FRACTAL MULTIWAVELETS RELATED TO THE CANTOR DYADIC GROUP

Orthogonal wavelets on the Cantor dyadic group are identified with multiwavelets on the real line consisting of piecewise fractal functions. A tree algorithm for analysis using these wavelets is described. Multiwavelet systems with algorithms of similar structure include certain orthogonal compactly supported multiwavelets in the linear double-knot spline space S1,2

To study the construction of wavelets and to study abstract harmonic analysis, we consider orthogonal wavelets on the locally compact Cantor dyadic group.In Lang [11], compactly sup- ported wavelets are constructed on this group; the construction proceeds similar to that of Meyer [16], Mallat [14] and Daubechies [2], via scaling filters.(See Holschneider [10] for general informa- tion about wavelets on locally compact groups; other constructions of wavelets on groups include Dahlke [1] and Lemarie [13].)The Cantor dyadic group may be ident!fied with the nonnegative real numbers as a measure space; harmonic analysis on the Cantor dyadic group corresponds to analysis using Walsh functions on the line.The wavelets constructed on the Cantor dyadic group turn out to be certain lacunary Walsh series on the line.
Here, we will continue study of these wavelets; we will consider these wavelets as wavelets on the real line.We will describe the form that the natural Mallat tree algorithm for these wavelets takes when used to analyze functions on the line.From the structure of the algorithm, we find that the Cantor dyadic group wavelets may be identified as multiwavelets on the line.In fact, they are multiwavelets consisting of piecewise fractal functions, in the sense of Massopust [15].It is possible to develop their properties without reference to the Cantor dyadic group.
Other wavelet systems with a tree algorithm with the same structure include certain com- pactly supported orthogonal multiwavelets in the linear double-knot spline space S 1,2 described in section 7 below; approximations with these multiwavelets take the form of piecewise linear, not necessarily continuous functions.
2. THE CANTOR DYADIC GROUP.We describe the locally compact Cantor dyadic group.This group, also known as the 2-series local field, consists of the countably infinite weak direct product of the group of integers modulo 2. We write G Z/(2) where Z/(2)= {0,1}.
Thus if x 6 G then x (x,)_o where x, 6 {0,1} and where x, # 0 for only finitely many _> 0. So we may identify x with the real number '___o x,2'.The Cantor dyadic group is thus identified with the nonnegative real numbers as a measure space (but not algebraically or topologically).
Translation on the Cantor dyadic group is as usual; we will write Ty(x) x + y for x, y 6 G.We consider a simple example.Let y (y,) where yo 1 and y, 0 for #-0.So y corresponds to the real number 1. Translation by this number corresponds to the function f x+l if2k:x<2k+lforsomeintegerk_>0 T,(z) (2.1) x-1 otherwise when G is identified with [0, oo).
Dilation on the group G is given by p(x), x,-l.This corresponds exactly to the map p(x) 2x when G is identified with [0, oo).We let pk(x) 2kx for k E Z.
The group characters of the Cantor dyadic group become Walsh functions when the group is identified with the nonnegative reals.We describe the Walsh functions on the real line.First we define the Rademacher functions rj for j E Z.We let to(x) 1 if2k _< x < 2k+l for some integer k, and ro(x) -1 otherwise.We then define rj(x) ro(23x).Now consider a real number y.We write y y,2' where y, 6 {0, 1}.We then define the Walsh function W, by Wy(x) 1-I,ez(r,(x))Y'; for each x there is only finitely many terms in the product different than 1. (This is the Paley denumeration of the Walsh functions.)For example, W3(x) ro(x)rl (x) if x 6 [k-1/4, k+ 1/4) for some integer k and-1 otherwise.For a function f on G (or a function on the line), we may define the Walsh transform .{(y)f f(x)W,(x)dx; this is the natural analogue of the Fourier transform for the group G, but we will make no further reference to this transform here.

WAVELETS ON THE CANTOR DYADIC GROUP.
First we describe multiresolution analyses on the Cantor dyadic group.Let A be the subgroup of the Cantor dyadic group corresponding to the nonnegative integers.We say that a sequence (V) of closed subspaces of L2(G) is a multiresolution analysis if: V c V+I for all j 6 Z; f 6 Vo == f o Tn 6 V0 for all n 6 A; f 6 V ==> f o p 6 V+I for all j 6 Z; CV {0} and is dense in L2(G); and there is f 6 V0 whose translates by A form a Riesz basis of Vo.
We may then construct compactly supported, orthogonal wavelets on the Cantor dyadic group.This may be done by following the method of Meyer, Mallat and Daubechies, using conditions on scaling filters.We omit the details of this construction; see Lang [11] and [12].If we consider length-4 scaling filters (consisting of trigonometric polynomials of four Wish functions): we obtain the following wavelets.Let 0 < a _< 1 and a + b 2 1.Let (x) f(x/2) where f 1/2110,1) (1 + aWl + abW3 + ab2W7 + ab3W5 +... ).
(Here 110,1) is the indicator function of [0, 1); it is 1 if x is in that set and 0 otherwise.)Then is continuous (in the sense of G) and compactly supported, and the translate of by A are orthogonal.Also if Vo is the space spanned by the translates of , then V0 forms a multiresolution analysis as above.The corresponding mother wavelet is the function where a0 (1 + a + b)/4, al (1 + a b)/4, a2 (1 a b)/4 and a3 (1 a + b)/4.The translates of by A span a space Wo where V1 V0 Wo; the translates and dilates of form an orthogonal basis of L2(G) in the usual way.(See Lang [11] for details.) If we consider scaling filters of length two, we obtain the familiar Haar wavelets.In Lang [12], length-8 wavelets are detailed; some of these also take the form of lacunary Walsh series.
4. THE ALGOI:tITHM FOR WAVELETS ON THE CANTOR DYADIC GROUP.
Here we detail the Mallat tree-type algorithm for the length-4 wavelets on the Cantor dyadic group.Let co, al, a2 and a3 be as in the previous section and let b0 -al, bl co, b2 -a3 and 53 a2.For f a function on G, j 6 Z, and k 6 A, let fG f(x)(p(x) k)2 3/2 dx and di fG f(x)'(/P (x)-k)2 j/2 dx.Then the reconstruction algorithm is and the decomposition algorithm is 4 21/2 E an-P(k) n-i-1 and d 21/2 E bn_.(k)c.+1 n6A r,6A Note here that the subscripts of the coefficients are treated as members of the group G.
When we identify G as [0, oo), the algorithm takes the following form: The decomposition algorithm produces the lower level (smaller j) coefficients from higher level coefficients; the reconstruction algorithm produces higher level coefficients k from the coefficients die and lower level k multiply both sides of (4.3) by A-1.
The following diagram shows the structure of the algorithm.Each rectangle represents multiplication of the four coefficients above it by A to obtain the four coefficients below it.
The algorithm has a 'matrix filter' structure reminiscent of Strang and Strela [21], and hence we are led to consider our wavelets as multiwavelets on the line.

MULTIWAVELETS ON THE LINE.
With ordinary wavelets, there is a single scaling function whose translates span a space V0 which generates a multiresolution analysis.The condition (e.g.) V-1 C V0 requires that (x/2) kez ak(x k) for some coefficients ha.In the case of multiwavelets, we would have several scaling functions 1,... ,. whose translates by integers span a space V0, the dilates of which form a multiresolution analysis.We would write Then the condition V-1 becomes (x/2) ]Ez P(x-k) where the coefficients P are now n x n matrices.See Goodman and Lee [6], Goodman et al. [7], Plonka and Strela  The Cantor dyadic group wavelets of section 3 can be interpreted as multiwavelets on the real line.Let and be as in (3.1) and (3.2).Let 1 , 2 o T1, '1 D and 2 T1, where T1 is as in (2.1) Define ) by 2k(x)=1(23x-2k)2 /2 and CJ (x) =2(2x-2k)2 /2 2k+l and define similarly.Suppose f is a function on the real line and let 4 f/(x)CZ(x) dx and dk f f(x)(x)dx for j,k e Z. Then the coefficients are related by (4.3).This follows since every translate of on G by a member of A is, as a function on the line, an (ordinary) translate of either 1 or 2.
In the previous section, we identified the Cantor dyadic group wavelets of section 3 with multiwavelets on the line.In the present section, we will show that these are piecewise fractal functions in the sense of Massopust [15] p. 137 and p. 258.
We begin by defining fractal functions.(This definition is actually a specialization of the general definition in Massopust [15].)Consider the Read-Bajraktarevid operator for real-valued f on [0, 1]: f A+sf(2x)   ifO_<x<l/2 (6.1) #+tf(2x-1) if 1/2_<x_<1   where A, , s, are fixed real numbers, with Is[ < 1 and It[ < 1.The domain and range of this operator is L([0, 1]); it may be shown (Proposition 6.3 below) that there is a unique fixed point f for this operator in that space.We call f a fractal function, motivated by the selfosimiliarity of the graph of f. (The fixed point f of obeys f(x) A4-sf(2x) on [0, 1/2) and f(x) #+tf(2x-1) on [1/2, 1], so the graph of f restricted to [0,1/2) and the graph of f restricted to [1/2, 1] are each affine linear images of the graph of f.)We note that Read-Bajraktarevid operators serve as a framework for studying functions with fractal graphs in terms of iterated function systems; see Massopust [15].
It is possible to write the fixed point f explicitly as a series; we consider the case when s t.Let f0 be the function constantly 1 on [0, 1] and let f, f,-1 for n _> 1, where sf(2x So each fn is piecewise constant and f, is bounded by max{Isln, It, l" }.We have by the linearity of , PROPOSITION 6.3.The fixed point of (6.1) is the function f a + bfl + bf2 + b f3 + bf4 +---, where a(1 s) + bs A and a(1 t) + bt #, provided s # t.
We remark that we were able to show that 1 and 2 were orthogonlal, using lemma 6.5.Of course, this was already known in Lang [11], using Fourier analysis on the Cantor dyadic group.
Other properties of these wavelets may be developed the techniques of this section, such as the scaling relations of theorem 5.1; but we do not pursue this here.
7. OTHER MULTI'WAVELET SYSTEMS WITH SIMILAR ALGORITHMS.
The Cantor dyadic group wavelets of section 3 have an algorithm with a particular structure as described by the diagram in section 4. That structure in part reflects the arithmetic of the Cantor dyadic group.Other multiwavelet systems unrelated to the Cantor dyadic group have algorithms with the same structure.We will describe one example, composed of multiwavelets in the double-knot spline space S1'2.This space consists of the functions, not necessarily continuous.
whose restrictions to each interval [k, k + 1) (k an integer) is a first degree polynomial; our notation resembles de Boor [3] and Plonka [17]. (Here, the first superscript refers to the degree of the basis functions and the second superscript refers to the decrement for regularity; thus the splines belong to C-1, meaning no continuity is required.We may describe this space as a linear spline space where pairs of 'knots' coincide.See de Boor [3].)The multiwavelets will be compactly supported, piecewise linear and orthogonal; they fit into the general treatment of Plonka [17] and Plonka [19], and Strang and Strela [21] for more information on multiwavelets.