Construction of multiscaling functions using the inverse representation theorem of matrix polynomials

Abstract Wavelet analysis deals with finding a suitable basis for the class of L functions. Symmetric basis functions are very useful in various applications. In the case of all wavelets other than the famous Haar wavelet, the simultaneous inclusion of compact supportedness, orthogonality and symmetricity is not possible. Theory of multiwavelets assumes significance since it offers orthogonal, compact frames without losing symmetry. We can also construct symmetric, compactly supported and pseudo-biorthogonal bases which are also possible only in the case of multiwavelets. The properties of a multiwavelet directly depends on the corresponding multiscaling function. A multiscaling function is characterized by a unique symbol function, which is a matrix polynomial in complex exponential. A matrix polynomial can be constructed from its spectral data. Our aim is to find the necessary as well as sufficient conditions a spectral data must satisfy so that the corresponding matrix polynomial is the symbol function of a compactly supported, symmetric multiscaling function UðxÞ. We will construct such a multiscaling function UðxÞ and its dual ~ UðxÞ such that the functions UðxÞ and ~ UðxÞ form a pair of pseudo-biorthogonal multiscaling functions.


Introduction
Wavelet bases can be constructed using the notion of multiresolution analysis (MRA). In order to generate an MRA, we need to find a function vector U ¼ ð/ i Þ n i¼1 , / i : R ! C which generates an MRA. A function vector U generates an MRA if it is L 2 stable, compactly supported and satisfies the multiscaling equation which is called a symbol function or a mask function. The existence of a solution to the multiscaling equation is determined by the nature of the corresponding symbol function. Moreover, the properties of this solution are determined by the nature of the symbol function. In fact, the symbol function HðnÞ is a matrix polynomial in complex exponential. Each matrix polynomial HðnÞ possesses a spectral pair or Jordan pair (X, T), where X is a matrix containing the generalized eigenvectors of HðnÞ and T is a block diagonal matrix where each block is a Jordan matrix corresponding to the eigenvalues of HðnÞ. Given the pair (X, T), we can construct a matrix polynomial having (X, T) as its spectral data. We have to find the properties of a spectral data so that the corresponding matrix polynomial is the symbol function of a compactly supported, symmetric multiscaling function UðxÞ. Also, we have to find a dual multiscaling functionŨðxÞ so that the functions UðxÞ andŨðxÞ form a pair of pseudo-biorthogonal multiscaling functions.

Preliminaries
Let where L p ðk 0 Þ is the p th derivative of LðkÞ at k 0 .
This is a generalization of the usual notion of a Jordan chain of a square matrix. Let T 2 C nlÂnl and T be a block diagonal matrix where each block is a Jordan matrix corresponding to an eigenvalue of LðkÞ, also let X 2 C nÂnl and column vectors of X are precisely the Jordan chains of LðkÞ corresponding to the eigenvalues of LðkÞ. The Jordan chains appear in X in the order the corresponding eigenvalues appear in T. Then the pair (X, T) is said to be a Jordan pair. Now we will give the definition of a decomposable pair.
where X 1 2 C nÂm , X 2 2 C nÂðnlÀmÞ and T 1 2 C mÂm , T 2 2 C ðnlÀmÞÂðnlÀmÞ with 0 m nl is called a decomposable pair of degree l if the matrix is nonsingular. A pair (X, T) satisfying this property is called a decomposable pair of the regular n Â n matrix polynomial LðkÞ ¼ Given a decomposable pair (X, T), we can construct a matrix polynomial LðkÞ having (X, T) as its decomposable pair using the inverse representation theorem of matrix polynomials which is stated as follows.
i¼0 . Then, for every n Â nl matrix V such that the matrix ð S lÀ2 V Þ is nonsingular, the matrix polynomial has (X,T) as its decomposable pair.
If (X, T) is a Jordan pair of a matrix polynomial LðkÞ, then it is a decomposable pair of LðkÞ [1]. We can construct a matrix polynomial for a given Jordan pair (X, T) using the inverse representation theorem. A sufficient condition on a Jordan pair (X, T) so that the corresponding matrix polynomial is the symbol function of a compactly supported multiscaling function has been derived by us in [2] and is as follows.
Theorem 2.2 [2] Let (X,T) = ð½X 1 X 2 ; T 1 È T 2 Þ be a Jordan pair such that the nl Â nl matrix ðI È T 2 Þ À ðT 1 È IÞ is of full rank. Then there exists a symbol function HðnÞ with Jordan pair (X,T) such that the corresponding multiscaling equation (1.1) has a solution vector U such thatÛ is continuous at 0 withÛð0Þ 6 ¼ 0.
Thus, by choosing a Jordan pair (X, T) such that ðI È T 2 Þ À ðT 1 È IÞ is of full rank, we can form a multiscaling function U. Now our aim is to find the additional conditions on (X, T) so that this multiscaling function is symmetric also. A multiscaling function vector U is symmetric if its each component function is symmetric about some point.
The symmetricity of U is closely related to the properties of the associated symbol function HðnÞ. A sufficient property of HðnÞ for U to be symmetric is given by the following Lemma.
then U is symmetric about the point a.
In the next section, we will find the conditions on the Jordan pair (X, T) so that the corresponding multiscaling function vector is symmetric based on these results.
A multiscaling function UðxÞ is said to be orthogonal if hUðx À kÞ; Uðx À tÞi ¼ In some situations, we use biorthogonal bases or pseudo biorthogonal bases instead of the orthogonal ones. Sometimes, biorthogonal bases with additional properties act more effectively than orthogonal bases. If we perform one analysis step followed by one synthesis step using a biorthogonal basis, we get the initial signal exactly. In the case of pseudo biorthogonal basis, an analysis step followed by the synthesis step will produce the initial signal multiplied by c. We can recover the signal exactly by rescaling by c at each synthesis step [5]. In the case of scalar wavelets, H(z) and F(z) are polynomials in z so that Hð1Þ ¼ Fð1Þ ¼ 1 and HðÀ1Þ ¼ FðÀ1Þ ¼ 0 [3]. Hence the case c 6 ¼ 1 is not possible in the case of scalar wavelets.
Our aim is to construct a symbol function of degree 3 by selecting a suitable Jordan pair so that the corresponding multiscaling function UðxÞ is symmetric, compactly supported and there exists a dual multiscaling functionŨðxÞ so that the pair fUðx À kÞ : k 2 Zg and fŨðx À kÞ : k 2 Zg form a pseudo-biorthogonal pair of bases. The condition on H(z) for pseudo-biorthogonality is given by Eq. (2.14). In this article, we will formulate the condition on H(z) for the symmetricity of the corresponding multiscaling function. Then, we will construct a compactly supported, symmetric multiscaling function UðxÞ. Finally, we will construct the dual multiscaling functionŨðxÞ so that UðxÞ andŨðxÞ form a pseudobiorthogonal pair of multiscaling functions.

Symmetry
In this section, we will define a symmetric matrix polynomial and will show that a multiscaling function vector U is symmetric if the corresponding symbol function HðnÞ is symmetric. We will then derive the necessary as well as sufficient properties a Jordan pair (X, T) must possess so that the corresponding matrix polynomial is symmetric.
Math Sci (2016) 10:95-104 97 Lemma 3.1 If the symbol function of degree l is symmetric, then the corresponding multiscaling function U is symmetric about the point l 2 . Proof Given that Our aim is to find the necessary as well as sufficient conditions on a Jordan pair such that the corresponding multiscaling function U is symmetric. We have shown that the multiscaling function corresponding to a symmetric symbol function is symmetric. Since the symbol function is a matrix polynomial, our problem changes to finding the properties of Jordan pair of a symmetric matrix polynomial. A crucial necessary property of Jordan pair of a symmetric matrix polynomial is given by the following Lemma.
Lemma 3.2 If LðkÞ is a symmetric matrix polynomial, then its Jordan pair (X, T) has the property that if k 0 6 ¼ 0 is an eigenvalue of LðkÞ with eigenvector x 0 , then 1 k 0 is also an eigenvalue with the same eigenvector x 0 . If 0 is an eigenvalue of LðkÞ, then LðkÞ will have an infinite eigenvalue with the eigenvector that of 0. Proof Given that is a symmetric matrix polynomial. If 0 is an eigenvalue of LðkÞ, then sinceLðkÞ ¼ k l Lð 1 k Þ ¼ LðkÞ, the matrix polyno-mialLðkÞ also has an eigenvalue 0. i.e. LðkÞ has an eigenvalue at infinity (By definition). Now, let k 0 6 ¼ 0 is an eigenvalue with eigenvector x 0 , then we have Since LðkÞ is a symmetric matrix polynomial, we have i.e. 1 k 0 is also an eigenvalue with the same eigenvector x 0 . h Lemma 3.2 states that for a symmetric matrix polynomial, it is necessary that the eigenvalues occur in reciprocals. Our attempt is to construct a symmetric matrix polynomial by selecting a suitable Jordan pair (X, T) with only finite eigenvalues. We will state the sufficient properties of a Jordan pair (X, T) such that the corresponding matrix polynomial is symmetric.
Theorem 3.1 Let (X,T) be a Jordan pair where X 2 n Â nl and T is a diagonal matrix of order nl with entries being eigenvalues, n 2 N, n ! 2 and l is even. T has only nonzero elements neither of which equals 1. Assume that the eigenvalues in T occur in reciprocals with same eigenvectors in X. i.e. if k 0 is an eigenvalue in T with eigenvector x 0 , then 1 k 0 is also an eigenvalue in T with same eigenvector x 0 . Then a matrix polynomial with Jordan pair (X,T) is symmetric.
Proof Given that (X, T) is a Jordan pair where X 2 n Â nl and T is a diagonal matrix such that T 2 nl Â nl. Let k i ði ¼ 1; 2 Á Á Á nl 2 Þ are the eigenvalues in T. Since the eigenvalues occur in reciprocals, it follows that 1 is a matrix polynomial with the Jordan pair (X, T). We have to show that LðkÞ is symmetric, i.e.
Assume the contrary that LðkÞ 6 ¼ k l Lð 1 k Þ, or the matrix polynomial NðkÞ ¼ LðkÞ À k l Lð 1 k Þ 6 ¼ 0. The sum of algebraic multiplicities of eigenvalues of a matrix polynomial will be the degree of its Determinant [1]. Since NðkÞ is a nonzero matrix polynomial of degree l and order n, the sum of algebraic multiplicities of the eigenvalues of NðkÞ will be less than or equal to nl . Now we will show that if NðkÞ 6 ¼ 0, then the total algebraic multiplicity exceeds nl, which is a contradiction.
We claim that k i and 1 k i ði ¼ 1; 2 Á Á Á nl 2 Þ are eigenvalues of NðkÞ with same eigenvectors they had for LðkÞ. To prove this, suppose that k i is an eigenvalue of LðkÞ with eigenvector x i for some i. Then we have Lðk i Þx i ¼ 0 and Thus k i and 1 k i are eigenvalues of NðkÞ for i ¼ 1; 2 Á Á Á nl 2 . Thus we get a total of nl 2 þ nl 2 ¼ nl eigenvalues for NðkÞ. Now NðkÞ is given by i.e. 1 is an eigenvalue of NðkÞ with algebraic multiplicity n. Then the sum of algebraic multiplicities of eigenvalues of NðkÞ is at least nl þ n (there can be other eigenvalues also), which is not possible since the sum of algebraic multiplicities of all eigenvalues of NðkÞ should not exceed nl. Hence our assumption that LðkÞ 6 ¼ k l Lð 1 k Þ is false. We can conclude that LðkÞ ¼ k l Lð 1 k Þ, i.e. the matrix polynomial LðkÞ is symmetric. h We can construct symmetric matrix polynomials of even degree using the above result. To construct symmetric matrix polynomials of odd degree, we have to select the Jordan pair (X, T) with minor changes. For any symmetric matrix polynomial LðkÞ of odd degree, we can easily verify that LðÀ1Þ ¼ 0. Then, we have LðÀ1Þp i ¼ 0, for linearly independent eigenvectors p i where i ¼ 1; 2 Á Á Á n. Hence -1 is an eigenvalue of LðkÞ with multiplicity n. Incorporating this change, we state the preceding result for odd values of l.
Theorem 3.2 Let (X,T) be a Jordan pair where X 2 n Â nl and T is a diagonal matrix of order nl, n 2 N, n ! 2 and l is odd. T has only nonzero elements neither of which equals 1. Assume that the eigenvalues in T occur in reciprocals with same eigenvectors in X. Also, -1 is an eigenvalue in T with multiplicity n. Then a matrix polynomial with Jordan pair (X,T) is symmetric.
Proof Given that -1 occurs n times in T, then there will be nl À n eigenvalues in T other than -1. Given that they occur in reciprocals, i.e. if k i is an eigenvalue in T, then 1 k i is also an eigenvalue in T. Thus we have, for i ¼ 1; 2 Á Á Á nlÀn 2 , k i and its reciprocal 1 k i are eigenvalues in T, and together they constitute nl À n eigenvalues (Here nl À n is always even since l is odd). Now, let LðkÞ ¼ A 0 þ A 1 k þ A 2 k 2 þ Á Á Á þ A l k l be a matrix polynomial with Jordan pair (X, T), we have to show that As we did in the last proof, assume the contrary that LðkÞ 6 ¼ k l Lð 1 k Þ, or the matrix polynomial NðkÞ ¼ LðkÞ À k l Lð 1 k Þ 6 ¼ 0. Since NðkÞ is a nonzero matrix polynomial of degree l and order n, the sum of algebraic multiplicities of the eigenvalues of NðkÞ can be maximum nl. We claim that k i and 1 k i ði ¼ 1; 2 Á Á Á nlÀn 2 Þ are eigenvalues of NðkÞ with same eigenvectors they had for LðkÞ. To prove this, suppose that k i is an eigenvalue of LðkÞ with eigenvector x i for some i. Then we have, Lðk i Þx i ¼ 0 and i.e. k i and 1 k i are eigenvalues of NðkÞ for i ¼ 1; 2 Á Á Á nlÀn 2 . Thus, we get a total of nlÀn 2 þ nlÀn 2 ¼ nl À n eigenvalues for NðkÞ. Since -1 is an eigenvalue of LðkÞ with algebraic multiplicity n, LðÀ1Þp i ¼ 0 for linearly independent eigenvectors p i , i ¼ 1; 2 Á Á Á n. Then we have, i.e. -1 is an eigenvalue of NðkÞ with algebraic multiplicity n. Then, the sum of algebraic multiplicities of eigenvalues of NðkÞ is at least nl À n þ n ¼ nl. Now, NðkÞ is given by i.e. 1 is an eigenvalue of NðkÞ with algebraic multiplicity n. Then, the sum of algebraic multiplicities of eigenvalues of NðkÞ is at least nl þ n (there can be other eigenvalues also), which is not possible since the sum of algebraic multiplicities of all eigenvalues of NðkÞ cannot exceed nl. Hence, our assumption that LðkÞ 6 ¼ k l Lð 1 k Þ is false. We conclude that LðkÞ ¼ k l Lð 1 k Þ, i.e. the matrix polynomial LðkÞ is symmetric. h

Construction of multiscaling function
We have obtained the properties of the spectral data of a matrix polynomial so that it is a symbol function of a symmetric multiscaling function U. Since each entry in this symbol function is a trigonometric polynomial (algebraic polynomial in z ¼ e Àin ), the associated multiscaling function is compactly supported [6]. We will obtain the multiscaling function by employing the cascade algorithm [3].
The cascade algorithm will converge if the multiscaling coefficients satisfy certain properties and if the initial functions are appropriately chosen. Let H 0 , H 1 , H 2 , H 3 2 C 2Â2 be the set of multiscaling coefficients and define the 8 Â 8 matrix D as Let D 0 be the 3 Â 3 sub block matrix of D at the top left, D k is the sub matrix 'k' steps to the left. Then and Theorem 4.1 [3] Assume that HðnÞ satisfies the sum rules of order 1, and the joint spectral radius where F 1 is the orthogonal complement of the common left Then, the cascade algorithm has a unique solution U which is Holder continuous of order a for any a\ À log m k.
A method to find HðnÞ which satisfies the sum rules of order 1 by suitably selecting the Jordan pair (X, T) is given in [2]. Based on that method, we construct the symbol function HðnÞ so that it satisfies the sum rules of order 1. While finding the multiscaling coefficients H k , we ensure that the value of the joint spectral radius qðD 0 jF 1 ; D 1 jF 1 . . .D mÀ1 jF 1 Þ is less than 1. Then by Theorem 4.1, the cascade algorithm converges for the set of multiscaling coefficients H k . An example of this construction is given as follows. We will start with a Jordan pair (X, T) satisfying the conditions 1. I À T is of full rank (Theorem 2.2) 2. The eigenvalues are nonzero and not equal to 1. They occur in reciprocals with same eigenvectors. Also, -1 is an eigenvalue with multiplicity n (in the following example, n = 2) (Theorem 3.2) Without loss of generality, we take X 2 C 2Â6 and T 2 C 6Â6 satisfying the above listed conditions, and are given by, It can be verified that I À T is of rank 6, then there exists a matrix polynomial with Jordan pair (X, T) which is a symbol function of a multiscaling function vector U (Theorem 2.2). Since (X, T) satisfies the conditions in Theorem 3.2, the matrix polynomial HðnÞ must be symmetric. By employing the procedure to find the multiscaling coefficients H k given in [2], we have obtained the multiscaling coefficients as follows.
Now choose the initial function U 0 as the piecewise linear function that interpolates at the points 0, 1, 2 and 3. Then, the cascade algorithm will converge to the solution U ( Fig. 1) which is compactly supported in [0,3] and is symmetric about the point 1.5.
The obtained multiscaling function is compactly supported and symmetric. Thus, we are able to construct a compactly supported multiscaling function which is symmetric about the point 1.5.

Construction of pseudo biorthogonal symmetric multiscaling functions
In the previous sections, we constructed a symbol function H(z) and the corresponding multiscaling function UðxÞ which is symmetric and compactly supported. Now, we have to construct the dual multiscaling functionŨðxÞ. For that, we have to find the dual symbol F(z) corresponding to H(z) so that the generalized condition of perfect reconstruction (2.14) holds. For a given symbol H(z), a dual symbol F(z) satisfying Eq. (2.14) exists if Determinants of H(z) and HðÀzÞ do not have common roots [5]. Now, the Jordan pair (X, T) that we selected for constructing H(z) is given by, The diagonal entries of T are the eigenvalues of the obtained symbol function H(z). Looking at the diagonal entries of T, it is clear that negative of an eigenvalue is not again an eigenvalue. Since eigenvalues of H(z) are precisely the roots of the Determinant of H(z), we can say that negative of a root of Determinant of H(z) is not again its Fig. 1 The two components / 0 and / 1 of the multiscaling function U corresponding to the obtained multiscaling coefficients. Here both components / 0 and / 1 are symmetric and are compactly supported in [0, 3] root. In other words, Determinants of H(z) and HðÀzÞ do not have common roots. Thus, there exists a dual symbol F(z) so that the generalized condition of perfect reconstruction is satisfied. Using the cofactor method given in [5], we got the dual symbol function F(z) corresponding to H(z) as Here The properties of the multiscaling coefficients H k which enable the symmetricity of UðxÞ are preserved and hence the dual multiscaling functionŨðxÞ will also be symmetric. Since the entries in the matrix coefficients F À2 , F À1 , F 4 , F 5 are very small, the support ofŨðxÞ will be almost similar to that of UðxÞ. The components of the dual multiscaling functionŨðxÞ are given in Fig. 2.
Thus we have obtained a symmetric and compactly supported dual multiscaling functionŨðxÞ so that the functions UðxÞ andŨðxÞ form a pair of pseudo-biorthogonal multiscaling functions.

1.
We defined a symmetric matrix polynomial analogous to symmetric scalar polynomials (Definition 3.1) 2. If the symbol HðnÞ is a symmetric matrix polynomial of degree l, then the corresponding multiscaling function U will be symmetric about the point l 2 (Lemma 3.1)

The eigenvalues of a symmetric matrix polynomial
LðkÞ occur in reciprocals with same eigenvectors, i.e. If k 0 is an eigenvalue of LðkÞ with eigenvector x 0 , then 1 k 0 is also an eigenvalue of LðkÞ with same eigenvector x 0 (Lemma 3.2) 4. Let (X, T) be a Jordan pair where X 2 n Â nl and T is a diagonal matrix of order nl, n 2 N, n ! 2 and l is even. Assume that T has only nonzero elements neither of which equals 1 and eigenvalues of T occur in reciprocals with same eigenvectors in X. Then a matrix polynomial with Jordan pair (X, T) is symmetric (Theorem 3.1) 5. In the above result, if l is an odd number then also the matrix polynomial is symmetric provided that -1 is an eigenvalue of T (or diagonal entry in T) with algebraic multiplicity n (Theorem 3.2)

Conclusions
We have found the necessary as well as sufficient conditions on a Jordan pair (X, T) such that the corresponding matrix polynomial HðnÞ is symmetric. We selected a Jordan pair satisfying these conditions and constructed a symmetric matrix polynomial HðnÞ. Using cascade algorithm, we found the multiscaling function U for which the matrix polynomial HðnÞ acts as a symbol function. Since HðnÞ is a symmetric matrix polynomial, we saw that U is also symmetric. Finally we constructed the dual multiscaling functionŨðxÞ which is symmetric Fig. 2 The two components/ 0 and/ 1 of the dual multiscaling functionŨ. Here/ 0 and/ 1 are symmetric and compactly supported and compactly supported so that the functions UðxÞ and UðxÞ form a pair of pseudo-biorthogonal multiscaling functions.
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