The bounds estimate of sub-band operators for multi-band wavelets

A concept of the sub-band operator of multi-band wavelets is introduced, the theory of d-circular matrices is developed and the upper bound and the lower bound of the norm of the sub-band operator are obtained. Examples are provided to illustrate the results proposed in this paper.

Let H be a separable Hilbert space, and E an indexing set. A sequence {f l } l∈E is called a frame in H if there exist constants 0 < A ≤ B < +∞ such that where A and B are called lower and upper frame bounds, respectively. If A = B, the frame is called tight frame. When H = L 2 (R), a wavelet ψ(x) ∈ H gives rise to a classical wavelet frame {a -j/2 ψ(a -j xbk), j, k ∈ Z} with parameters a > 1, b > 0. Chui and Shi [17] established the relationship between the parameters and the frame bounds, Equation (1.2) shows that the energy of a biorthogonal wavelet transform is controllable although it is not conservative. Moreover, B -A or B -A are smaller, the performance of a biorthogonal wavelet transform may be better, i.e., the energy is amplified in some cases and decreased in other cases. The classical biorthogonal wavelets are not tight frames, so obtaining the exact values of their bounds is difficult. Instead of estimating the bounds in (1.2), we can try to obtain the upper bound and the lower bound of the norm of the sub-band operator.
This paper is organized as follows. In Sect. 2, we define the sub-band operator, and obtain the limit form of the norm of the sub-band operator. We present a method for computing the upper bound and the lower bound of the norm of the sub-band operator based on the theory of circular matrix. Section 3 gives some examples to illustrate the results proposed in this paper.

Sub-band operator and d-circular matrix
Recall the sub-band coding scheme or Mallat algorithm associated to a d-band real biorthogonal wavelets. There are 2d filters h = (h n ) n∈Z , . . , g d-1 } are used for decomposition and { h, g 1 , g 2 , . . . , g d-1 } for reconstruction. Starting from a data sequence x = (x n ) n∈Z , we convolve with {h, g 1 , g 2 , . . . , g d-1 }, The reconstruction operation is The constraint conditions for biorthogonal d-band filter banks with perfect reconstruction property are: a. the low-pass and high-pass condition where δ j denotes the Dirac sequence such that δ j = 1 for j = 0 otherwise δ j = 0; c. the perfect reconstruction condition x = x. In order to define (2.1) and (2.2) as operators, namely, sub-band operators, we assume that the input signal x ∈ l 2 (-∞, +∞). Now consider the separable Hilbert space l 2 (-∞, +∞). Define Throughout this paper, we assume h, g 1 , g 2 , . . . , g d-1 , h, g 1 , g 2 , . . . , g d- 1 have only finitely many nonzero elements.
The proof of Theorem 2.1 is trivial. As is well known, a bounded linear operator on l 2 (-∞, +∞) can be expressed by an infinite-dimensional matrix. Using matrix notations, we have a more helpful expression of the operator T. Let Then A = [s n,k ] and A = [ s n,k ] (-∞ < n, k < +∞) are infinite block circular matrices along four directions (up, down, left and right). At this time, where x and y are doubly infinite column vectors to fit the matrix operation. Hence T can be viewed as an infinite matrix A, i.e., T = A. If the sub-band decomposition has the perfect reconstruction condition, then it is obvious that A and A should satisfy where I denotes the infinite identity matrix and A * denotes the transpose of complex conjugate of A. Thus, T -1 = A * . Since (Ax, y) = (x, A * y), the adjoint operator of T is T * = A * . Let Q = T * T, we have the following lemma.

Lemma 2.1 Retaining the definitions and notations as above, we have
Proof Items (1) and (2) are trivial according to the operator theory [18]. Item (3) follows the fact that Q is a self-adjoint operator due to Q * = Q. Q = T * T is called a frame operator in general [2]. Let ν n and ν n denote the nth rows of A and A, respectively. Clearly, (ν j , ν k ) = δ j-k , where δ j is the Dirac sequence, i.e., δ j = 1 for j = 0 otherwise δ j = 0. Therefore, {ν n } and { ν n } are dual biorthogonal bases in l 2 (-∞, +∞).
For an arbitrary x ∈ l 2 (-∞, +∞), Let m and M denote the lower bound and the upper bound of T , respectively. Then Similarly, where m and M are the lower bound and the upper bound of T , respectively. Therefore, (2.4) and (2.5) are the counterparts of (1.1) in l 2 (-∞, +∞). Now we define a finite matrix A n as the partial matrix of the infinite matrix A, its row index and column index are finite with the following form: · · · · · · · · · · · · · · · · · · · · · · · · · · · Since Q n is a finite-dimensional self-adjoint compact operator, we have Extend [y] n to y (n) ∈ l 2 (-∞, +∞) by appending 0s to the tuples which are not defined by [y] n . Clearly, [y] n = y (n) = 1, and It implies that Hence, Q = lim n→∞ max{τ (n) 1 , τ (n) 2 , . . . , τ (n) dn }. The proof is complete.
Theoretically, Theorem 2.2 gives an exact value for Q and T = √ Q is used to compute the norm of T. However, the eigenvalues are not easy to compute for generalized block Toeplitz matrices. We shall use the theory of circular matrix to compute the norm of Q. The block circular matrix is defined as follows [19]: -5 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · Clearly, A n is different from B n . Let C n = B n -A n , then only finitely many (fixed) elements in C n are not 0 no matter how large the dimension 4n is. We have A * n A n = B * n B n -C * n B n -B * n C n + C * n C n . (2.6) We have Theorem 2.3.

Theorem 2.3
Let P n = B * n B n and {λ (n) 1 , λ (n) 2 , . . . , λ (n) dn } be all eigenvalues of P n . Then Proof Let [x] n be a dn-dimensional vector. We first prove that for [x] n = 1, In fact, the operator B n is bounded. For [x] n = 1, there exists a positive real number M such that Note that C n [x] n only contains dn nonzero components. Hence By Theorem 2.2, we have The proof is complete.
A so-called d-circular matrix [20], which is generated by the filters h, g 1 , g 2 , . . . , g d-1 , is denoted as M n . For d = 4, M 3 is as follows: Clearly, B n and M n are not so different. One can be obtained by exchanging the places of some rows of another, i.e., there exists an orthonormal matrix E n such that M n = E n B n . The purpose is to facilitate the calculation of the eigenvalues.
Similarly, let M n be a d-circular matrix generated by h, g 1 , g 2 , . . . , g d- 1 . The perfect reconstruction condition is that there exists an integer N 0 , such that, for all n ≥ N 0 (in what follows, n sufficiently large is in this sense), where I dn is a dn × dn identity matrix.

Theorem 2.4
Let M n be a d-circular matrix generated by h, g 1 , g 2 , . . . , g d- 1 . For sufficiently large n, then Proof Define Since A dn B dn ∞ ≤ A dn ∞ × B dn ∞ , · ∞ is a compatible matrix norm. Note that A dn = M n M T n positive definite matrices, and all of the eigenvalues of M n M T n are positive. According to the theory of matrices [19], we have We have Firstly, we verify that, for n ≥ N 0 ,