Moment Computation in Shift Invariant Spaces David A. Eubanks Patrick J.Van Fleet y Jianzhong Wang ẓ Abstract An algorithm is given for the computation of moments of f 2 S, where S is either a principal h-shift invariant space or S is a nitely generated h-shift invariant space. An error estimate for the rate of convergence of our scheme is also presented. In so doing, we obtain a result for computing inner products in these spaces. As corollaries, we derive Marsden-type identities for principal h-shift invariant spaces and nitely generated h-shift invariant spaces. Applications to wavelet/multiwavelet spaces are presented. AMS(MOS) 99 subject classication: 4A25. Keywords and Phrases: (Principal) h-shift invariant space, nitely generated h-shift invariant space, moment, Marsden's identity, wavelet, multiwavelet. Introduction We consider the computation of the moment m (f) of a function f 2 L 2 (). To this end, we project f into either an h-principal invariant subspace or a nitely generated shift invariant subspace. The approximation order and other characteristics of such spaces have been studied extensively in the fundamental paper [] and again in [, 5, ]. Our main result deals with the computation of inner products in shift invariant spaces. The advantage of utilizing these spaces is the fact that we can often construct stable bases whose elements are integer translates of a compactly supported function or nite compactly supported functions. Thus the computation of the inner product is easily implemented on a computer. As corollaries to our main result, we obtain as a special case the ability to compute moments of functions f 2 V.We then show how the process can be rened to obtain Dept. of Mathematics, Coker College, Hartsville, SC 2955 USA, eubanks@coker.edu. y Department of Mathematics, Sam Houston State University, Huntsville, T 7734 USA, vanfleet@guy.shsu.edu, research partially supported by National Science Grant DMS953282 and The Texas Regional Institute for Environmental Studies. z Department of Mathematics, Sam Houston State University, Huntsville, T 7734 USA, jwang@galois.shsu.edu, research partially supported by National Science Foundation Grant DMS953282
moments of the function f h 2 V h. The idea is to construct a sequence of shift invariant spaces V h approximating L 2 () in hopes of eventually approximating the momentoff 2 L 2 () by the moment off h 2 V h. In the case where the vector that generates the nitely generated shift invariant space V is renable, we use a result of Cohen, Daubechies, and Plonka [5] in order to obtain an estimate of the error jm (f), m (f h )j. As a consequence of our main result, we characterize a Marsden's identity for nitely generated shift invariant spaces. Recall Marsden's identity gives the explicit representation of x n in terms of B-splines [2]. A multivariate analog for box splines is given in [3]. The outline of this paper is as follows: In Section 2, we give basic denitions and elementary results necessary to the sequel. An inner product theorem and related corollaries concerning moment computation and Marsden's identity are given in Section 3. In addition, we give an error estimate for the dierence between the moment off 2 L 2 () and the moment of its orthogonal projection f h in a shift invariant subspace of L 2 (). The nal section contains moment recursion formulas for renable functions and vectors as well as examples of illustrating our results. 2 Notation, Denitions, and Basic Results In this section we introduce notation, denitions, and basic results used throughout the remainder of the paper. Let us begin by dening various types of shift invariant spaces. Suppose V h is a linear space and h>. Then V h is said to be an h-shift invariant space if f 2 V h =) f(,h) 2 V h : V h is a principal h-shift invariant space if V h is an h-shift invariant space generated by a compactly supported function 2 V h. That is f 2 V h =) f(x) = k2 c k (x, hk): When h =,we will suppress the h in the denitions above and refer to the spaces as shift invariant and principal shift invariant, respectively. As a matter of convention, we will denote an h-shift invariant space generated by by V h (). We will also be interested in shift invariant spaces generated by several functions. That is, we dene a nitely generated shift invariant space V h by insisting that f 2 V h =) f(x) = k2 (a k ) T (x, hk); where now (x) = B @. r C A (x); 2
and the a k 2 r. Such a space generated by will be denoted by V h (). We readily observe the following properties: () If V is a shift invariant space, then V h = ff() :f(h) 2 V g is an h-shift invariant space. (2) If V is a shift invariant space generated by, then V h is an h-shift invariant space generated by ( h ). We will say that the h-shift invariant space V h is of degree n and write deg(v h )=nif x k 2 V h, k =;:::;n, but x n+ =2 V h. Since the degree of a polynomial is invariant under dilation, we observe that deg(v h ) = deg(v ). If deg(v ()) = n, we will call any identity of the form x` = k2 c` k(x, k); ` =;:::;n a Marsden's identity. The identity is easily generalized to the case where V is generated by. Suppose generates the principal shift invariant space V. In this paper, we always assume that is integrable. Recall that if is compactly supported and integrable, then is also in L 2.Nowwesay a function 2 V is the dual of if < (,k);(,j) >= kj ; where <f;g>= R f(x)g(x)dx is the standard inner product and ( if k = j kj = otherwise. Analogously, ifvector generates the space V, then we will dene its dual 2 V as the vector of functions that satify < `(,k); m (,j) >= `m kj : We will say is stable if k2 j^(! +2k)j 2 > 8! 2 and vector is stable if the r r matrix ^ := (^ ij (!)) r i;j= is positive denite where ^ ij (!) = k2 ^ i (! +2k)^ j (! +2k): 3
Recall that if is stable, then also generates a stable basis for V. Suppose f`(,k)g, k 2, ` =;:::;r forms a basis for V. Then it also forms a basis for V \ L 2 (). For convenience, we will use V to denote V \ L 2 (). We say that V h provides L 2 -approximation order m if, for each suciently smooth function f 2 L 2 (), jjf, Proj V fjj 2 Ch m : When is a compactly supported vector in L 2, then V () provides L 2 -approximation order m if and only if V () contains m,, the set of all polynomials of degree m, []. Remark. If is renable, i.e., (x) = N k= P k (2x, k) () where the P k are rr matrices, then m, V () is equivalent to the existence of solutions to two systems of equations involving the renement mask of (see [5,, 5]). When is renable in the sense of (), we can obtain estimates on the accuracy of approximating moments of f 2 L 2 () using projections of f into V h (). Proposition 2. Suppose 2 is a renable vector and assume f j (,k) :k 2 ;j = ;:::rg forms a linearly independent basis for the space V () with deg(v ()) = n,. Further suppose that f 2 L 2 () is compactly supported and suciently smooth. We have: jm (f), m (Proj Vh f)j C f h n (2) where =;:::;n, and C f isaconstant that depends on the support width of f. Proof. The result of [5] guarantees that V provides approximation order n. If S denotes the support of f, wehave 3 Main Results jm (f), m (Proj Vm f)j = j <x S (x); (f, Proj Vm f) > j jjx S (x)jjjjf, Proj Vm fjj C f h m : 2 The main goal of this paper is to provide an algorithm for computing moments in principal (or nitely generated) shift invariant spaces. Once the algorithm is in place, we will use it to attempt to approximate the moment m (f) off 2 L 2 (). In order to obtain formula for computing moments of f = Proj V f and subsequent moments in rened spaces, we establish the following result. 4
Theorem 3. Suppose is compactly supported, stable and that is its dual. Denote by V () the space generated by. Assume f 2 V () satises the decay condition jf(x)j and g 2 V () satises the growth condition where C is an absolute constant. Let C ; >; (3) +jxj jg(x)j C( + jxj );, > (4) f(x) = (a k ) T (x, k); g(x) = k2 k2 (b k ) T (x, k): (5) Then <f;g>= P k2 (a k ) T b k. Proof. Without loss of generality, assume that supp() =[;L]. (By the support of a vector 2 r,we mean supp() =[ r supp( k= k)). First note that since is compactly supported, is of exponential decay. That is, for k =;:::;r: for some >. j k (x)j C e,jxj (6) We shall now estimate the decay rate of a k i and b k i, k 2, and i =;:::r. First we show that for suciently large k and constant C 2 >, we have ja k i jc 2 jkj, : (7) Since is compactly supported and stable, the set f (,k)g k2 forms an unconditional basis for V (). Therefore, a k = i f(x) i (x, k): Then for suciently large k, we have ja k i j = j C C 2 jkj, f(x + k) i (x)dxj dx jxj ln jkj +jx + kj + Using an analogous argument, we have for constant C 3 > jxj> ln jkj e,x dx! jb k i j ( C3 ( + jk + Lj ) k C 3 ( + jkj ) k< (8) 5
Now for M 2 and M>dene f M (x) = (a k ) T (x, k) g M (x) = (b`) T (x, `): k>m `>M In an analogous manner dene f,m and g,m. Then f(x) = g(x) = M k=,m M `=,M (a k ) T (x, k)+f,m (x)+f M (x) (b`) T (x, `)+g,m (x)+g M (x) Then f(x)g(x)dx = = @ M + + + M k=,m @ M k=,m @ M k=,m `=,M (a k ) T (x, k) A @ M `=,M (b`) T (x, `) A dx (a k ) T (x, k) A (g,m (x)+g M (x)) dx (b`) T (x, `) A (f,m (x)+f M (x)) dx (f,m (x)+f M (x))(g,m (x)+g M (x))dx (a k ) T b + (f,m (x)+f M (x))(g,m (x)+g M (x))dx Using (7), (8) and the fact that, >we see that the second term in the above sum tends to as M!so that we obtain the desired result. 2 We obtain the following moment formulas as immediate corollaries of Theorem 3.. Corollary 3.2 Let V () be generated byacompactly supported and stable and let be the dual of. Further assume that deg(v ()) = n, x = k2 (c k; ) T (x, k); =;:::;n; (9) and that f 2 V () satises the decay condition jf(x)j C ; () +jxjn+ where, >. Then m (f) = k2 (a k ) T c k; () 6
Corollary 3.2 illustrates how wemay compute the moments of order or less of f 2 V (). Note that in order to implement (), we must have the coecient vector c k; for x. We will discuss a procedure for obtaining c k; later in the section. The following corollary describes howwe can rene our procedure and obtain moments m (f) where f 2 V h (). Corollary 3.3 Let f 2 V h () and have the representation f(x) = k2 (a k ) T ( x h, k); with h>. Further assume that deg(v h ()) = n and that f satises the growth condition () from Corollary 3.2. Let x be as in (9). Then m (f) =h + k2 (a k ) T c k; (2) We continue by deriving explicit representations for c k;. Assume that deg(v ()) = n. Then for =;:::;n, j =;:::;r, and k 2 we have: c k; j = = = = `= `= j (x, k)x dx j (x)(x + k) dx ` `! k,`! x j (x)dx k,`m ( j ) (3) Thus to compute the c k;,we need only the moments of order less than of the components of. We shall see in the nal section of the paper that in the case where is renable then this task can be performed recursively. Once we have these moments, we can use Corollary 3.2 or Corollary 3.3 to compute moments of functions in (nitely generated) principal shift invariant subspaces of L 2 (). In addition Proposition 2. provides a means to estimate moments from these spaces should we intend to use them to approximate moments of functions in L 2 (). We conclude this section by noting that in light of (9) and (3) we have the means for establishing a Marsden's identity in any nitely generated shift invariant space of degree n. Of course for computational purposes, we must also obtain explicit formula for the moments m ( j ), =;:::;n and j =;:::;r. Proposition 4. illustrates how we can obtain these values in the case where r = and the function solves (4). We give examples of particular vector functions in the next section. 7
4 Renable Functions and Vectors In the nal section of the paper, we discuss various methods for computing the initial moments m ( j )asgiven in (3). One of the most popular ways to obtain classes of (nitely generated) principal shift invariant spaces is to use ideas from wavelet or multiwavelet theory (see [2, 6] for wavelets, [8, 9] for multiwavelets). The idea is to construct a nested ladder of principal shift invariant subspaces of L 2 (). This ladder is constructed by nding a function (oravector ) who along with its integer translates forms a Reisz basis for a space V. For k 2, the space V k is formed using the translates (2 k x, n), n 2, of(2 k x). Other requirements are made of the nested ladder to ensure existence of a wavelet. The property we are particularly interested in is the renement property (). Our rst result of this section shows that if the generator is renable, then we need only compute R (x)dx and then use this value to recursively generate all moments needed in (3). Proposition 4. Assume that, 2, and that R (x)dx 6=. Furthermore, suppose that there exists real numbers p ;:::;p N so that Then Proof. obtain m () = (x) = 2 N k=, 2(2, ) `= p k (2x, k): (4) `!" N k= p k k,`# m`(): Multiply both sides of (4) by x and integrate over. Upon simplication, we m () = (2 +, P N k= p k) `= `!" N k= p k k,`# m`(): Take Fourier transforms of both sides of (4) and evaluate at. Since R (x)dx 6=,we have N p k = 2 k= from which the result follows. 2 It is possible to generalize this result to the vector case. To this end, we introduce some new notation. Let m () 2 rr be the vector whose components are given by m ()` = m (`), ` =;:::;r. In addition we dene the r r matrix P by P = 2 N k= where the P k satisfy the renement condition (). 8 P k ;
Proposition 4.2 Assume, 2 and that saties (). obtained via recursion with m () as an initial starting point. Then m () can be Proof. Multiply both sides of () by x and integrate over. Upon simplication we have: m () =2,, j= j!" N k= P k k,j # m j (): Further simplication yields (I, 2, P )m () = 2, j= j!" N k= P k k,j # m j (): It is shown in [3] that P, the spectral radius of P satises P =. Thus (I, 2, P ), exists. 2 The propositions above illustrate that we can use functions from wavelet theory and multiwavelet theory to approximate moments of functions in L 2 (). Daubechies ([6]) has created a family of functions that can possess arbitrary regularity. Chui and Wang ([4]) have derived wavelets from cardinal B-splines. In terms of multiwavelets, one could use Proposition 4.2 with the spline multiwavelets of Goodman and Lee ([9]) or the fractal multiwavelets given in ([7]). We conclude the paper with two examples from the scaling functions listed in the previous paragraph. In both cases, we shall attempt to estimate moments of the Dirichlet density f(x; a; b) = ( xa (, x) b =B(a; b); x 2 [; ] otherwise, (5) where B(a; b) is the beta function and a; b 2 with a; b >. Example 4.3 We consider the family of Daubechies scaling functions 2 ;:::; 5 [6] (see Figure below). V ` is the closed linear span of the integer translates of `, ` =2;:::;5. We take as V ` j the closed linear span of the set f2,j=2 `(2 j x, k)g k2, ` =2;:::;5. Note that Vj ` V ` j+ and that deg(vj ` )=`, 2. 9
.4.2.8.6.4.2.8.6.4.2.2 -.2.5.5 2 2.5 3 x -.2 2 3 4 5 x -.4 -.4 The functions 2 (left) and 3..8.8.6.6.4.4.2.2 -.2 2 3 4 5 6 7 x -.2 2 4 6 8 x The functions 4 (left) and 5. Figure. Scaling functions for Example 4.3. We will estimate the =; 2; 3; 4 moments of f(; 2 ; 2 ) using each of 2;:::; 5. In order to do so, we must calculate the order moment for each scaling function. We can then use the recursion formula in Proposition 4. to compute the higher order moments that are used to form the c k;. The next step is to obtain the a k 's in Corollary 3.3. We provide our results for h =2,j, where j =6; 8;. We have provided three dierent methods for obtaining the a k. In the rst case, we simply sample f. In the second case, we approximate f by a piecewise quadratic polynomial and then use the precomputed c k; to form a Newton-Cotes type integration scheme. The third method for computing the a k uses the numerical integration from the prior method but uses
a more sensitive approximation to f at the breakpoints x = and x =. Our results are given in the tables below. The numbers in parentheses represent the error between the approximation and the exact value. Note that we have used 2, 3, and 4 even in cases where Corollary 3.2 does not apply (that is, the order of the moment is larger than the degree of the space). The error in these cases is larger since x is not a member of the spaces generated by the corresponding scaling functions. In addition, since we must approximate the fa k g in some fashion the errors are dependent on the function f. Since f is only C, it it natural that the C function 2 does an adequate job approximating the moments. Since the support of 2 is less than that of any other scaling function we use, the expansions for f h consist of fewer terms. Thus, the computational cost of using 2 is less than that incurred by the other scaling functions. The actual moments of f for this example are m (f) =:5, m 2 (f) =:325, m 3 (f) =:2875, and m 4 (f) =:64625. Daubechies' 2 function. Daubechies' 3 function. Daubechies' 4 function. Daubechies' 5 function..523275(+4.6e-3).555824(+.2e-3).53547(+2.7e-4) 2.372956(-.67e-2).3233(-4.4e-3).32482(-.3e-3) 3.24235(-3.35e-2).2684828(-8.69e-3).2826775(-2.2e-3) 4.564446(-4.83e-2).6284(-.25e-2).635469(-3.7e-3).489597(-2.e-2).4973674(-5.27e-3).49933799(-.32e-3) 2.2948378(-5.65e-2).379563(-.46e-2).33586(-3.67e-3) 3.2283(-8.57e-2).2389255(-2.22e-2).2752234(-5.6e-3) 4.457952(-.2e-).592646(-2.92e-2).628469(-7.4e-3).4767935(-4.64e-2).4949456(-.6e-2).49854499(-2.9e-3) 2.28276572(-9.5e-2).348785(-2.46e-2).35636(-6.2e-3) 3.893(-.35e-).298329(-3.55e-2).2678298(-8.99e-3) 4.3589634(-.72e-).5657257(-4.57e-2).625836(-.6e-2).4648879(-7.8e-2).492384(-.8e-2).4977525(-4.49e-3) 2.272926(-.33e-).3777(-3.45e-2).397726(-8.73e-3) 3.7867438(-.83e-).28555(-4.87e-2).264598(-.24e-2) 4.2662787(-2.28e-).539947(-6.8e-2).647259(-.58e-2)
Table : a k sampled from f. Daubechies' 2 function..554693(+3.8e-2).538863(+7.77e-3).597426 (+.95e-3) 2.3238575(+2.52e-2).344497(+6.24e-3).3298696(+.56e-3) 3.2235858(+2.2e-2).299594(+5.53e-3).295338(+.39e-3) 4.673742(+2.2e-2).649249(+5.2e-3).642745(+.29e-3) Daubechies' 3 function. Daubechies' 4 function. Daubechies' 5 function..57963(+3.59e-2).5386543(+7.73e-3).59773(+.94e-3) 2.322994(+3.36e-2).344292(+6.7e-3).3298446(+.55e-3) 3.22624(+3.32e-2).2993897(+5.44e-3).29587(+.38e-3) 4.6964373(+3.4e-2).648822(+5.e-3).642762(+.27e-3).579752(+3.6e-2).5385567(+7.7e-3).59756(+.94e-3) 2.3235(+3.36e-2).34496(+6.4e-3).3298329(+.55e-3) 3.22664(+3.34e-2).2992972(+5.39e-3).29497(+.37e-3) 4.69756(+3.44e-2).6487322(+4.94e-3).642747(+.27e-3).5794667(+3.59e-2).5384989(+7.7e-3).596988(+.94e-3) 2.32298667(+3.36e-2).34439(+6.2e-3).3298263(+.54e-3) 3.2268993(+3.36e-2).2992478(+5.37e-3).29494(+.37e-3) 4.6977257(+3.48e-2).64869(+4.92e-3).642698(+.26e-3) Table 2: a k obtained by numerical integration. Daubechies' 2 function. Daubechies' 3 function..599295(+.98e-2).53884(+7.76e-3).597352(+.95e-3) 2.3482643(+7.44e-3).3444359(+6.22e-3).3298622(+.56e-3) 3.2857369(-8.6e-4).2995342(+5.5e-3).29526(+.29e-3) 4.629963(-6.97e-3).6489665(+5.8e-3).6427338(+.29e-3) 2
Daubechies' 4 function. Daubechies' 5 function..53953(+2.78e-2).535675(+7.3e-3).593456(+.87e-3) 2.3972(+2.3e-2).34372(+5.24e-3).3294749(+.43e-3) 3.22229573(+.62e-2).2964928(+4.e-3).29395(+.2e-3) 4.66264(+.9e-2).645948(+3.24e-3).6423478(+.5e-3).599577(+.99e-2).532468(+6.25e-3).587968(+.76e-3) 2.3583967(+.7e-2).337336(+3.88e-3).328939(+.26e-3) 3.29393(+.78e-3).29234(+2.2e-3).2895985(+9.59e-4) 4.6359(-6.2e-3).648253(+7.32e-4).64886(+7.2e-4).556894(+.3e-2).525799(+5.6e-3).58227(+.62e-3) 2.324977(-.2e-3).33275(+2.25e-3).3282678(+.5e-3) 3.25773(-.39e-2).2872938(-9.43e-5).2889378(+6.57e-4) 4.5993477(-2.52e-2).636997(-2.26e-3).6452(+3.2e-4) Table 3: a k obtained by adaptive numerical integration. We now consider an example illustrating our methods with nitely generated shift invariant spaces. To this end, we employ the scaling vector comprised of fractal interpolation functions given in ([7]). We also note that in the case of these functions, a recursion formula for the moments exists (see [4]) and could be used in place of Proposition 4.2. Example 4.4 We consider the closed linear space V spanned by the fractal interpolation functions and 2 as derived in ([7]). We choose ; 2 so that deg(v )=3. We obtain the V j spaces by taking the closed linear span of the set f2,j=2 `(2 j x, k);` =; 2g k2. As in Example 4.3, we approximate moments of the beta distribution. 3 2.5 2.5.5.5.5.5 2 2.5 3 3
Figure 2. The fractal interpolation functions and 2. Note that the accuracy is about the same as that of the Daubechies 2 function. In the rst table, the a k were function samples; in the second table the a k were obtained using numerical integration; in the third table the a k were obtained using adaptive numerical integration. The adaptive integration is not as eective here since one of the scaling functions has the same support as does f. The actual values for m (f), =; 2; 3; 4, are given in the previous example. a k from function samples: hline.58292(+2.6e-2).536346(+6.3e-3).5794(+2.6e-3) 2.356433(+.e-2).33472(+3.e-3).32937(+.4e-3) 3.299522(+.58e-3).289827(+.5e-3).2894533(+8.93e-4) 4.63788(-5.39e-3).6395625(-6.48e-4).6447(+4.73e-4) a k obtained via numerical integration:.5494589(+2.99e-2).5382728(+7.65e-3).596687(+.93e-3) 2.399697(+2.39e-2).3439355(+6.6e-3).3297975(+.54e-3) 3.2239232(+2.3e-2).299482(+5.28e-3).294625(+.35e-3) 4.6772(+.79e-2).648494(+4.79e-3).642678(+.25e-3) a k obtained via adaptive numerical integration: References.544434(+2.89e-2).53763(+7.52e-3).595837(+.92e-3) 2.3946792(+2.23e-2).343265(+5.84e-3).329724(+.5e-3) 3.2226942(+.8e-2).2983786(+4.97e-3).293774(+.32e-3) 4.6652(+.49e-2).647823(+4.39e-3).6425857(+.2e-3) Table 2. Moment computation from Example 4.4. [] C. deboor, R. DeVore, and A. Ron, \Approximation from shift- invariant subspaces of L 2 ( d ), Trans. Amer. Math. Soc., 34 (994), 787{86. [2] C.K. Chui, An Introduction to Wavelets, Academic Press, San Diego, 992. 4
[3] C.K. Chui and M.J. Lai, \A multivariate analog of Marsden's identity and a quasiinterpolation scheme", Constr. Approx., 3 (987), {22. [4] C.K. Chiu and J.. Wang, \A general framework of compactly supported splines and wavelets", J. Approx. Th., 7(992), 263-34. [5] A. Cohen, I. Daubechies, G. Plonka, \Regularity of renable function vectors", (preprint). [6] I. Daubechies, Ten Lectures on Wavelets, SIAM CBMS Series No. 6, Philadelphia, 992. [7] G. Donovan, J. Geronimo, D. Hardin, and P. Massopust, Construction of orthogonal wavelets using fractal interpolation functions", (to appear SIAM J. Math. Anal.). [8] J. Geronimo, D. Hardin, and P. Massopust, \Fractal functions and wavelet expansions based on several scaling functions", J. Approx. Theory, 78(3) (994), 373-4. [9] T.N.T Goodman, and S. L. Lee, \Wavelets of multiplicity r", Trans. Amer. Math. Soc., Vol. 342, Number, March 994, 37-324. [] C. Heil, G. Strang, V. Strela, \Approximation by translates of renable functions, Numer. Math., (to appear). [] Jia, R.Q., \Shift-Invariant spaces on the real line", Proc. Amer. Math. Soc., (to appear). [2] M.J. Marsden, \An identity for spline functions with applications to variationdiminishing spline approximation", J. Approx. Th., 3 (97), 7{49. [3] P.R. Massopust, D.K. Ruch, and P.J. Van Fleet, \On the support properties of scaling vectors", Comp. and Appl. Harm. Anal., 3(996), 229{238. [4] P.R. Massopust and P.J. Van Fleet, \On the moments of fractal functions and Dirichlet spline functions", (preprint). [5] G. Plonka, \Approximation properties of multi-scaling functions: A Fourier approach", Constr. Approx. (to appear). 5