A CLASS OF M-DILATION SCALING FUNCTIONS WITH REGULARITY GROWING PROPORTIONALLY TO FILTER SUPPORT WIDTH

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 26, Number 2, December 998, Pages S (98) A CLASS OF M-DILATION SCALING FUNCTIONS WITH REGULARITY GROWING PROPORTIONALLY TO FILTER SUPPORT WIDTH XIANLIANG SHI AND QIYU SUN (Communicated by J. Marshall Ash) Abstract. In this paper, a class of M-dilation scaling functions with regularity growing proportionally to filter support width is constructed. This answers a question proposed by Daubechies on p.338 of her book Ten Lectures on Wavelets (992).. Introduction Let M 2 be a fixed integer. A multiresolution analysis for dilation M consists of a sequence of closed subspaces V j of L 2 (R) that satisfy the following conditions (see [C], [D], [M]): i) V j V j+, j Z; ii) j Z V j = L 2 (R); iii) j Z V j = {0}; iv) f V j f (2 j ) V 0 ; v) there exists a function φ in V 0 such that {φ( n); n Z} is an orthonormal basis of V 0. The function φ is called an M-dilation scaling function. It is easy to see that φ satisfies the refinement equation () φ(x) = n Zc n φ(mx n), where the sequence {c n } satisfies c n = M. n Z Received by the editors November 20, Mathematics Subject Classification. Primary 42C5. The first author is supported by the Texas Higher Education Coordinating Board under Grant Number The second author is partially supported by the National Natural Sciences Foundation of China # , the Tian Yuan Foundation, the Doctoral Bases Promotion Foundation of National Educational Commission of China # the Zhejiang Provincial Sciences Foundation of China # 96083, by the Wavelets Strategic Research Program, National University of Singapore, under a grant from the National Science Technology Board the Ministry of Education, Singapore. 350 c 998 American Mathematical Society

2 3502 XIANLIANG SHI AND QIYU SUN In this paper we shall only deal with compactly supported M-dilation scaling functions. In this case the sequence {c n } must have finite length. The function H(ξ) = (2) c n e inξ M is called a symbol corresponding to the refinement equation (). The filter support width W (φ) ofanm-dilation scaling function φ is defined as the difference of the largest the smallest indices of the nonzero c n. The regularity R(φ) ofφis defined as the supremum of α such that φ C α,wherec α denotes the Hölder class of index α. In her book [D, p.338], Daubechies remarks that: At present, I know of no explicit scheme that provides an infinite family of m 0 (i.e., symbol H), for dilation 3 (i.e., M = 3), with regularity growing proportionally to the filter support width. To our knowledge, this question is still open. The purpose of this paper is to construct a class of M-dilation scaling functions φ N for which there exists a constant λ M independent of N such that (3) n Z R(φ N ) λ M W (φ N ), where M 3. On the other h it is already known that (see [DL]) W (φ N ) R(φ N ). These facts give an affirmative answer to Daubechies question. The regularity of φ has been studied widely; see for instance [CL], [BDS], [D], [HW], [HW2], [So], [S] [WL]. In general, to study the regularity of φ we need to consider the symbol (2) first. By the Fourier transform, we see that all the symbols H satisfy (4) H(ξ + 2lπ M ) 2 =. The solutions H of the equation (4) are determined by (see [BDS], [H], [HSZ]) H(ξ) 2 = ( sin 2 N (Mξ/2)) N M (5) M 2 sin 2 Na(s)sin 2s ξ ξ/2 2 +(sinmξ 2 )2N R(ξ), s=0 where M N a(s) = s + +s =s j= ( N +sj ) s j sin 2sj jπ/m R is a real-valued trigonometric polynomial such that R(ξ+2lπ/M) =0 the right h side of (5) is nonnegative. By the Riesz Lemma (see [D, p.72]), such symbol H exists. Let N H be a solution of (5) with R = 0, let N φ be the solution of () corresponding to the symbol NH. In [BDS], Bi, Dai Sun prove the following estimates on the regularity of Nφ: R( N φ) ln N 4lnM C

3 ACLASSOFM-DILATION SCALING FUNCTIONS 3503 when M is odd, R( N φ) 4N ln ( ) sin Mπ 2M+2 +lnn C 4lnM when M is even. A more precise estimate of R( N φ) can be found in [S]. For the special cases M =3,4,5,similar results are obtained by Soardi ([So]) Heller Wells ([HW2]) independently. This result shows that for these special N φ the regularity does not grow proportionally to the filter support width when M is odd. To construct M-dilation scaling functions with regularity growing proportionally to the filter support width we use the symbol H N determined by H N (ξ) 2 (MN M +)! = α N (k 0,,k ) k 0! k! k 0+ +k =MN M+ ( sin Mξ/2 ) 2kl, M sin(ξ/2+lπ/m) where N, α N (k 0,,k ) is defined by { 0, if k0 N, α N (k 0,,k )= #(E), if k 0 N, where E = {j : k j N} #(E) is the cardinality of E. Letφ N be the solution of () corresponding to a symbol H N. Then we have the following Theorem. Let M 3 N 2 be any natural numbers. Then φ N is an M- dilation scaling function there exists a constant C independent of N such that ( (M ) ln( + ) )N ln N 2 2lnM 4lnM C R(φ N ) ( 2 (M ) ln( + M ) )N ln N 2lnM 4lnM +C. Remark. Observe that W (φ N ) 2(M )MN. Therefore the regularity R(φ N ) of φ N grows proportionally to the filter support width W (φ N ), i.e., (3) holds. Remark 2. Let D(φ) =R(φ)/W (φ) be the rate of regularity filter support width of a scaling function φ. Then D(φ N ) when M is odd, 4M() ( (M ) ln( + ln M D( N φ) ln N 4NM ln M + C N D( N φ) ln ( sin Mπ/(2M +2)) Mln M when M is even. Therefore we get when M is odd, D(φ N )/D( N φ) D(φ N )/D( N φ) ) + C ln N N ) C ln N N, N ln N ( ln M M ln( + M )) C ln M (M ) ln( + ) N Cln 4(M ) ln(sin Mπ/(2M +2)) N

4 3504 XIANLIANG SHI AND QIYU SUN when M is even. This shows that D(φ N )ofthem-dilation scaling function φ N is larger than the one of N φ even when M is an even integer larger than Proof of the Theorem To prove the Theorem, we estimate H N (ξ) first.let h(ξ)= sin2 Mξ/2 M 2 sin 2 ξ/2 B N (ξ) = ( h(ξ) ) N HN (ξ) 2. Then for all real valued ξ we have (MN M +)! B N ( ξ) = α N (k 0,k,,k ) k 0!k! k! k 0+ +k =MN M+ (h(ξ)) k0 N (h(ξ + 2lπ M ))k l l= = B N (ξ) B N (ξ) 0. Therefore by the Riesz Lemma ([D, p.72]) we obtain the existence of H N (ξ) with H N (ξ) = ( e imξ )N M( e iξ HN (ξ) ) H N (ξ) 2 = B N (ξ). From the definition of α N (k 0,,k ) from h(ξ + 2lπ M )=, we get α N (k 0,k,,k )+α N (k,k 2,,k,k 0 ) + +α N (k,k 0,,k M 2 )= = H N (ξ + 2lπ M ) 2 k 0+ +k =MN M+ = ( =. + +α N (k,k 0,,k M 2 ) ) h(ξ + 2lπ M )) MN M+ ( αn (k 0,k,,k )+α N (k,k 2,,k,k 0 ) (MN M +)! k 0!k! k! ( h(ξ + 2lπ M )) k l

5 ACLASSOFM-DILATION SCALING FUNCTIONS 3505 Therefore (4) holds for H N (ξ). Recall that H N (ξ) 0when ξ π/m. Hence the solution φ N of () corresponding to the symbol H N (ξ) isanm-dilation scaling function by an elementary argument ([D, p.82, Theorem 6.3.] with K =[ π, π]). To estimate the regularity of φ N, we need some estimates on B N (ξ). From the Stirling formula, which says that n! is equivalent to n n e n n, from /(M ) α N (k 0,,k ) whenk 0 N from h(2π/(m )) = /M 2,weget B N (ξ) = B N (2π/(M )) Therefore we get k 0+ +k =MN M+,k 0 N 0 k 0 (N )() (h(ξ)) k0 N (MN M +)! k 0! k! (h(ξ + 2lπ M ))k l l= (MN M +)! (k 0 +N)!((N )(M ) k 0 )! (h(ξ)) k0 ( h(ξ)) (N )() k0 (MN M +)! N!((N )(M ))! CMN ( + M )()N N /2 M k + +k =()(N ) l= (h(2π/(m ) + 2lπ M ))k l (MN M +)! N!k! k! (MN M +)! ( ) (N )() M N!((N )(M ))! M 2 CM N ( + M )()N N /2. ) R(φ N ) ( (M ) ln( + 2 2lnM )N ln N 4lnM C by an argument as in [D, p.27]. Observe that 2Mπ/(M ) = 2π/(M ) + 2π. By an argument similar to that on p.220 of [D] we obtain R(φ N ) ( (M ) ln( + /M ) )N ln N 2 2lnM 4lnM C. This completes the proof of the Theorem. References [C] C. K. Chui, An Introduction to Wavelets, Academic Press, 992. MR 93f:42055 [CL] C. K. Chui J. Lian, Construction of compactly supported symmetric antisymmetric orthonormal wavelets with scale = 3, Appl. Comput. Harmonic Anal., 2(995), 2-5. MR 95m:42042 [BDS] N. Bi, X. Dai Q. Sun, Construction of compactly supported M-b wavelets, Appl. Comput. Harmonic Anal., Toappear. [D] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics 6, Philadephia, 992. MR 93e:42045

6 3506 XIANLIANG SHI AND QIYU SUN [DL] I. Daubechies J. Lagarias, Two-scale difference equation I: existence global regularity of solution, SIAM J. Math. Anal., 22(99), MR 92d:3900 [H] P. N. Heller, Rank M wavelets with N vanishing moments, SIAM J. Matrix Anal. Appl., 6(995), MR 95k:42058 [HW] P. N. Heller R. O. Wells, Jr., The spectral theory of multiresolution operators applications, In Wavelets: Theory, Algorithms, Applications, editedbyc.k.chui, L. Montefusco, L. Puccio, Academic Press, 994, pp MR 96a:42046 [HW2] P. N. Heller ad R. O. Wells Jr., Sobolev regularity for rank M wavelets, CML Technical Report TR 96-08, Computational Mathematics Laboratory, Rice University, 996. [HSZ] D. Huang, Q. Sun Z. Zhang, Integral representation of M-b filter of Daubechies type, Chinese Sciences Bulletin, 42(997), CMP 97:7 [M] Y. Meyer, Ondelettes et Opérateurs,I: Ondelettes, Hermann, Paris, 990. MR 93i:42002 [So] P. M. Soardi, Hölder regularity of compactly supported p-wavelets, Constr. Approx., To appear. [S] Q. Sun, Sobolev exponent estimates asymptotic regularity of M b Daubechies scaling functions, Preprint, 997. [WL] G. V. Well M. Lundberg, Construction of compact p-wavelets, Constr. Approx., 9(993), MR 94i:42049 Department of Mathematics, Texas A&M University, College Station, Texas address: xshi@math.tamu.edu Center for Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang 30027, People s Republic of China Current address: Department of Mathematics, National University of Singapore, 0 Kent Ridge Crescent, Singapore 9260, Singapore address: matsunqy@leonis.nus.edu.sg