Approximation Properties of Multi{Scaling Functions: A Fourier Approach Gerlind Plona Fachbereich Mathemati Universitat Rostoc 1851 Rostoc Germany Dedicated to Prof. L. Berg on the occasion of his 65th birthday Abstract In this paper, we consider approximation properties of a nite set of functions ( ::: r; 1) which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vector =( ) r;1 is renable, we setch a new way, how to derive necessary and sucient conditions for the renement mas in Fourier domain. 1. Introduction For applications of multi{wavelets in nite element methods, the problem occurs, how to construct renable vectors := ( ) r;1 (r 2 N) of functions with short support, such that algebraic polynomials of degree < m(m 2 N) can be exactly reproduced by a linear combination of integer translates of ( ::: r; 1). In Heil, Strang and Strela [9] and in Plona [13], the approximation properties of renable function vectors := ( ) r;1 were studied in some detail. In particular, new necessary and sucient conditions for the renement mas of could be derived. In [13], it could even be shown that the function vector can only provide approximation order m if its renement mas factorizes in a certain manner. For nding these results, [9] as well as [13] strongly used properties of doubly innite matrices determined by the matrix coecients occuring in the renement equation (in time domain). Now wewant tosetch away, how the necessary and sucient conditions for the renement mas of can completely be derived in the Fourier domain. 1
As in [13], the functions are allowed to have a noncompact support if they have a suitable decay rate. The main tool of our new approach is the so called superfunction, which iscontained in the span of the integer translates of ( ::: r; 1) and already provides the same approximation order as. The results are applied to some multi{scaling functions 1 rst considered by Donovan, Geronimo, Hardin and Massopust [6, 7]. 2. Notations Let us introduce some notations. Consider the Hilbert space L 2 = L 2 (R) ofall square integrable functions on R. The Fourier transform of f 2 L 2 (R) is dened by ^f := R 1 ;1 f(x)e;ix dx: The function vector with elements in L 2 (R) isrenable, if satises a renement equation of the form = l2z P l (2 ;l) (P l 2 R rr ) or equivalently, if satises the Fourier transformed renement equation with ^ := (^ ) r;1 ^ = P (=2) ^(=2) (1) and with the renement mas (two{scale symbol) P = P := 1 2 l2z P l e ;il : (2) Note that P is an (r r){matrix of 2{periodic functions. The components of a renable function vector are called multi{scaling functions. Let BV (R) be the set of all functions which are of bounded variation over R and normalized by lim f(x) f(x) jxj1 =1 2 lim (f(x + h)+f(x ; h)) h (;1 <x<1): If f 2 L 1 (R) \ BV (R), then the Poisson summation formula l2z f(l) e;iul = j2z ^f(u +2j) is satised (cf. Butzer and Nessel [3]). By C(R), we denote the set of continuous functions on R. For a measurable function f on R and m 2 N let f p := ( Z 1 ;1 jf(x)j p dx) 1=p jfj m p := D m f p f m p := 2 m D f p :
Here and in the following, D denotes the dierential operator with respect to x D:= d= dx. Let W m p (R) be the usual Sobolev space with the norm m p. The l p {norm of a sequence c := fc l g l2zis dened by c l p := ( P l2z jc lj p ) 1=p : For m 2 N, let E m (R) be the space of all functions f 2 C(R) with the decay property sup x2r fjf(x)j (1 + jxj)1+m+ g < 1 ( >): Let l 2 ;m := fc := (c ): P 1 =;1 (1 + jj2 ) ;m jc j 2 < 1g beaweighted sequence with the corresponding norm c l 2 ;m := 1 l=;1 (1 + jlj 2 ) ;m jc l j 2 1=2 Considering the functions 2 E m (R) ( ::: r; 1), we call the set B() := f (;l) : l 2 Z ::: r ; 1g L;m 2 {stable if there exist constants <A B<1 with A r;1 c 2 l 2 ;m r;1 r;1 l2z c l (;l) 2 B L 2 ;m : c 2 l 2 ;m for any sequences c = fc l g l2z2 l;m 2 ( ::: r; 1). Here L 2 ;m denotes the weighted Hilbert space L 2 = ;m ff : f L 2 := ;m (1+jj2 ) ;m=2 f 2 < 1g. Note that, if the functions are compactly supported, then the (algebraic) linear independence of the integer translates of ( ::: r; 1) yields the L 2 ;m{stability ofb(). For m,we obtain the well{nown L 2 {stability (Riesz stability). For 2 E m (R) ( ::: r; 1), we say that provides controlled L p {approximation order m (1 p 1), if the following three conditions are satised: For each f 2 Wp m (R) there are sequences c h = fc h g l l2z( ::: r; 1 h>) such that for a constant c independent ofh we have: r;1 P P ;1=p (1) f ; h l2z ch l (=h ; l) p ch m jfj m p. (2) Furthermore, c h l p c f p ( ::: r; 1): (3) There is a constant independent ofh such that for l 2 Z dist (lh supp f) > ) c h l ( ::: r; 1): This notation of controlled L p {approximation order, rst introduced in Jia and Lei [11], is a generalization of the well{nown denition of approximation order for compactly supported functions. In [11], the strong connection of controlled approximation order provided by and the Strang{Fix conditions for was shown. Note 3
that, instead of using the denition of Jia and Lei [11], we also could tae the denition of local approximation order by Halton and Light [8]. For our considerations the equivalence to the Strang{Fix conditions is important. The theory of closed shift{invariant subspaces of L 2 (R), spanned by integer translates of a nite set of functions has been extensively studied (cf. e.g. de Boor, DeVore and Ron [1, 2] Jia [1]). In particular, it has been shown that the approximation order provided by a vector can already be realized by a nite linear combination f = r;1 We call f superfunction of. l2z a l (;l): (a l 2 R): 3. Approximation by renable function vectors In this section we shall give a new approach to necessary and sucient conditions for the renement mas of a renable vector ensuring controlled L p {approximation order m. In particular, we show, how a superfunction f of (providing the same approximation order as ) can be constructed by the coecients which occur in the linear combinations of reproducing the monomials. In the following, let r 2 N and m 2 N be xed. First we want to recall the result in [13] dealing with the connection between controlled L p {approximation order, reproduction of polynomials and Strang{Fix conditions. Theorem 1 (cf. [13]) Let =( ) r;1 beavector of functions 2 E m (R)\BV (R). Further, let B() be L 2 ;m{stable. Then the following conditions are equivalent: (a) The function vector provides controlled approximation order m (m 2 N). (b) Algebraic polynomials of degree <mcan be exactly reproduced by integer translates of,i.e., there arevectors y n l 2 R r (l 2 Z n ::: m; 1) such that the series P l2z (yn l )T (;l) are absolutely and uniformly convergent on any compact interval of R and l2z (yn l ) T (x ; l) =x n (x 2 R n ::: m; 1): (c) The function vector satises the Strang{Fix conditions of order m, i.e., there is a nitely supported sequence ofvectors fa l g l2z, such that f := l2z at l (;l) satises ^f() 6= D n ^f(2l) (l 2 Znfg n ::: m; 1): 4
The equivalence of (a) and (c) is already shown in Jia and Lei [11], Theorem 1.1. Further, (b) follows from (c) by [11], Corollary 2.3. For showing that (b) yields (c), in [13] the function f := m;1 a T ( + ) (3) is introduced. Here, the coecient vectors a are determined by (a ::: a m;1 ):=(y ::: ym;1 ) V ;1 with the Vandermonde matrix V := ( n ) m;1. Hence we have n y n = m;1 By Fourier transform of (3) we obtain ^f(u) =A(u) T ^(u) with n a (n ::: m; 1): (4) A(u) := m;1 a e iu : (5) That means, A(u) isan(rr){matrix of trigonometric polynomials. Observe that by (4) (D n A)() = m;1 (i) n a = i n y n (n ::: m; 1): Using the Poisson summation formula it can be shown that f satises the conditions (D ^f)(2l) l (l 2 Z ::: m; 1) and hence the Strang{Fix conditions of order m (cf. [13]). Observe that f in (3) is a superfunction of. In the new proof for the following theorem, this superfunction will be the main tool. Theorem 2 Let =( ) r;1 bearenable vector of functions 2 E m (R)\BV (R). Further, let B() be L 2 ;m {stable. Then the function vector provides Lp {controlled approximation order m if and only if the renement mas P of in (2) satises the following conditions: There arevectors y 2 Rr y 6= ( ::: m; 1) such that for n ::: m; 1 we have n n (y ) T (2i) ;n (D n; P )() = 2 ;n (y n ) T (6) n n (y ) T (2i) ;n (D n; P )() = T (7) where denotes the zero vector. 5
Proof: Note that the conditions (6){(7) can also be written in the form D n [A T (2u) P (u)]j u = (D n A) T () (n ::: m; 1) (8) D n [A T (2u) P (u)]j u= = T (n ::: m; 1) (9) where A, dened in (5), is the symbol of a superfunction f of in (3). From Theorem 1 we now that provides controlled approximation order m if and only if f satises the Strang{Fix conditions of order m. Hence we onlyhave to prove: The relations (8){(9) are satised if and only if f satises the Strang{Fix conditions of order m, i.e., (D n ^f)(2l) =cn l (n ::: m; 1) (1) with constants c n 2 R and c 6. 1. We show that the relations (8){(9) are satised if we have (1). Note that by (1) ^f(2u) =A T (2u) ^(2u) =A T (2u) P (u)^(u): Taing the derivatives, it follows on the one hand and on the other hand (D n ^f)(u) = D n [A T (u) n ^(u)] n = (D A T )(u)(d n; ^)(u) 2 n (D n ^f)(2u) = D n [A T (2u) P (u) n ^(u)] n = D [A T (2u) P (u)] (D n; ^)(u): 2. Let us rst show that the conditions (9) are satised. For all l 2 Zwe nd by (1) that = ^f(4l +2) =A T (4l +2) P (2l + ) ^(2l + ) = A T () P () ^(2l + ): Hence, linear independence of the sequences f^ ( +2l)g l2zfor ::: r; 1 gives A T () P () T : Note that the linear independence of the sequences f^ (u +2l)g l2zfor all u 2 R and for ::: r; 1 is satised if and only if the integer translates of form a L 2 {stable basis of their closed span (cf. Jia and Micchelli [12]). This was the rst step of the induction proof. 6
Let now D [A T (2u) P (u)]j u= = T be satised for ::: n; 1(n<m), and observe that by assumption (1) (D n ^f)(4l +2) = for all l 2 Z. Then, by the linear independence of f^ ( +2l)g l2zfor ::: r; 1 and by n n = 2 n (D n ^f)(4l +2) = D [A T (2u) P (u)]j u= (D n; ^)( +2l) = D n [A T (2u) P (u)]j u= ^( +2l) it follows that D n [A T (2u) P (u)]j u= = T : Thus, the relations (9) are satised. 3. Now we show that (1) yields (8). Let u =2l (l 2 Z). Then we have on the one hand by the Strang{Fix conditions and on the other hand ^f(4l) =A T () P () ^(2l) =c l ^f(2l) =A T () ^(2l) =c l : By linear independence of f^ (2l)g l2zfor ::: r; 1we obtain A T () P () = A T (): Again, we proceed by induction. Let now D [A T (2u) P (u)]j u =(D A T )() be satised for ::: n; 1(n<m) and observe that by assumption (D n ^f)(2l) = c n l (l 2 Z c 6= ). Then we nd for all l 2 Z n n 2 n (D n ^f)(4l) = D [A(2u) T P (u)]j u (D n; ^)(2l) n;1 n = (D A T )() (D n; ^)(2l) On the other hand, for l 2 Z (D n ^f)(2l) = n Hence, a comparison yields +D n [A T (2u) P (u)]j u ^(2l) =cn l n (D A T )() (D n; ^)(2l) =cn l : D n [A T (2u) P (u)]j u ^(2l) =(D n A T )() ^(2l) 7
By linear independence of f^ (2l)g l2zfor ::: r; 1we obtain D n [A T (2u) P (u)]j u =(D n A T )(): Now the proof by induction is complete. 4. We are going to prove the reverse direction. Assume that the relations (8){(9) are satised. We show that then the conditions (D n ^f)(2l) =cn l (n ::: m; 1) hold, where c 6. For the -th derivative of ^f we nd 2 (D ^f)(4l) = D [A T (2u) P (u)]j u (D ; ^)(2l) = (D A)() (D ; ^)(2l) = (D ^f)(2l) and 2 (D ^f)(4l +2) = = : D [A T (2u) P (u)]j u= (D ; ^)(2l + ) Thus, we indeed obtain (D n ^f)(2l) =cn l. It only remains to show that c 6. By Poisson summation formula and using the L 2 {stability of we have ^() = A T () ^() = (y ^() )T = (y )T (;l) 6= : l2z Hence f satises the Strang{Fix conditions of order m. Remar 3 For proving the second direction in Theorem 2 we do not need any stability condition if we assume that (y ) T ^() 6=. Since y and ^() are a left and a right eigenvector of P (), respectively, this assumption is satised if the eigenvalue 1 of P () is simple. 4. The GHM{multi{scaling functions We consider the example of a vector of two multi{scaling functions := ( 1 ) T treated in Donovan, Geronimo, Hardin and Massopust ([6, 7]). In the special case s = s = s 1 (with s 2 [;1 1]) of their construction, the renement equation of is given by (x) =P (2x)+P 1 (2x ; 1) + P 2 (2x ; 2) + P 3 (2x ; 3) (11) 8
where P := P 2 := ; s2 ;4s;3 1 ; 3(s;1)(s+1)(s2 ;3s;1) 4(s+2) 2 ; 3(s;1)(s+1)(s2 ;s+3) 4(s+2) 2 3s 2 +s;1 3s 2 +s;1 P 1 := P 3 := ; s2 ;4s;3 ; 3(s;1)(s+1)(s2 ;s+3) 4(s+2) 2 1 ; 3(s;1)(s+1)(s2 ;3s;1) 4(s+2) 2 : For the renement mas P we have P (u) := 1 2 (P + P 1 e ;iu + P 2 e ;2iu + P 3 e ;3iu ): Applying the result of Theorem 2 we can show that provides the controlled L p { approximation order m = 2: Observing that P () = ;s 2 +4s+3 ;3(s;1) 3 (s+1) 2 1 2 3s 2 +2s+1 (DP )() = i s 2 ;4s;3 4(s+2) 9(s;1) 3 (s+1) ;(3s 2 +2s+1) 4(s+2) 2 and P () = 1 2 3(s2 ;1) (DP )() =i ;s 2 +4s+3 4(s+2) 3(s 2 ;1)(;s 2 +4s+3) ;3(s 2 ;1) 4(s+2) 2 we nd with the relations and y ;3(s2 ; 1) = s +2 1 y 1 ;3(s2 ; 1) = 2(s +2) 1 (y ) T P ()=(y ) T (y ) T P () T (2i) ;1 (y ) T (DP )() + (y 1 ) T P () = 2 ;1 (y 1 ) T (2i) ;1 (y ) T (DP )()+(y 1 ) T P () = T : Hence, (8){(9) are satised for m =2. Knowing y superfunction f of (as dened in (3)) by and y 1,we can construct a f(x) =(y ; y 1 ) T (x)+(y 1 ) T (x +1) obtaining f(x) = 3(1 ; s2 ) 2(s +2) ( (x)+ (x +1))+ 1 (x +1): 9
Application of the renement equation (11) on the right hand side yields f(x) = 9(1 ; s2 ) 4(s +2) ( (2x)+ (2x + 1)) + 3(1 ; s2 ) 4(s +2) ( (2x ; 1) + (2x + 2)) + 1 2 ( 1(2x +2)+ 1 (2x)) + 1 (2x +1) = 1 2 f(2x ; 1) + f(2x)+1 f(2x +1): 2 That means, f itself satises the renement equation of the hat{function h(x) := maxf(1 ;jxj) g. Hence, taing a proper normalization constant, the superfunction f coincides with the hat function h. Indeed, in [6] the approximation order 2 provided by was derived by showing that the hat{function h lies in the span of the integer translates of 1. References [1] de Boor, C., DeVore, R. A., Ron, A.: Approximation from shift{invariant subspaces of L 2 (R d ). Trans. Amer. Math. Soc. 341 (1994) 787{86. [2] de Boor, C., DeVore, R. A., Ron, A.: The structure of nitely generated shift{ invariant spaces in L 2 (R d ). J. Funct. Anal. 119(1) (1994) 37{78. [3] Butzer, P. L., Nessel, R. J.: Fourier Analysis and Approximation. Basel: Birhauser Verlag, 1971. [4] Chui, C. K.: An Introduction to Wavelets. Boston: Academic Press, 1992. [5] Daubechies, I.: Ten Lectures on Wavelets. Philadelphia: SIAM, 1992. [6] Donovan G., Geronimo, J. S., Hardin, D. P., Massopust, P. R.: Construction of orthogonal wavelets using fractal interpolation functions, preprint 1994. [7] Geronimo, J. S., Hardin, D. P., Massopust, P. R.: Fractal functions and wavelet expansions based on several scaling functions, J. Approx. Theory 78, 373{41. [8] Halton, E. J., Light, W. A.: On local and controlled approximation order. J. Approx. Theory 72 (1993) 268{277. [9] Heil, C., Strang, G., Strela, V.: Approximation by translates of renable functions. Numer. Math. (to appear). [1] Jia, R. Q.: Shift{Invariant spaces on the real line, Proc. Amer. Math. Soc. (to appear). [11] Jia, R. Q., Lei, J. J.: Approximation by multiinteger translates of functions having global support. J. Approx. Theory 72 (1993) 2{23. 1
[12] Jia, R. Q., Micchelli, C. A.: Using the renement equations for the construction of pre{wavelets II: Powers of two, Curves and Surfaces (P.J. Laurent, A. Le Mehaute, L.L. Schumaer, eds.), pp. 29{246. [13] Plona, G.: Approximation order provided by renable function vectors. Constructive Approx. (to appear). [14] Strang, G., Strela, V.: Short wavelets and matrix dilation equations. IEEE Trans. on SP, vol. 43, 1995. 11