdot \overline{X_m}+\overline{X_m}= \Theta _j\cdot \mathbf{1} \end{aligned}$$

(we assume that $\Theta _m=I_{N+1}$, i.e., the right-hand side is equal to $\mathbf{1}$ when $j=m$) or, equivalently,

$$\begin{aligned} (\Theta _1\cdot \overline{\Theta _1}+\Theta _2\cdot \overline{\Theta _2} +\cdots +\Theta _{m-1}\cdot \overline{\Theta _{m-1}}+I_{N+1})\cdot \overline{X_m} =\Theta _j\cdot \mathbf{1}. \end{aligned}$$

(50)

This is the same system as (21) [see (20)]. Since each $\Theta _i$ is symmetric, $\Theta _i=\Theta _i^T$, we have that $\Theta _i\overline{\Theta _i}=\Theta _i\Theta _i^*, i=1,2,\ldots ,m-1$, are non-negative definite and the coefficient matrix (20) of the system (50) (which is the same for each $j=1,2,\ldots ,m$) is positive definite (with all eigenvalues larger than or equal to 1). Consequently, $\Delta$ is nonsingular, $\det \Delta \ge 1$, and the system (50) has a unique solution for each $j$.

Finding the vector $\overline{X_m}\in \mathbb {F}^{N+1}$ from (50) and then determining $X_1,X_2,\ldots ,X_{m-1}$ from (49), we get the unique solution of $\mathbb {S}_j$. To indicate its dependence on $j$, we denote the solution of $\mathbb {S}_j$ by $(X_1^j,X_2^j,\ldots ,X_{m-1}^j,X_m^j)$,

$$\begin{aligned} X_i^j:=\left( \alpha _{i0}^j,\alpha _{i1}^j,\ldots , \alpha _{iN}^j\right) ^T, \quad i=1,2,\ldots ,m, \end{aligned}$$

(51)

so that the vector functions (40), where

$$\begin{aligned} v_{ij}(z)=\sum _{k=0}^N \alpha _{ik}^j z^k, \quad 1\le i,j\le m, \end{aligned}$$

are $m$ solutions of the system (31) (see Remark 5). These vector functions $\mathbf {v}_1(z), \mathbf {v}_2(z), \ldots , \mathbf {v}_m(z)$ are linearly independent since the linear transformation $\mathbf{L}:\mathcal {P}_N^\oplus [\mathbb {F}](m)\rightarrow \mathbb {F}^m$ which maps $(x_1(z),x_2(z), \ldots ,\widetilde{x_m}(z))$ into the 0th coefficients of the functions (or their adjoints) standing on the left-hand side of the system (31), viz. into $\big (c_0\{\widetilde{\zeta _1}(z)\widetilde{x_m}(z)-{x_1(z)}\}, c_0\{\widetilde{\zeta _2}(z)\widetilde{x_m}(z)-{x_2(z)}\}, \ldots , c_0\{\zeta _1(z)x_1(z)+\zeta _2(z)x_2(z)+\cdots +\widetilde{x_m}(z)\}\big )$, transforms $m$ vector functions $\mathbf {v}_1(z), \mathbf {v}_2(z), \ldots , \mathbf {v}_m(z)$ into linearly independent standard bases of $\mathbb {F}^m$. Namely, $\mathbf{L}\big (\mathbf {v}_i(z)\big ) =(\delta _{i1},\delta _{i2},\ldots ,\delta _{im}), i=1,2,\ldots ,m$, because of (47). Consequently $V(1)$ is nonsingular (see (41)) since if $\mathbf{w}\cdot V(1)\!=\!\mathbf{0}\in \mathbb {F}^m$ for some $\mathbf{0}\not \!=\!\mathbf{w}=(w_1,w_2,\ldots ,w_m) \in \mathbb {F}^m$, then $\mathbf{w}\!\cdot V(z) = \sum \nolimits _{k=1}^mw_k\mathbf {v}_k(z)={\mathbf {0}}$ by virtue of Corollary 1 of Lemma 3, which contradicts the independence of $\mathbf {v}_1(z), \mathbf {v}_2(z), \ldots , \mathbf {v}_m(z)$.

Let $U(z)$ be defined by (24). Then it has the form (10) and its column vectors $\mathbf {u}_1(z), \mathbf {u}_2(z), \ldots , \mathbf {u}_m(z)$,

$$\begin{aligned} U(z)=({\mathbf {u}_1}(z),{\mathbf {u}_2}(z),\ldots , {\mathbf {u}_m}(z)), \end{aligned}$$

(52)

are $m$ solutions of (31) satisfying

$$\begin{aligned} \mathbf {u}_i(1)=\mathbf{e}_i,\quad i=1,2,\ldots ,m, \end{aligned}$$

(53)

[since (12) holds because of (24)]. Therefore, the matrix function $U(z)$ is paraunitary since $\widetilde{U}(z)U(z)$ is a constant matrix, by virtue of Lemma 3, and this constant is $I_m$ since (12) holds. By virtue of Lemma 4, the determinant of $U(z)$ is also a constant which is equal to 1 since $\det U(1)=1$. Hence the relations (9)–(12) are valid for (52). Observe also that (17) holds because of (43) (see Remark 4) and (24). Hence it remains to show only the relation (13) in order to complete the proof of the first part of Theorem 1. Meanwhile we are ready to formulate and prove a corollary of the above discussion which will be used in the next section.

Corollary 2

Let (30) hold, let $F(z)$ be the matrix function defined by (15), and let (52) be the corresponding (according to Theorem 1) $U(z)\in \mathcal {P}\mathcal {U}_1(m,N,\mathbb {F})$.

If $\mathbf{b}=(b_1,b_2,\ldots ,b_m)\in \mathbb {F}^m$, then $\mathbf {u}_\mathbf{b}(z)$ is a solution of the system (31) satisfying

$$\begin{aligned} \mathbf {u}_\mathbf{b}(1)=\mathbf{b}, \end{aligned}$$

if and only if

$$\begin{aligned} \mathbf {u}_\mathbf{b}(z)=\sum _{i=1}^mb_i\,\mathbf {u}_i(z),\quad z\in \mathbb {C}\backslash \{0\}. \end{aligned}$$

Proof

Since $\mathbf {u}_1(z),{\mathbf {u}_2}(z),\ldots , {\mathbf {u}_m}(z)$ [see (52)] are solutions of the system (31) (recall that the set of its solutions is a linear subspace) and (53) holds, the first part of the corollary is clear. The second part follows from the first part and from Corollary 1. $\square$

In order to prove (13), let

$$\begin{aligned} \Psi (z)=\big (\psi _{ij}(z)\big )_{i,j=1}^m:=F(z)U(z), \end{aligned}$$

(54)

which, as has already been mentioned, belongs to $\mathcal {P}^+(m\times m)$ [see (17)]. Since the determinants of $F(z)$ and $U(z)$ are 1 for each $z\in \mathbb {C}\backslash \{0\}$, we can conclude that

$$\begin{aligned} \det \Psi (z)=1\quad \text { for each } z\in \mathbb {C}. \end{aligned}$$

(55)

Because of the structure of the matrix (15), we have

$$\begin{aligned} \psi _{ij}(z)=u_{ij}(z),\quad 1\le i\le m-1,\quad 1\le j\le m. \end{aligned}$$

Since $\widetilde{U}(z)=U^{-1}(z)$ and (11) holds, we have

$$\begin{aligned} u_{mj}(z)=\mathrm{cof}u_{mj}(z)=\mathrm{cof}\psi _{mj}(z),\quad 1\le j\le m. \end{aligned}$$

Thus, by virtue of (55),

$$\begin{aligned} 1=\det \Psi (z)=\sum _{j=1}^m\psi _{mj}(z)\mathrm{cof}\psi _{mj}(z)= \sum _{j=1}^m\psi _{mj}(z)u_{mj}(z) \end{aligned}$$

for each $z\in \mathbb {C}$ including 0. Consequently (13) holds.

Thus, the constructed matrix function $U(z)$ has all the desired properties, i.e., (16) and (17) hold. Note that a brief way of construction of the entries of $U(z)$ is described by the formulas (18)–(24).

The uniqueness of (16) follows from the uniqueness of spectral factorization (see Sect. 2) since $F(z)U(z)$ is the spectral factor of $F(z)\widetilde{F}(z)$.

Let us now show the converse part of Theorem 1. If we have a matrix function (16), then $u_{mj}(0)\not =0$ for some $j$ since (13) holds. Let us determine functions $\zeta _i$ by the formula (25), construct the matrix function (15), and show that (17) is valid. For this we need only to check that the entries of the last row of $\Psi (z)$ [see (54)] belong to $\mathcal {P}^+[\mathbb {F}]$. For $1\le n\le m$, we have

$$\begin{aligned}&\mathcal {P}\ni \sum _{i=1}^{m-1}\zeta _i(z)u_{in}(z)+\widetilde{u_{mn}}(z)= \sum _{i=1}^{m-1} \left( \frac{\widetilde{u}_{ij}(z)}{u_{mj}(z)} -\left[ \frac{\widetilde{u}_{ij}(z)}{u_{mj}(z)}\right] ^+\right) u_{in}(z)+\widetilde{u_{mn}}(z)\nonumber \\&\quad =\frac{1}{u_{mj}(z)}\left( \sum _{i=1}^{m-1} \widetilde{u}_{ij}(z)u_{in}(z)+u_{mj}(z)\widetilde{u_{mn}}(z)\right) - \sum _{i=1}^{m-1} \left[ \frac{\widetilde{u}_{ij}(z)}{u_{mj}(z)}\right] ^+ u_{in}(z)\nonumber \\&\quad =\frac{\delta _{nj}}{u_{mj}(z)}- \sum _{i=1}^{m-1} \left[ \frac{\widetilde{u}_{ij}(z)}{u_{mj}(z)}\right] ^+ u_{in}(z), \end{aligned}$$

(56)

which is analytic in a neighborhood of 0 and yields that this function belongs to $\mathcal {P}^+$. Thus (17) holds.

Let us now show the uniqueness of the desired $F(z)$.

Recall that $F^{-1}(z)$ can be obtained by replacing each $\zeta _i$ by $-\zeta _i$ in (15). Hence $F^{-1}(z)\in \widetilde{\mathcal {P}_N^+}[\mathbb {F}](m\times m)$. Since $\Psi (z)\in \mathcal {P}^+(m\times m)$ [see (54) and (17)] and (55) holds, we have $\Psi ^{-1}(z)\in \mathcal {P}^+(m\times m)$.

If $F_1(z)$ has the same form (15) with the last row $[\zeta _1'(z),\zeta _2'(z),\ldots ,\zeta _{m-1}'(z),1]$ and $\Psi _1(z):=F_1(z)U(z)\in \mathcal {P}^+(m\times m)$, then $U(z)=F^{-1}(z)\Psi (z)=F_1^{-1}(z)\Psi _1(z)$, which yields

$$\begin{aligned} \widetilde{\mathcal {P}^+}(m\times m)\ni F_1(z)F^{-1}(z)=\Psi _1(z)\Psi ^{-1}(z)\in \mathcal {P}^+(m\times m). \end{aligned}$$

Thus $F_1(z)F^{-1}(z)$ is a constant matrix while, on the other hand, this product has the form (15) with the last row $[\zeta _1'-\zeta _1,\zeta _2'-\zeta _2,\ldots ,\zeta _{m-1}'-\zeta _{m-1},1]$, which implies that $\zeta '_i=\zeta _i, i=1,2,\ldots ,m-1$.

5 Wavelet Matrices with Given First Rows

If we know the first row of a nonsingular wavelet matrix (7), then one can uniquely determine each $P_i$ and $Q_i$ in (7) and then recover ${\mathbf{A}}(z)$ by the formula (7), i.e., find the remaining $m-1$ rows of ${\mathbf{A}}(z)$. The algorithm for this procedure is well known (see e.g., [12, Th. 4.4.17]). We describe a new method of reconstruction of ${\mathbf{A}}(z)$ based on the proposed parametrization of nonsingular compact wavelet matrices. This description is included in a constructive proof of the following

Theorem 3

Let ${\mathbf{A}}_1(z)=\big ({a}_{11}(z),{a}_{12}(z),\ldots ,{a}_{1m}(z)\big )\in \mathcal {P}^+_N[\mathbb {F}](1\times m)$ be a row function of order $N \big (\sum \nolimits _{j=1}^m|\alpha _{Nj}|>0$, where ${a}_{1j}(z)=\sum \nolimits _{k=0}^N\alpha _{kj}z^k, j=1,2,\ldots ,m$ $\big )$ such that

$$\begin{aligned} {\mathbf{A}}_1(z)\widetilde{{\mathbf{A}}_1}(z)=1, \end{aligned}$$

$z\in \mathbb {C}\backslash \{0\}$. If ${\mathbf{A}}_1(1)=\mathrm{v}=(v_1,v_2,\ldots ,v_m)$ and $V\in \mathbb {F}^{m\times m}$ is a unitary matrix with the first row $\mathrm{v}$, then there exists a unique wavelet matrix of rank $m$ and of order and degree $N, {\mathbf{A}}(z)\in \mathcal {W}\mathcal {M}(m,N,N,\mathbb {F})$, which has the first row ${\mathbf{A}}_1(z)$ and ${\mathbf{A}}(1)=V$.

Remark 6

The existence of a unitary matrix $V\in \mathbb {F}^{m\times m}$ with the first row $\mathrm{v}$ is necessary for the existence of required ${\mathbf{A}}(z)$ since ${\mathbf{A}}(1)$ should be a unitary matrix from $\mathbb {F}^{m\times m}$. Obviously such $V$ exists if  $\mathbb {F}$ has the additional property that $\sqrt{x}\in \mathbb {F}$ when $x\in \mathbb {F}$. The row extension problem up to a unitary matrix over any general subfield of $\mathbb {C} ($including an algebraic number field as a special case$)$ is considered in recent paper [6]. We are grateful to the anonymous reviewer for indicating this reference.

Proof

For notational convenience, we will find ${\mathbf{A}}^T(z)$ instead of ${\mathbf{A}}(z)$. Thus we assume that the $\mathrm{v}=(v_1,v_2,\ldots ,v_m)^T$ is a column vector and ${\mathbf{A}}_1(z)=\big ({a}_{11}(z),\ldots ,{a}_{1m}(z)\big )^T\in \mathcal {P}^+_N(m\times 1)$, which satisfies $\widetilde{{\mathbf{A}}_1}(z){{\mathbf{A}}_1}(z)=1$ and ${\mathbf{A}}_1(1)=\mathrm{v}$. Furthermore, without loss of generality (we interchange the rows if necessary), we assume that $\alpha _{Nm}\not =0$ where we recall ${a}_{1m}(z)=\sum \nolimits _{k=0}^N\alpha _{km}z^k$.

Consider now $U_1(z)=\big (u_{11}(z),u_{21}(z),\ldots ,\widetilde{u_{m1}}(z)\big )^T \in \mathcal {P}^\oplus _N(m)$, where $u_{j1}(z)=a_{1j}(z), j=1,2,\ldots ,m-1$, and $\widetilde{u_{m1}}(z)=z^{-N}a_{1m}(z)$. Then $u_{m1}(0)\not =0$, and we can define $\zeta _i, i=1,2,\ldots ,m-1$, by a formula like (25)

$$\begin{aligned} \zeta _i(z)=\left[ \frac{\widetilde{u}_{i1}(z)}{u_{m1}(z)}\right] ^-= \left[ \frac{\widetilde{u}_{i1}(z)}{u_{m1}(z)}\right] _N^-, \quad i=1,2,\ldots ,m-1, \end{aligned}$$

and construct the matrix function $F(z)$ defined by (15). Consider now the corresponding (according to Theorem 1) $U(z)\in \mathcal {P}\mathcal {U}_1(m,N,\mathbb {F})$ whose columns $\mathbf {u}_1(z), \mathbf {u}_2(z), \ldots , \mathbf {u}_m(z)$, as we remember, are solutions of the system (31) and also satisfy (53). We can make sure by direct computations like (56) that $U_1(z)$ satisfies the last condition of (31), and taking into account the relations

$$\begin{aligned}&\zeta _i(z)u_{m1}(z)-\widetilde{u_{i1}}(z)=\left( \frac{\widetilde{u}_{i1}(z)}{u_{m1}(z)} -\left[ \frac{\widetilde{u}_{i1}(z)}{u_{m1}(z)}\right] ^+\right) u_{m1}(z)-\widetilde{u_{i1}}(z)\\&\quad =-\left[ \frac{\widetilde{u}_{i1}}{u_{m1}}\right] ^+ u_{m1}\in \mathcal {P}^+, \end{aligned}$$

$i=1,2,\ldots ,m-1$, we see that $U_1(z)$ is a solution of the system (31). Consequently, according to Corollary 2,

$$\begin{aligned} U_1(z)=U(z)\cdot \mathrm{v}=\sum _{i=1}^mv_i\,\mathbf {u}_i(z), \end{aligned}$$

(as this equation holds for $z=1$) and it is the first column of $U(z)\cdot V$. This matrix function is paraunitary as well (since $V$ is unitary) and satisfies the condition $U(1)V=V$. Hence

$$\begin{aligned} {\mathbf{A}}(z)=\mathrm{diag}[1,1,\ldots ,1,z^N]U(z)V \end{aligned}$$

will be the desired matrix.

The uniqueness of ${\mathbf{A}}(z)$ is also valid since $F(z)$ described in the above proof is the same as the matrix function corresponding to ${\mathbf{A}}(z)\cdot V^{-1}\in \mathcal {W}\mathcal {M}_1(m,N,N,\mathbb {F})$ by the to one-to-one map (14) and Theorem 1, and such $F(z)$ is unique. $\square$

6 Other Possible Applications

In the end, we would like to discuss briefly some possible applications of the proposed method of wavelet matrices construction. It should be emphasized that the intent of this section is largely motivational and we consider three separate topics.

6.1 Rational Approximations to Compact Wavelet Matrices

Most of compact wavelet matrices used in practice, for example Daubechies wavelet matrices, obey certain additional restrictions. The proofs of the existence of such matrices and the ways of their construction are highly non-linear and thus the obtained coefficients are in general irrational. In actual calculations on a digital computer, these coefficients should be quantized and hence approximated by rational numbers. It may happen during this approximation that the basic property of the wavelet matrices (2) will not be preserved exactly. Using the proposed parametrization and taking $\mathbb {Q}$ in the role of $\mathbb {F}$, we can provide an approximation of any nonsingular compact wavelet matrix $\mathcal {A}$ by $\mathcal {A}'$ with rational coefficients preserving exactly the shifted orthogonality condition (2). Indeed, it is just sufficient to note in the proof of Theorem 1 that the maps described in (26) are continuous. Hence, if we find the point in $\mathbb {R}^{(m-1)N}$ corresponding to $\mathcal {A}$, approximate it by rational coordinates and go back into the space of wavelet matrices, then we will get the desired $\mathcal {A}'$ (in a similar manner, Vaidyanathan [14] used the parametrization (8) for the same purposes). Such various approximations for Daubechies wavelet matrices of different genus are explicitly constructed in [5].

6.2 Cryptography

As said in Remark 1, the factorization (7) much resembles the factorization for ordinary polynomials into linear terms. On the other hand, an efficient way of construction of paraunitary matrix polynomials associated to the given points in $\mathbb {F}^{(m-1)N}$ expressed by the diagram (26), which helps to handle such matrix polynomials easily, can be compared to natural parametrization of ordinary polynomials by their coefficients. An application of polynomial factorization theory in the cryptography is widely known. Having the above similarities (and the advantages mentioned in Remark 1), one can also expect certain applications of the developed theory in the cryptography.

6.3 Selection of Best Wavelet Matrices

Which wavelet matrix is most suitable to apply in a given practical situation represents frequently a certain optimization problem (which is sometimes very hard to solve) or should be obtained empirically by computer simulations. Having a quick access to the complete bank of nonsingular compact wavelet matrices due to the parametrization (26) gives an opportunity to choose the best possible wavelet matrix by nearly complete screening.