MRA Frame Wavelets with Certain Regularities Associated with the Refinable Generators of Shift Invariant Spaces

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1 Chapter 6 MRA Frame Wavelets with Certain Regularities Associated with the Refinable Generators of Shift Invariant Spaces Tian-Xiao He Department of Mathematics and Computer Science Illinois Wesleyan University, Bloomington, IL USA [3mm] the@sun.iwu.edu Abstract In this paper, we will start the discussion with the refinable generators of the shift invariant SI) spaces in L R) that possess the largest possible regularities and required vanishing moments. For the pseudo-scaling generators, the corresponding MRA frame wavelets with certain regularities are constructed. In addition, the stability of the refinable SI spaces and the corresponding complementary spaces, biorthogonality of the SI spaces, and the approximation property of the spaces are also discussed. Keywords: Shift invariant spaces, frames, tight frame wavelets, MRA frame wavelets, pseudo-scaling generators functions), stability, biorthogonality, stable basis. 6.1 Introduction [10] pointed out that to achieve higher regularity by increasing vanishing moments of scaling functions and wavelets is not efficient, because 80% of zero moments are wasted. In this paper, we give a method for constructing pseudo-scaling functions and the corresponding MRA frame wavelets with the largest possible regularities and required vanishing moments. We start by setting some notations. Define low-pass filter as m 0 ξ) = 1 n h n e inξ. 1.1) 165

2 166 Tian-Xiao He Here, we assume that only finitely many h n are non-zero. However, some of our results can be extended to infinite sequences that have sufficient decay for n. Next, we define φ by ˆφξ) = Π j=1m 0 j ξ). 1.) This infinite product converge only if m 0 0) = 1 i.e., if n h n =. In this case, the infinite products in 1.) converge uniformly and absolutely on compact sets, so that ˆφ is well-defined C functions. Obviously, ˆφξ) = m 0 ξ/) ˆφξ/), 1.3) or, equivalently, φt) = n h nφt n) at least in the sense of distributions. From Lemma 3.1 in [3], φ has a compact support. A shift invariant SI) space is a closed subspace of L R) that is invariant under the operator S k f) := f k) k Z). For φ L R), we say that V = Sφ) := span{φ k) k Z} is generated by φ. In addition, if φ is refinable, then φ is said to be a refinable generator of Sφ), and Sφ) is called a refinable SI space. Each element φ Φ is a refinable generator of the corresponding SI space Sφ). A refinable generator is said to be a pseudo-scaling refinable) generator see [19]) if it satisfies Equation 1.3) and m 0 ξ) + m 0 ξ + π) = ) We now consider the simplest possible masks m 0 ξ) of refinable generators with the following form. Definition 1. Denote by Φ the set of all functions φt) that have Fourier transform ˆφξ) = m 0 ξ/) ˆφξ/). The filter m 0 ξ) = 1 n h ne inξ is in set M that contains all filters with the form where ) 1 + e m N iξ N 0 ξ) = F ξ), 1.5) F ξ) = e ik ξ k a j e ijξ. 1.6) Here, all coefficients of F ξ) are real, F 0) = 1; N and k are positive integers; and k Z. Hence, the corresponding φ can be written as follows. ) N 1 + e ˆφξ) iξ/ = F ξ/) ˆφξ/). 1.7) Clearly, φ is a B-spline of order N if F ξ) = 1. The vanishing moments of φ ) are completely controlled by the exponents of its spline factor, 1+e iξ N. In

3 MRA Frame Wavelets with Certain Regularities Associated with addition, the regularity of φ is justified by the factors F ξ), and are independent of its vanishing moments. A frame in a Hilbert space H is a family {f n, n I} of elements in H for which there exist two positive constants, 0 < A B <, such that A f n I f, f n B f, 1.8) for all f H. If A = B, {f n } is called a tight frame. For tight frames, the frames constants A = B can be assumed to be 1, simply by dividing each frame generator f n by A. In other words, the tight frames can be defined by f, f n = f. 1.9) n I The index set I for the family {f n } can be quite general. We assume it to be countable and, in particular, we are often dealing with the case where {f n } is the sequence of translates {φ n)} of a function φ H L R), n Z, or the sequence of { j/ φ j k)} with the index set consists of the pairs j, k) Z Z. Definition. [14, 17] A function ψ L R) is a tight frame wavelet TFW) if the system {ψ jk } j,k Z, ψ jk x) = j/ ψ j x k), is a tight frame for L R); that is, for all f L R), f, ψ jk = f. 1.10) j,k z The applications of TFW are based on the following expression that is equivalent to the condition 1.10) see [15], p.334). ft) = f, ψ jk ψ jk t), 1.11) j,k z where the equation holds unconditionally for all f L R). Definition 3. A TFW ψ is an MRA Multiresolution Analysis) TFW if there exists a pseudo-scaling generator defined by Equation 1.3) such that ˆψξ) = e iξ/ m 0 ξ/ + π) ˆφξ/); 1.1) i.e., the symbol of ψ is m 1 ξ) = e iξ/ m 0 ξ/ + π). Since vanishing moment conditions x l ψx)dx = 0, l = 0, 1,, L, are equivalent to dl ˆψ dξ l ξ=0 = 0, l = 0, 1,, L, we immediately know that the maximum order of vanishing moment for ψ is N 1 if ψ is given by Equation 1.1) and φ is defined by Equation 1.7). Therefore, the vanishing moments of φ are completely controlled ) by the exponents of its respective spline factors, 1+e iξ N. In addition, the regularities of ψ is justified by the factors F ξ), and are independent of their vanishing moments.

4 168 Tian-Xiao He From [14, 17], we have the following result regarding the characterization of TFWs. Theorem 4. [14, 17] A function ψ L R) is a TFW if and only if ψ satisfies and ˆψ j ξ ) = 1 a.e. 1.13) j Z ˆψ j ξ ) ˆψ j ξ + qπ)) = 0 a.e., 1.14) j 0 for all q Z + 1; i.e., for all odd integers, q. In [], [3], and [11], the following concepts were introduced that are important in our discussion. Definition 5. Denote T = [0, π). The bracket operator [, ] : L R) L R) L 1 T) is defined by [f, g] = k Z fξ + πk)gξ + πk). 1.15) For f L R), the function [f, f] L 1 T) is called the auto-correlation of f. If f, g are compactly supported, then [ ˆf, ĝ] is a trigonometric polynomial and has the Fourier expansion [ ˆf, ĝ]ξ) = k Z f ), g + k) e ikξ. 1.16) Definition 6. Let Sφ) be a shift invariant space that is generated by φ. {φ k)} k Z is called a stable basis of Sφ) if there exist constants 0 < A B < such that for every c = {c k } k Z l Z) A c l Z) c k φ k) k Z L R) B c l Z). 1.17) Obviously, a stable basis of Sφ) is a basis of Sφ). Theorem 7. [17] Let φ L R) and let 0 < A B <. Inequality 1.17) is equivalent to [ A ˆφ, ˆφ] B, a.e. In Section, we will give the conditions for the coefficients {a j } such that the corresponding refinable pseudo-scaling generator φ is in L R) and the corresponding ψ defined by Equation 1.1) is an MRA TFW. Section 3 discusses the stability of the SI space, Sφ), generated by φ and the stability of the corresponding complementary spaces. We then construct stable scaling function φ Φ such that {φ k)} k Z and { φ k)} k Z are biorthogonal. Thus, {φ k)} k Z and { φ k)} k Z are

5 MRA Frame Wavelets with Certain Regularities Associated with stable bases in the subspace that they generate. In addition, biorthogonal wavelets that possess the largest possible regularities and required vanishing moments can be obtained accordingly. We will also discuss the L approximation from Sφ) in the section. 6. MRA TFWs with certain regularities We construct pseudo-scaling generators and MRA TFWs using the following theorem. Theorem 8. Let φ Φ be defined by Definition 1; i.e., φ = Π j=1 mn 0 j ξ), where ) N F ξ) m N 0 ξ) M is defined by Equations 1.5) and 1.6): mn 0 ξ) = 1+e iξ and F ξ) = e ik ξ k a je ijξ, N, k Z + and k Z, where F 0) = 1. If F also satisfies F π) 1 and its coefficients satisfy and N+k+k j=k k k l=0 l=0 k + 1) N j l k k a j < N 1,.1) ) ) N j + n l k a la l = N 1 δ n0,.) where δ n0 is the Kronecker symbol and n = 0, ±1, ±,, then φ L R) and is a pseudo-scaling generator. In addition, The corresponding ψ defined by 1.1) is an MRA TFW in C α. Here α is more than N 1 k log k + 1). Condition.1) can be replaced by the following weaker condition. a j C{a j }, k) < N 1,.1) where C{a j }, k) equals k k a j if k 1 and equals a 0 if k = 0. Hence, the corresponding regularities of ψ is determined by ψ C α, where α is more than N 1 log C{a j }, k)). We shall break up the proof of Theorem 8 into several lemmas, in which Lemma 9 is given in [13]. Here, we provide a simpler alternative proof. Lemmas 10 and 11 are also shown in [13]. However, we list them as follows for readers convenience and the completeness of the paper. Lemma 9. Assume that φ Φ that is defined by Equation 1.7) in Definition 1. If F π) 1 and C{a j }, k) < N 1,

6 170 Tian-Xiao He then φ = Π j=1 mn 0 j ξ) is in L R). Proof. It is sufficient to prove the boundedness of the following integral = = ξ π C C C l=1 l=1 ˆφξ) dξ = Π j=1 l 1 π ξ l π l 1 π ξ l π l=1 l=1 l 1 π ξ l π 1 ln 4 ln l=1 ) N 1 + e i j ξ F j ξ) 1 e iξ N iξ Π j=1 F j ξ) dξ l 1 π ξ l π l 1 π ξ l π 1 ξ N Π j=1 F j ξ) dξ dξ Π l j=1 F j ξ) Π j=1 F l j ξ) dξ Π l j=1 F j ξ) dξ..3) We now prove the boundedness of the last integral in inequality.3). Denote T fξ) = ξ F ) f ) ξ ) + ξ F + π f Hence, for any π-periodic continuous function f, we have f l ξ)π l j=1 F j ξ) dξ = π π l 1 π ξ l π ) ξ + π..4) T l fξ)dξ π T l f L π f L T l.5) Let ρt ) be the spectral radius of the operator T. Since F 0) = 1 and F π) 1, we have ρt ) > 0 also see [8]). In fact, considering the Fourier expansion where k is a positive integral and F ξ) = b t = k t k l= k b l e ilξ, a k t j a k j

7 MRA Frame Wavelets with Certain Regularities Associated with t = k,, k), we find that the matrix of T restricted to is E k = { k l= k c l e ilξ : c k,..., c k ) C k+1 } b k b k b k 1 b k M T = b i j ) i,j= k,...,k = b k b k+1 b k+ b k..6) 0 0 b k b k b k Noting that F 0) = k l= k b l = 1 and F π) = k l= k 1)l b l = α 1, it follows that b l = b l+1 = α + 1)/, l l and for the vector u = 1,..., 1) C k+1, T u = um = α + 1)u. Thus, T has at least one eigenvalue α For every ɛ > 0, there is an integer lɛ) such that T l ρt ) + ɛ) l, l > lɛ). It follows from.3) that ξ π lɛ) ˆφξ) dξ C 4 Nl T l + C l=1 l=lɛ)+1 4 Nl ρt ) + ɛ) l, so ρt ) must be estimated if we are to choose an ɛ > 0 small enough for the series to converge. Regardless of how small an ɛ > 0 is chosen, the contribution lɛ) C 4 Nl T l C 4 Nl T l l=1 is finite, although possibly very large. To evaluate ρt ), we consider the matrix of T, M T, which was shown in Equation.6). It is clear that ρt ) = ρm T ). We write M T = H, where lɛ) l=1 H = b i j ) i,j= k,,k.

8 17 Tian-Xiao He Obviously, b β can be written as b β = k β a k β j a k j, β = k,, k. Hence, b β = b β, for all β = k,, k. It is also clear that b k is an eigenvalue of H with multiplicity. To estimate bounds of the eigenvalues of H, we establish b β b 0, β = k,, k. In fact, b β k β a k β j a k j k β [ 1 a k β j + 1 a k j ] = = 1 k a k j + 1 k β a k j j=β k a k j = b 0. It is obvious that the spectral radius of H is b 0 if k = 0. For k 1, the characteristic polynomial of H is b k λ)b k λ) multiplied by the characteristic polynomial of the core matrix, H c, which consists of all rows and columns of H except its first and last rows and columns. Hence, the spectral radius of H is = max{b k, ρh) = max{b k, ρh c )} max{b k, H c 1 } k 1 i= k+1 b i j : j = k + 1,, k 1} kb 0. Therefore, ρt ) = ρh) C{a j }, k). Here, C{a j }, k) was defined in Theorem 7. If C{a j }, k) < N 1, then ρt ) < N, so we choose Thus and we obtain the estimation ξ π ɛ = 1 N ρt ) ). ρt ) + ɛ < N, ˆφξ) lɛ) dξ C 4 Nl T l + l=1 l=lɛ)+1 ) ρt ) + ɛ l. The tail of the series is a convergent geometric series, thus completing the proof of φ L R) if condition.1), C{a j }, k) < N 1, holds. 4 N

9 MRA Frame Wavelets with Certain Regularities Associated with Obviously, C{a j }, k) k + 1) k a j. Hence, we have proved the Lemma 9 and the first part of Theorem 8; i.e., φ L R) under condition.1) or condition.1). It is well-known see [1, 4]) that ψ C r and r = N 1 log ρt )), where ρt ) is the spectral radius of the matrix representing operator T, which is defined by equation.4). Hence, we obtain the following result. Lemma 10. ψ C α, where α is more than N 1 log C{a j }, k)). Obviously, if φt), φt n) = 0, then 1 h j h j+n = δ n0,.7) j where n is an integer. From equations.7), noting that h j = k l=0 a l )/ N 1, j = k,, N + k + k, we immediately obtain the following lemma. Lemma 11. If φ defined by 1.7) satisfies φt), φt n) = δ n0, then N+k+k j=k k k l=0 l=0 N j l k N j l k ) ) ) N j + n l k a la l = N 1 δ n0,.8) where δ n0 is the Kronecker symbol and n = 0, ±1, ±,. For sufficient conditions of φt), φt n) = δ n0, we have the following result. Lemma 1. Let φ be the function defined by 1.7). Assume that φ satisfies.8) and either or k i) a j < N 1/, k ii) k + 1) a j < N 1. Then φt), φt n) = δ n0, and ˆφξ) C1 + ξ ) 1/ α, where α = N 1/ log B n, B n Π = max n 1 ξ F j ξ) 1/n. Proof. If B n = max ξ F j ξ) 1/n < N 1/.9) Π n 1 for some integer n > 0, then by using Propositions 4.8 and 4.9 in [3] we have φt), φt n) = δ n0 and ˆφξ) C1 + ξ ) 1/ α.

10 174 Tian-Xiao He On the other hand, B n max ξ F ξ) = k a j. Hence, by using the Cauchy- Schwartz inequality, we obtain B n k + 1) where the equal sign does not hold if {a j } satisfies condition.8), because all {a j } can not be the same. Hence, if k + 1) k a j < Ñ 1, then B n < Ñ 1/. This completes the proof of Lemma 1. Proof of Theorem 8. Let φ Φ be the function defined by 1.7) in Definition 1) that satisfies F π) 1 and the Condition.1). Then, from Lemma 9, φ is in L R). Noting Lemmas 11 and 1, we have that Conditions.1) and.) imply φt), φt n) = δ n0, which is equivalent to k a j 1/ ˆφξ + kπ) = 1 for a.e. ξ R. k Z By using 1.3), the above equation can be written as an equivalent form 1.4). Hence, φ is a pseudo-scaling generator. To prove that ψ defined in 1.1) is an MRA TFW, it is sufficient to prove that ψ is a TFW; i.e., it satisfies Equations 1.13) and 1.14). From 1.1) and 1.4), = j Z ˆψ j ξ ) = j Z m0 j 1 ξ + π ) ˆφ j 1 ξ ), = lim n n j= n = lim n n j= n = lim n m0 j 1 ξ + π ) ˆφ j 1 ξ ) = [ 1 m0 j 1 ξ ) ] ˆφ j 1 ξ ) = [ ˆφ n 1 ξ ) ˆφ n ξ) ]. Since φ L R), lim ˆφ n ξ) = 0 for a.e. ξ. From condition F 0) = 1, we have n lim m0 n ξ ) = 1. Hence, taking limit n on the both sides of Equation n 1.) and noting that lim ˆφξ) = lim n n Π j=1 m 0 j ξ) 0,

11 MRA Frame Wavelets with Certain Regularities Associated with ˆψ j ξ ) = 1. There- we obtain see [19]) lim ˆφ n ξ ) = 1, which derives n j Z fore, Equation 1.13) holds for our ψ. For any odd integer q, j 0 ˆψ j ξ ) ˆψ j ξ + qπ)) = = j>0 e ij 1ξ m 0 j 1 ξ + π) ˆφ j 1 ξ ) e ij 1ξ m 0 j 1 ξ + π) ˆφ j 1 ξ + j qπ)+ +e i 1ξ m 0 1 ξ + π) ˆφ 1 ξ ) e i 1 ξ+π m 0 1 ξ) ˆφ 1 ξ + qπ) = = j>0 m0 j 1 ξ + π ) ˆφ j 1 ξ ) ˆφ j 1 ξ + j π) m 0 1 ξ + π) ˆφ 1 ξ ) m 0 1 ξ ) ˆφ 1 ξ + π) = = j>0 [ 1 m0 j 1 ξ ) ] ˆφ j 1 ξ ) ˆφ j 1 ξ + j π) ˆφξ) ˆφξ + π) = = [ ˆφ j 1 ξ ) ˆφ j 1 ξ + j π) ˆφ j ξ ) ] ˆφ j ξ + j+1 π) ˆφξ) ˆφξ + π) = j>0 = lim ˆφ n ξ) ˆφ n ξ + n+1 π) = 0, n where ˆφ j 1 ξ + j qπ ) = ˆφ j 1 ξ + j π ), q Z + 1, because m 0 ξ) is π periodic. Therefore, 1.14) also holds, and the proof of Theorem 8 is complete. We now give a general algorithm to construct the pseudo-scaling generator φ such that the corresponding MRA TFW ψ is of the largest possible regularity and the required vanishing moments. In fact, this method can be described as an optimization problem of finding suitable F ξ), or, equivalently, suitable coefficient set, a = {a 0,, a k }, of F ξ), such that k a j is the minimum under conditions.1) and.) of Theorem 8. Thus, the above optimization problem can be written as

12 176 Tian-Xiao He follows. min a subject to k a j,.10) k 1) j a j + 1 > 0,.11) k a j < N 1 /k + 1),.1) N+k+k j=k k l=0 N j + n l k ) ) k a l l=0 N j l k ) a l = N 1 δ n0, n = 0, ±1, ±,,.13) where object.10) will give the largest possible regularity, condition.11) is from the definition of F π) 1, and conditions.1) and.13) come from conditions.1) and.) of Theorem 8. Problem.10)-.13) can be written in a form without the inequality conditions by defining parameters s, t 0 as s = N 1 /k + 1) k a j and t = k 1)j a j + 1), respectively. Hence, the optimization problem becomes min a k a j + 1 s + 1 t, k subject to t = 1) j a j + 1 s + N+k+k j=k k a j = N 1 /k + 1) k l=0 N j + n l k = N 1 δ n0, n = 0, ±1, ±,, ) ) k a l l=0 N j l k ) a l = As examples, we choose N = 1 and k = k = 0, then the solution of problem.10)-.13) is a 0 = 1 and we obtain the Haar function. If we choose N =, k = 1, and k = 0, then the solutions of the problem are a 0 = 1± 3 and a 1 = 1 3. Hence, the corresponding φ is defined by ˆφξ) = Π j=1 m 0 j ξ), where 1 e iξ ) 1+e iξ m 0 ξ) = 1± 3 log 4)/ = 1; i.e., ψ C 1. ). In addition, the regularities of ψ is more than

13 MRA Frame Wavelets with Certain Regularities Associated with Stable generators of an SI space and its complementary space We now discuss the stability of φ. From [16], we obtain a necessary and sufficient condition for a refinable function defined by Definition 1 to be stable. Here, a function φ is stable if it is a stable generator of space Sφ), or, equivalently, the integer translates of φ form a stable basis of the space. Lemma 13. The function φ defined by Definition 1 is stable if and only if F ξ) satisfies the following two conditions. i) F ξ) does not have any symmetric zeroes on T = [0, π); ii) For any odd integer m > 1 and a primitive mth root ω = e inπ/m of unity i.e., n is an integer relatively prime to m), there exists an integer d, 0 d < ord m, such that F d+1 nπ/m ) 0, where p = ord m is the smallest positive integer with p 1mod m). In addition, if φ is stable, then ˆφξ) and F ξ) = Π j=1 F j ξ + πu) ) /Π j=1 Π j=1 k l=0 a le il j ξ+πu) = k l=0 a le il j ξ Π j=1 F j ξ ) 3.1) have no roots in T = [0, π). Proof. By using Lemma 6.6 of [1] and Theorem 1 of [16], we obtain that φ is stable. In addition, since we have ˆφξ) = 1 e iξ iξ N ˆφξ + πu) = F ξ) ˆφξ) where F ξ) is defined as 3.1). Therefore, Π j=1 F j ξ ), 3.) [ ˆφ, ˆφ] = u Z ˆφξ + πu) = = ˆφξ) F ξ)ξ) N u Z ξ N ξ + πu) N, 1 ξ+πu) N. By using formula 4..7) in [6], we can rewrite the last expression as [ ˆφ, ˆφ] = = ˆφξ) F ξ)ξ/) N sin N ξ/) u= ˆB N ξ + πu) l 3.3)

14 178 Tian-Xiao He where ˆB N ξ) is the Fourier transform of the B-spline of order N. In addition, the sum on the right-hand side of Equation 3.3) can be evaluated by using formula 4..10) in [6]: u= ˆB N ξ + πu) = sinn ξ N 1)! d N 1 cot ξ. dξn 1 Thus, noting that φ is stable and applying Theorem 7 to the above [ ˆφ, ˆφ], we immediately know that ˆφξ), F ξ) 0 for all ξ T. This completes the proof of Lemma 13. From Theorem 8 and Lemma 13, we have the following result. Theorem 14 Let φ Φ be defined as in Definition 1. If F ξ) satisfies F π) 1, conditions i) and ii) in Lemma 13, and.1) or.1), then the corresponding φ is in L R) and is stable. Let V j := span{φ j t k) : k Z}. Following [11], for any φ L R), we define the natural) dual φ by its Fourier transform, φ := ˆφ [ ˆφ, ˆφ], where we interpret 0/0 = 0. Thus, from Equation 3.3), the dual function s Fourier transform of φ is φ = [ ˆφ ˆφ, ˆφ] = sin N ξ/) ξ/) N ˆφ F ξ) ˆB N ξ + πu). u= The properties of the dual function φ can be found in [6, 11]. It is clear that Sφ) V 1 = span{φ k) : k Z}. We now consider the complementary space of Sφ) in V 1, which is generated by a function ψ V 1. We say a function f L T) is in W, the Wiener Algebra, if its Fourier series k Z f ke ikξ satisfies {f k } l 1 Z). From [11], we can establish the following lemma. Theorem 15. Let φ Φ satisfy all conditions of Theorem 14, where Φ is defined in Definition 1, and let ψ V 1 := span{φ k) : k Z} have the symbol m 1 ξ) W, the Wiener Algebra, such that m 1 ξ) + m 1 ξ + π) > 0, ξ T. 3.4) Then ψ is stable i.e., a stable generator for Sψ)). Proof. Since ψ V 1, using the two-scale relation of ψ yields

15 MRA Frame Wavelets with Certain Regularities Associated with [ ˆψ, ˆψ] ξ) = = ) ξ ) m 1 + πi ξ ˆφ + πi = i Z = ) ξ ) m 1 + πi ξ ˆφ + πi + i Z + ) ξ ) m 1 + π + πi ξ ˆφ + π + πi = i Z ) = ξ [ ) )] ξ m 1 ˆφ, ˆφ ξ + ) + ξ [ ) )] ξ m 1 + π ˆφ + π, ˆφ ξ + π. From Theorem 8 and Lemma 13, φ is in L R) and is stable. Thus, Theorem 7 shows there exist 0 < A B < such that [ A ˆφ, ˆφ] B a.e. Thus we can bound the auto-correlation of ψ by [ AM 1 ξ) ˆψ, ˆψ] ξ) BM 1 ξ), where M 1 ξ) := m 1 ) ξ + m 1 Since m 1 ξ) C T), from the condition 3.4) we have Ā := min ξ T M 1ξ) > 0. ) ξ + π. On the other hand, B := m1 ξ) CT) <. Therefore, [ ] 0 < ĀA ˆψ, ˆψ ξ) BB <, By using Theorem 7, we have proved that ψ is a stable generator for Sψ). We can also construct a biorthogonal system from φ Φ as follows. Here, the set Φ was defined in Definition 1. Assume that φ Φ; i.e., φt) = h n n φt n) or equivalently, ˆ φξ) = m 0 ξ ) ˆ φ ξ ) with m 0ξ) = 1 h n n e inξ M, which is defined in Definition 1. Therefore, we can write 1 + e iξ m 0 ξ) = mñ0 ξ) = ) Ñ a.e. F ξ). 3.5)

16 180 Tian-Xiao He Here F ξ) is defined by F ξ) = e i k ξ k ã j e ijξ, where, F 0) = 1; all coefficients of F ξ) are real; Ñ and k are positive integers; and k Z. Hence, the corresponding φ and φ can be written as follows. ) N 1 + e ˆφξ) iξ/ = F ξ/) ˆφξ/), 1 + e ˆ φξ) iξ/ = ) Ñ We also define the corresponding ψ and ψ by ˆψξ) = e iξ/ m 0 ξ/ + π) ˆφξ/) F ξ/) ˆ φξ/). 3.6) ˆ ψξ) = e iξ/ m 0 ξ/ + π) φξ/), 3.7) or, equivalently, ψx) = n 1) n 1 h n 1 φx n) ψx) = n 1) n 1 h n 1 φx n). 3.8) Similar to φ, φ is also a B-spline of order Ñ if F ξ) = 1. Since vanishing moment conditions x l ψx)dx = 0, l = 0, 1,, L, are equivalent to dl ˆψ dξ l ξ=0 = 0, l = 0, 1,, L, we immediately know that the maximum orders of vanishing moment for ψ and ψ are N 1 and Ñ 1, respectively. Therefore, following the similar argument in Section, the vanishing moments of φ and φ are completely controlled ) by the exponents of their respective spline factors, 1+e iξ N ) and 1+e qiξ Ñ. In addition, the regularities of φ and φ are justified by the factors F ξ) and F ξ), respectively, and are independent of their vanishing moments. Similar to φ, if φ defined in 3.6) satisfies F π) 1 and k + 1) k ã j < Ñ 1, 3.9) then φ L R). From [8], the stability of φ, φ, ψ, and ψ are implied by their biorthogonality. In fact, [8] gave the following results.

17 MRA Frame Wavelets with Certain Regularities Associated with Lemma 16. If φ, φ L R) satisfy φt), φt n) = δ n0, then {φt k)} k Z and { φt k)} k Z are stable; i.e., they are stable bases Riesz bases) in the subspace that they generate. In addition, the corresponding biorthogonal wavelet functions ψ and ψ are also stable; i.e., they are stable bases Riesz bases) in the subspace that they generate. Hence, we have the following results. Theorem 17. Let φ, φ Φ be defined in Definition 1; that is, φ = Π j=1 mn 0 j ξ) and φ = Π j=1 mñ0 j ξ), where m N 0 ξ) and mñ0 ξ) are defined by Equations 1.5) and 3.5). Assume F π), F π) 1, and conditions.1) and 3.9) are satisfied by m N 0 ξ) and mñ0 ξ) and ν k k j=µ l=0 l=0 ) ) Ñ N j l k j + n l k ã la l = N+Ñ 1 δ n0 3.10) holds for all n Z, where µ = min{k, k }; ν = max{n + k + k, Ñ + k + k } and δ n0 is the Kronecker symbol, then φ, φ L R) satisfy φt), φt i) = δ i,0 for all i Z, and thus they are stable. In addition, The corresponding biorthogonal wavelets ψ and ψ are in C α and C α, respectively. Here α and α are more than N 1 k log k + 1) and Ñ 1 log k + 1) a j k ã j, 3.11) respectively. Proof. By using Lemma 9, we have φ, φ L R). Similar to the proofs of Lemmas 11 and 1, from Propositions 4.8 and 4.9 in [9], φ and φ are biorthogonal if.1), 3.9), and the following conditions hold. 1 h j h j+n = δ n0, n Z, 3.1) j where h j and h j are respectively the coefficients of two-scale relations of φ and φ. From equations 3.1), noting that k ) N h j = )a j l k l / N 1 j = k,, N + k + k ), and h j = l=0 k l=0 Ñ ) j l k ã l /Ñ 1

18 18 Tian-Xiao He j = k, ), Ñ + k + k, we immediately obtain 3.10). It is well-known see [7, 9]) that ψ C r and r = N 1 log ρt )), where ρt ) is the spectral radius of the matrix representing operator T, and T is defined by equation.4). Hence, we obtain that the corresponding biorthogonal wavelets ψ and ψ are in C α and C α, respectively, and α and α are more than the quantities shown in 3.11). Remark 1. Conditions.1) and 3.9) can be replaced by the following weaker conditions. C{a j }, k) < N 1, C{ã j }, k) < Ñ 1, 3.13) where C{a j }, k) equals k k a j if k 1 and equals a 0 if k = 0 while C{ã j }, k) equals k k ã j if k 1 and equals ã 0 if k = 0. Hence, the corresponding regularities of φ and φ are determined by φ C α and φ C α, where α and α are more than and N 1 log C{a j }, k)) Ñ 1 log C{ã j }, k ), respectively. Remark. If F ξ) = 1 i.e., k, k = 0), then condition 3.10) can be written as k l=0 max{n,ñ+ k+ k } j= k 3.14) ) ) Ñ N j l k ã l = N+Ñ 1 δ n0, 3.15) j + n where n = 0, ±1,. Condition 3.15) can be reduced again as follows for Ñ = N k and k = 0: k N ) ) N k N ã l = N k 1 δ n0, 3.16) j l j + n l=0 where n = 0, ±1, ±,. We now give a general algorithm to construct biorthogonal scaling functions φ and φ with the largest possible regularity and the required vanishing moments. Similar the algorithm shown in Section, this method can be described as an optimization problem of finding suitable F ξ) and F ξ), or, equivalently, suitable coefficient sets, a = {a 0,, a k } and ã = {ã 0,, ã k }, of F ξ) and F ξ), respectively, such that k k a j and ã j are the minimum under conditions.1) and 3.9). From the wavelet analysis of spline approximation, we assume F ξ) = 1; i.e., the corresponding φ is the B-spline of order N. Then, the above optimization problem

19 MRA Frame Wavelets with Certain Regularities Associated with can be written as follows. min a k ã j, 3.17) subject to k k l=0 1) j ã j + 1 > 0, 3.18) ã j < Ñ 1 / k + 1), 3.19) max{n,ñ+ k+ k k } j= k ) ) Ñ N j l k j + n = N+Ñ 1 δ n0, 3.0) where n = 0, ±1,, object 3.17) will give the largest possible regularity, condition 3.18) is from the definition of F, and conditions 3.19) and 3.0) come from conditions 3.9) and 3.10). Problem 3.17)-3.0) can be written in a form without the inequality conditions by defining parameters s, t 0 by s = Ñ 1 / k + 1) k ã j and t = k, 1)j ã j + 1) respectively. Hence, the optimization problem becomes ã l min a k ã j + 1 s + 1 t, subject to t = k l=0 k 1) j ã j + 1 ã j + s = Ñ 1 / k + 1), max{n,ñ+ k+ k k } j= k = N+Ñ 1 δ n0, where n = 0, ±1,. As examples, we consider m 0 ξ) = 1+e iξ ) ) Ñ N j l k j + n ã l ) ; i.e., φ is B-spline of order 1. If we choose Ñ = 1 and k = k = 0, then the solution of problem 3.17)-3.19) is ã 0 = 1 and we obtain the Haar function. If we choose Ñ =, k = 1, and k = 0, then the solutions of the problem are ã 0 = 3/ and ã 1 = 1/. Hence, the corresponding φ is defined by ˆ φξ) ) = Π j=1 m 0 j ξ), where m 0 ξ) = 1+e iξ ) 3 e iξ. Clearly,

20 184 Tian-Xiao He Both φ and φ satisfy all the conditions of Theorem 17. Hence, they are in L R) and are stable. In addition, the regularities of φ and φ are respectively 1 and log 5)/ = We now discuss the error of L approximation from Sφ). Denote Ef, Sφ)) L := inf f g L g Sφ). We have the following result. Theorem 18. Let φ Φ defined by Definition 1; i.e., φ = Π j=1 mn 0 j ξ), where ) m N 0 ξ) M is defined by Equations 1.5) and 1.6): mn 0 ξ) = 1+e iξ N F ξ) and F ξ) = e ik ξ k a je ijξ, N, k Z + and k Z, where F ξ) satisfies Conditions i) and ii) of Lemma 13, F 0) = 1, and F π) 1. If inequality.1) or.1) holds, then for any function f W N+1 R), the Sobolev space, E f, Sφ) h) = CN φ f W N R) + O h N+1), where and Sφ) h := {f /h) f Sφ)} Cφ N = 1 N! ˆφ N) πu). u 0 Proof. Following [4], we define the error kernel Γ φ := 1 [ ˆφ ˆφ, ˆφ] 1, where 0/0 is interpreted as 0. From Equation 3.3), we have ˆφ [ ˆφ, ˆφ] = sin N ξ/) ξ/) N F ξ) ˆB N ξ + πu). u= Noting Equation 3.1) and the inequality see [6]) u= ˆB N ξ + πu) 1, we obtain ξ N Γ φ ξ) L T). Thus, from Theorems 1.6 and.0 of [4], Sφ) provides approximation order N; i.e., E f, Sφ) h) CN φ hn f W N R),

21 MRA Frame Wavelets with Certain Regularities Associated with where C N φ can be found as follows. By observing that φ E N := {f fx) C1 + x ) N+1+ɛ), ɛ > 0} is stable and using the following lemma shown in [5], we immediately obtain the Theorem 18. Lemma 19. [5] Assume φ E m is stable while ˆφ0) = 1 and provides L approximation order m. Then for any function f W m+1 R), E f, Sφ) h) = Cm φ hm f W m R) + O h m+1), where Cφ m! m = 1 u 0 ˆφ m) πu).

22 186 Tian-Xiao He

23 Bibliography [1] A.S. Cavaretta, W. Dahmen, and C.A. Micchelli Stationary Subdivision, Mem. Amer. Math. Soc., 453, Amer. Math. Soc., Providence, RI, [] C. de Boor, R. DeVore, and A. Ron Approximation from shift invariant subspaces of L R d ), Trans. Amer. Math. Soc., ). [3] C. de Boor, R. DeVore, and A. Ron The structure of finitely generated shift-invariant spaces in L R d ), J. of Functional Analysis, ). [4] C. de Boor, R. DeVore, and A. Ron Approximation orders of FSI spaces in L R d ), Const. Approx ). [5] T. Blu and M. Unser Approximation error for quasi-interpolators and multi-) wavelet expansions, App. Comp. Harmonic Anal., 6, ). [6] C.K. Chui An Introduction to Wavelets, Academic Press, Boston, 199. [7] C.K. Chui and X. Shi Orthonormal wavelets and tight frames with arbitrary real dilations, Appl. Comp. Harmonic Anal ). [8] A. Cohen, and I. Daubechies A stability criterion for biorthogonal wavelet bases and their related subband coding scheme, Duke Math J., ). [9] A. Cohen, I. Daubechies, and J. C. Feauveau Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math., ). [10] I. Daubechies Ten Lectures on Wavelets, CBMS-NSF Series in Appl. Math., SIAM Publ., Philadelphia, 199. [11] S. Dekel, D. Leviatan Wavelet decompositions of non-refinable shift invariant spaces, Appl. Comp. Harmonic Anal., to appear. [1] T.X. He Short time Fourier transform, integral wavelet transform, and wavelet functions associated with splines, J. Math. Anal. & Appl., ). 187

24 188 Tian-Xiao He [13] T.X. He Biorthogonal wavelets with certain regularities, Appl. Comp. Harmonic Anal ). [14] T.X. He Stable refinable generators of shift invariant spaces with certain regularities and vanishing moments, Approximation Theory X: Wavelets, Splines, and Applications, C. K. Chui, L. L. Schumaker, and J. Stöckler eds.), Vanderbilt University Press, Nashville, TN, 00, [15] E. Hernández and G. Weiss A First Course on Wavelets, CRC Press, New York, [16] R.Q. Jia and J. Wang Stability and linear independence associated with wavelet decompositions, Proc. Amer. Math. Soc., ). [17] Y. Meyer Wavelets and Operators, Cambridge, 199. [18] M. Papadakis, H. Sikić, and G. Weiss The characterization of low pass filters and some basic properties of wavelets, scaling functions and related concepts, [19] M. Papadakis, H. Sikić, G. Weiss, and S. Xiao Generalized low pass filters and MRA frame wavelets, 000. [0] G. Strang and G. Fix A Fourier analysis of the finite element variational method in Constructive Aspects of Functional Analysis, G. Geymonat ed.) C.I.M.E., II Ciclo 1971,