Connectivity, homotopy degree, and other properties of [alpha]-localized wavelets on R
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Publicacions Matem`atiques, Vol 43 (1999), 303–340. CONNECTIVITY, HOMOTOPY DEGREE, AND OTHER PROPERTIES OF α-LOCALIZED WAVELETS ON R Gustavo Garrig´ os Abstract In this paper, we study general properties of α-localized wavelets and multiresolution analyses, when 1 2<α≤∞. Related to the latter, we improve a well-known result of A. Cohen by showing that the correspondence m→ ϕ=∞ 1m(2−j·), between lowpass filters in Hα(T) and Fourier transforms of α-localized scaling functions (in Hα(R)), is actually a homeomorphism of topological spaces. We also show that the space of such filters can be regarded as a connected infinite dimensional manifold, extending a theorem of A. Bonami, S. Durand and G. Weiss, in which only the case α=∞is treated. These two properties, together with a careful study of the “phases” that give rise to a wavelet from the MRA, will allow us to prove that the space Wα,of α-localized wavelets, is arcwise connected with the topology of L2((1 + |x|2)αdx) (modulo homotopy classes). This last result is new even for the case α=∞, as well as the considerations about the “homotopy degree” of a wavelet. 1. Introduction Awavelet is a function ψ∈L2(R) such that the system ψj,k(x)=2 j/2ψ(2jx−k),j,k∈Z, is an orthonormal basis for L2(R). Elementary properties, examples, and standard notation can be found in the texts: [5] and [7] (Part II). Keywords. Wavelet, MRA, Sobolev space, homotopy degree, phase. 1991 Mathematics subject classifications: 42C15.
304 G. Garrig´ os In this paper, we shall be interested in those wavelets having a certain smoothness and decay at infinity. The smoothness will be given in terms of Sobolev spaces, and the decay in terms of “L2-localization”. We say that a measurable function f:R→Cis α-localized whenever (1.1) R |f(x)|2(1 + |x|2)αdx < ∞. A function ffor which the integral in (1.1) is finite for every α∈Z+ is said to be ∞-localized or to have polynomial decay. We will suppose 1 2<α≤∞, so that, in particular, the Fourier transform of f, given by ˆ f(ξ)=R f(x)e−ix·ξdx, ξ ∈R, represents a continuous function in R. Note, further, that a function f is α-localized if and only if ˆ fbelongs to the Sobolev space Hα(R), since the expression in (1.1) is precisely ˆ f2 Hα(R). In the sequel, we will consider the following class of wavelets: Definition 1.2. Let 1 2<α≤∞. We say that ψ∈W αif (i) ψis a wavelet. (ii) ψis α-localized (equivalently, ψ∈Hα(R)). (iii) There exists a positive εsuch that ψ∈Hε(R). We endow Wαwith the topology of L2((1 + |x|2)αdx) (respectively, of the Fr´echet space ∩∞ k=1L2((1 + |x|2)kdx), if α=∞). That is, for a sequence ψ,ψ1,ψ 2,... ∈W α, the statement “ψn→ψin Wα”, means that “ ψn→ ψin Hα(R)”. In this setting, one of the main theorems in this paper can then be stated as follows: Theorem 1.3. Let 1 2<α≤∞, and ψ∈W α. Then there exists an integer k=k(ψ)such that (1.4) R x|ψ(x)|2dx =k+1 2. Moreover, for each k∈Z, the set W(k) α=ψ∈W α:R x|ψ(x)|2dx =k+1 2 is a connected, open and closed topological subspace of Wα. In particular, Wαcan be written in connected components as the disjoint union: Wα= ∪k∈ZW(k) α.
Connectivity in the set of α-localized wavelets on R305 As interesting as the result itself, is the technique we use to prove it, which includes a detailed study of the α-localized multiresolution analyses (precise definitions will be postponed to subsequent sections). Recall that, by a theorem of P. G. Lemari´e([8]), every wavelet in Wαmust arise from one of these MRA’s. That is, given ψ∈W α, there is an α-localized scaling function ϕ, with associated low-pass filter m, and a unimodular 2π-periodic function ν(sometimes referred to as “phase”) such that (1.5) ψ(ξ)=e−iξ/2ν(ξ)m(ξ/2+π)ϕ(ξ/2),ξ∈R. The “mild” condition given in (iii) of Definition 1.2 plays a crucial role in the proof of Lemari´e’s theorem. However, it is not too restrictive since the converse is also true ([4]): every α-localized wavelet arising from an MRA satisfies (iii) and, therefore, belongs to Wα. This characterization for wavelets in Wαproduces a whole variety of examples, including those of Daubechies, Lemari´e-Meyer, or the wavelets obtained in [6]by “smoothing” discontinuous low-pass filters. Moreover, general questions, like the connectedness of the set Wα, can be understood from their study on the different elements of an α-localized MRA: filters, scaling functions, and phases. In section 2, we study the set, Eα, of low-pass filters associated with α-localized scaling functions. We show that it is an arcwise connected manifold when endowed with the topology of the periodic Sobolev space Hα(T). This set was introduced and studied as a manifold for the first time by A. Bonami, S. Durand and G. Weiss ([1]), in the case α=∞. Our contribution here consists mainly in adapting the definitions and proofs for other values of α. In section 3, we take care of the α-localized scaling functions, showing that the set of all these is homeomorphically equivalent to Eα. Like in the α=∞case, treated by the authors of [1], the homeomorphism is given by the usual assignment to each scaling function of its associated low-pass filter, which according to Cohen’s theorem, can be written as: (1.6) m∈E α−→ ϕ(ξ)= ∞ j=1 m(2−jξ). However, the machinery required to prove the continuity of this map when α<∞, is much more sophisticated than for the case α=∞. This is due, on the one hand, to the use of Sobolev norms, but also to the sharpness of the result, which is no longer true, for instance, if one enlarges slightly the domain (see Example 3.12 below). Convergence questions involving filters and scaling functions have also been considered by P. G. Lemari´ein[9], although only in the α=∞case, leaving open the study for other values of α.
306 G. Garrig´ os In section 4, we turn to the study of the “phase” νin (1.5), left aside by many authors in the wavelet literature, but key to obtain our general result in Theorem 1.3. In [1], for instance, only wavelets that admit the phase ν≡1 are considered and, therefore, only those can be shown to be connected by arcs. So, the general connectivity statement we present in Theorem 1.3 is new even in the case of the space W∞. Formula (1.4), relating the homotopy degree of the phase, and the “center of mass” of the wavelet plays a central role here, since from it we can easily see that Wαis not a connected topological space (with the topology considered in Definition 1.2). Indeed, any continuous path t→ ψtin Wαmust leave the integral in (1.4) constant and, therefore, a wavelet ψcan never be joined to its shift ψ(·−1), continuously in L2(|x|dx). This formula was announced for the case α=∞in [8], and amounts to the work of L. Villemoes ([11]) on compactly supported wavelets, but the simple and general proof we give here (which covers all the values of α>1 2) seems to be new. To conclude section 4, we classify wavelets according to the homotopy degree of their phases, and establish, then, our final connectivity result. In the appendix, we study a bit further the decomposition of a wavelet in terms of a phase and a filter as in (1.5). This question was previously considered by P. G. Lemari´ein[8], where it was proved that every wavelet ψ∈W (0) ∞can be written as: (1.7) ψ(ξ)=ce −iξ/2m(ξ/2+π) ∞ j=2 m(2−jξ),ξ∈R, with a filter m∈C∞(T) and a unimodular constant c∈C. In other words, a constant phase ν≡cis “admissible” in (1.5). These results are no longer true in the range 1 2<α<∞, as we illustrate with different counter-examples. However, when ψ∈W (0) α, we show how to find a filter of maximum regularity: m∈∩ β<αHβ(T), and how the choice of this filter is unique (up to multiplication by eiξ,∈Z) only when α>1. Preliminary warning and acknowledgements. The content of this article forms part of the Ph. D. thesis of the author ([3]), completed in Washington University, St. Louis, in May 1998, under the direction of Guido Weiss. The thesis is available at http://www.math.wustl.edu/∼gustavo, and it contains several technical results that are (only) referred to in this presentation. We have not included this material here since it also appears in [1]or[7]; however, the reader might find the write-up in [3] more accessible since the notation and terminology are closer to the ones used in this article.
Connectivity in the set of α-localized wavelets on R307 The author is indebted to Guido Weiss for his constant support and advice during the completion of the thesis and the write-up of this article. We also wish to thank A. Bonami and M. Taibleson for some very useful remarks during different stages of this work. 2. The space Eαof low-pass filters We defined above Eαas the set of low-pass filters associated with α-localized scaling functions. Since in this section we won’t use any of the terminology of multiresolution analyses, but instead will concentrate only on topological and manifold-like properties of Eα, we might as well consider the equivalent definition of this set given in terms of Cohen’s Theorem. That is, a function mis in Eαif and only if belongs to the Sobolev space Hα(T) and satisfies the conditions: (2.1) m(0) = 1. (2.2) |m(ξ)|2+|m(ξ+π)|2=1,ξ∈R. (2.3) ∃K⊂Rcompact such that 0 ∈◦ K,∈ZχK(ξ+2π)=1,a.e. ξ∈R, and m(2−jξ)=0,j≥1, ξ∈K. Of course, as a topological space Eαwill be endowed with the topology of Hα(T). Our goal is to establish the following result: Theorem 2.4. Let 1 2<α≤∞. Then Eαis an infinite dimensional manifold, in the sense that for every m∈E αthere exists a neighborhood of m,U⊂E α, a neighborhood of 0,V⊂Hα(T), and a homeomorphism Φ:U→V, such that Φ(m)=0. Moreover, Eαis an arcwise connected topological space. The proof of this theorem is not much different to the one presented by A. Bonami et al. in [1], for the case α=∞. For the sake of brevity, and in order to move quickly to the new material in subsequent sections, we just indicate how to adapt the arguments in [1] to the case 1 2<α<∞. The reader interested in more details is referred to the monograph [3]. Before getting started, however, we would like to illustrate with an example (also from [1]) the intuitive idea underlying Theorem 2.4. Example 2.5. Trigonometric polynomials of degree ≤3inEα.
308 G. Garrig´ os Consider the family of functions (2.6) m(a,c)(ξ)=1+eiξ 2(a+beiξ +cei2ξ),ξ∈R, where a,b,care real numbers such that (2.7) b=1−a−cand a2+c2=a+c. An easy computation shows that (2.1) and (2.2) hold for m(a,c), if and only if the triad (a, b, c) is taken as in (2.7). These equalities can also be interpreted as (a, c) lying in a circle C={(a, c)∈R2:(a−1 2)2+(c−1 2)2= 1 2}(see Figure 2.1, below). c (0,1) (1,1) (0,0) (1,0) a Figure 2.1. Circle Cof low-pass filters in Example 2.5. When (a, c)=(1,1) we obtain m(1,1)(ξ)=1+ei3ξ 2, which is known not to belong to Eα(see Example F in Chapter 2 of [5]). On the other hand, when (a, c)∈C\{(1,1)}, an elementary calculation shows that, with the choice K=[−π,π], m(a,c)satisfies Cohen’s condition (2.3) and, hence, belongs to Eα. Clearly, the set {m(a,c),(a, c)∈C\{(1,1)}} is a connected topological space with a 1-dimensional manifold structure. We now concentrate in the proof of Theorem 2.4, which will be splitted into two parts: the manifold condition, and the connectivity property.
Connectivity in the set of α-localized wavelets on R309 2.1. The manifold condition. Following the ideas in [1], we denote by Fαthe topological subspace of Hα(T) whose elements satisfy (2.1) and (2.2). That is, we have removed from the space Eαthe intricate condition (2.3). Then we have the following proposition. Proposition 2.8. Let α>1 2. Then Eαis an open subset of Fα. Once Proposition 2.8 is proved, it will suffice to show that Fαis a manifold to establish the first part of Theorem 2.4. But this can be done proceeding exactly as in the proof of Theorem 2.1 of [1], after replacing the space C∞(T)byHα(T), and noticing that the last one is a Banach algebra (when α>1 2) and hence, closed under multiplication and the action of holomorphic functions (that is, f∈Hα(T) and Fholomorphic in a neighborhood of f(T), imply that F◦f∈Hα(T)). The verification of these facts is left to the interested reader (who may consult [3] for further details). Note that as an immediate corollary of our results we obtain: Corollary 2.9. Let α>1 2. Then Eαand Fαare locally connected topological spaces. The proof of Proposition 2.8, however, requires some more modifications, and we present it here for completeness. In fact, it is an immediate consequence of the following lemma, and the Sobolev Embedding Theorem. We will denote by Λβ(T) the usual Lipschitz space on the torus, 0<β<1. Lemma 2.10. Let α>1 2and 0<β<min{1,α−1 2}.Letm∈F α, Kbe a compact set on R, and ε>0. Then there exists a positive δ=δ(β, ε, K)such that for every F∈F αwith F−m Λβ(T)<δwe have ∞ j=1 F(2−jξ)− ∞ j=1 m(2−jξ)<ε, for all ξ∈K. Proof: Let us denote by ϕF(ξ)= ∞ j=1 F(2−jξ) and ϕ(ξ)= ∞ j=1 m(2−jξ),ξ∈R.
310 G. Garrig´ os It is well-known that the infinite products above converge uniformly on compact sets and represent continuous functions in R(see, e.g., Lemma 4.5 in Chapter 7 of [5]). Now, note that for every ξ∈R, (2.11) ϕF(ξ)−ϕ(ξ)= ∞ =1 −1 j=1 m(2−jξ) [F(2−ξ)−m(2−ξ)] ϕF(2−ξ), where the series converges uniformly and absolutely on K. Indeed, the partial sums of the series satisfy N =1 −1 j=1 m(2−jξ) [F(2−ξ)−m(2−ξ)] ϕF(2−ξ) =ϕF(ξ)− N j=1 m(2−jξ) ϕF(2−Nξ)→ϕF(ξ)−ϕ(ξ),as N→∞, the convergence in the last step following from properties of the infinite product and the continuity of ϕFat ξ= 0. Suppose now that the compact set K⊂[−M,M], for some M>0, and let j0be a positive integer such that 2−j0M≤π. Then, for all ξ∈K, and using that |F|,|m|≤1, F(0) = m(0) = 1, we have: |ϕF(ξ)−ϕ(ξ)|≤ ∞ =1 |F(2−ξ)−m(2−ξ)| ≤ j0 =1 F−m ∞+ ∞ =j0+1 |2−ξ|β F−m Λβ(T). Then, if we take δ< ε 2j0and δ<ε(2β−1) 2(2M)β, we have |ϕF(ξ)−ϕ(ξ)|<ε. 2.2. The connectivity. To end this section, we need to show that both Fαand Eαare connected, establishing then Theorem 2.4. But the proof of this fact, once again, does not differ much from the α=∞case studied in [1]. There, it was shown that every filter (in F∞,orE∞) of the form (2.12) m0(ξ)= P0(ξ) |P0(ξ)|2+|P0(ξ+π)|2,ξ∈T,
Connectivity in the set of α-localized wavelets on R311 where P0is a trigonometric polynomial, can be joined with a continuous path: t∈[0,1] −→ mt=Pt |Pt|2+|Pt(·+π)|2, consisting of functions of the same type, to the Haar filter m1(ξ)= 1+eiξ 2∈E ∞. This is essentially due to an argument using the F´ejer-Riesz lemma for positive trigonometric polynomials (see section 3 of [1]). From here and Corollary 2.9, the connectivity question reduces to show that the set of filters of the form (2.12) is dense in the spaces Fαand Eα. The proof of this fact is simple, but again somewhat different from the one given in [1], so we present it below for completeness. Proposition 2.13. Let α>1 2and F∈F α. Then there exists a sequence of trigonometric polynomials {Pn}∞ n=1 such that Pn(0)=1, Pn(π)=0,n=1,2,... and (2.14) Pn |Pn|2+|Pn(·+π)|2→Fin Fα,as n→∞. Proof: Let {Qn}∞ n=1 be a sequence of trigonometric polynomials such that Qn→Fin Hα(T) (e.g., take the nth-symmetric partial sum of the Fourier series of F). Then, by the Sobolev Embedding Theorem: lim n→∞ Qn(0) = F(0) = 1 and lim n→∞ Qn(π)=F(π)=0. Therefore, for large n, we have |Qn(0) −Qn(π)|>1 2. Define, in these cases, Pn(ξ)=Qn(ξ)−Qn(π) Qn(0) −Qn(π),ξ∈R. Then, Pnare trigonometric polynomials converging to Fin the topology of Hα(T). Moreover, Pn(0) = 1 and Pn(π) = 0. Once again, the Sobolev Embedding Theorem tells us that |Pn(ξ)|2+|Pn(ξ+π)|2→1, uniformly in T, and therefore, for nlarge enough we must have |Pn(ξ)|2+|Pn(ξ+π)|2>1 2,∀ξ∈T. Now, the properties of the Banach algebra Hα(T) imply that (|Pn(·)|2+|Pn(·+π)|2)−1/2∈Hα(T) and, consequently, (2.14) holds. Remark 2.15. Note that if F∈E α, the trigonometric polynomials {Pn}in the previous proposition can be taken to satisfy Cohen’s condition (2.3). This follows from the fact that Eαis an open subset of Fα, and implies in particular that the former set is connected.
318 G. Garrig´ os where the series converges absolutely. Using (3.19) with ϕnand ϕ, and subtracting, we obtain (ϕn−ϕ)(ξ)−(ϕn−ϕ)(ξ+η) = ∞ L=1 L−1 h=1 h−1 j=1 m(2−j(ξ+η))[(mn−m)(2−h(ξ+η))] L−1 j=h+1 mn(2−j(ξ+η)) ×[mn(2−Lξ)−mn(2−L(ξ+η))]ϕn(2−Lξ) + ∞ L=1 L−1 j=1 m(2−j(ξ+η))[(mn−m)(2−Lξ)−(mn−m)(2−L(ξ+η))]ϕn(2−Lξ) + ∞ L=1 L−1 j=1 m(2−j(ξ+η))[m(2−Lξ)−m(2−L(ξ+η))][(ϕn−ϕ)(2−Lξ)] =A+B+C. It is enough to show that each of the terms ωα(A), ωα(B) and ωα(C) tends to 0, as n→∞. We start with A. Estimating the first three multiplicative blocks in Awith the · ∞-norm, and changing variables in ξand η, we obtain ωα(A)≤ mn−m ∞ ∞ L=1 L−1 h=1 2−L(α−1 2) ×RR |mn(ξ)−mn(ξ+η)|2|ϕn(ξ)|2dξ dη |η|1+2α1 2 ≤C mn−m Hα(T) ϕn ∗RT |mn(ξ)−mn(ξ+η)|2dξ dη |η|1+2α1 2 where in the second inequality we have periodized the integral in ξand used the Sobolev Embedding Theorem. Now, it follows from Lemma 3.13 that ϕn ∗is uniformly bounded and, therefore, ωα(A)≤C mn−m Hα(T)→0,as n→∞.
Connectivity in the set of α-localized wavelets on R319 One deals similarly with B, obtaining: ωα(B)≤ ∞ L=1 2−L(α−1 2) ϕn ∗TT |mn(ξ)−mn(ξ+η)|2dξ dη |η|1+2α +8π|η|>π mn−m 2 ∞ dη |η|1+2α1 2 ≤C mn−m Hα(T)→0,as n→∞. Finally, the third term is bounded by ωα(C)≤ ∞ L=1 2−L(α−1 2) ϕn−ϕ ∗RT |mn(ξ)−mn(ξ+η)|2dξ dη |η|1+2α1 2 ≤C ϕn−ϕ ∗→0,as n→∞, where the convergence to 0 follows from Lemma 3.13. This shows (3.18) and, together with (3.17), completes the proof of case 1. Case 2: α=k∈Z+. We need to show that: (3.20) D(k)ϕn−D(k)ϕ L2(R)→0,as n→∞. Recall that D(k)ϕis the distributional derivative of an infinite product of periodic functions. Note that, when 0 ≤h≤k−1, the distribution m(h) may be considered as 2π-periodic function in Ck−h−1(R), while m(k)is (a.e.) a function in L2(T). Moreover, the following formula, appearing in Chapter 4 of [7], holds (here δ jdenotes the usual Kr¨onecker symbol): Lemma 3.21. Let k∈Z+.Ifm∈Hk(T), with m(0)=1, |m|≤1, and ϕis defined by the infinite product formula (3.6) then, the kth distributional derivative of ϕcan be written as: (3.22) D(k)ϕ(ξ)= ∞ L=1 ∈(Z+)k sup1≤i≤ki=L #1 2$||L j=1 m(ε j)(2−jξ)ϕ(2−Lξ), where ||=1+···+k,ε j=k i=1 δi j,=(1,... , k)∈(Z+)k,j≥1, and the series in (3.22) converges absolutely for a.e. ξ∈R.
320 G. Garrig´ os The notation in (3.22) can be interpreted in the following way: for a fixed L≥1 and a given multi-index =(1,... , k) we spread the k derivatives among the Lfirst terms of the infinite product, where each iindicates that the th i-term is differentiated one time. The condition sup1≤i≤ki=Lguarantees that the Lth-term is always considered and, therefore, that there are no repetitions in the process. Note that, when =(L,... ,L), all the derivatives are concentrated in one term and, therefore, the function m(k)(which is in L2(T), but may not be continuous) comes into play. This will give rise to different cases when proving (3.20) above. Using (3.22), we can write: D(k)ϕn(ξ)−D(k)ϕ(ξ) = ∞ L=1 sup i=L =(L,... ,L) 2−|| L h=1 h−1 j=1 m(ε j)(2−jξ)[(mn−m)(ε h)(2−hξ)] × L j=h+1 mn(ε j)(2−jξ)ϕn(2−Lξ) + L j=1 m(ε j)(2−jξ)[ϕn(2−Lξ)−ϕ(2−Lξ)] +1 2kL L−1 h=1 h−1 j=1 m(2−jξ)[(mn−m)(2−hξ)] × L−1 j=h+1 mn(2−jξ)mn(k)(2−Lξ)ϕn(2−Lξ) + L−1 j=1 m(2−jξ)[(mn−m)(k)(2−Lξ)]ϕn(2−Lξ) + L−1 j=1 m(2−jξ)m(k)(2−Lξ)[(ϕn−ϕ)(2−Lξ)] =[I+II]+[III +IV +V].
Connectivity in the set of α-localized wavelets on R321 An upper bound for the L2(R)-norm of the first two terms can be obtained easily, estimating the periodic functions with · ∞, and changing variables: I L2(R)≤ ∞ L=1 sup i=L =(L,... ,L) 1 2L L h=1 2L/2 h−1 j=1 m(ε j) ∞ (mn−m)(ε h) ∞ × L j=h+1 m(ε j) n ∞ ϕn L2(R) ≤C m k Hα(T) mn−m Hα(T) mn k Hα(T) ∞ L=1 Lk+1 2L/2 ≤C mn−m Hα(T)→0,as n→∞, and II L2(R)≤ ∞ L=1 Lk+1 2L/2 m k Hα(T) ϕn−ϕ L2(R)→0,as n→∞, where the convergence to 0 in the second case follows from (3.17). For the other three terms, we cannot take the · ∞-norm of m(k), and need to periodize the integral, obtaining: III L2(R)≤ ∞ L=1 1 2kL L−1 h=1 2L/2 mn−m ∞ ϕn ∗ mn(k) L2(T) ≤C mn−m Hα(T)→0,as n→∞, IV L2(R)≤ ∞ L=1 1 2kL 2L/2 ϕn ∗ mn(k)−m(k) L2(T) ≤C mn−m Hα(T)→0,as n→∞,
322 G. Garrig´ os and V L2(R)≤ ∞ L=1 1 2kL 2L/2 ϕn−ϕ ∗ m(k) L2(T)→0,as n→∞, where in the last limit we use Lemma 3.13. This shows (3.20) and completes the proof of case 2. Case 3: α=k+ε, k ∈Z+,0<ε<1. This case is a little more tedious to write, mainly because of the cumbersome notation needed; however, all the estimations follow from the same type of arguments we used previously. We only present here those that require major modifications, leaving the others to the reader (or referring to section 3.3 of [3] for further details). We have to show that (3.23) ωε(D(k)ϕn−D(k)ϕ) =RR |D(k) (ϕn−ϕ)(ξ)−D(k) (ϕn−ϕ)(ξ+η)|2dξ dη |η|1+2ε1 2 →0,as n→∞. By using (3.22), we obtain a similar decomposition to the one in (3.19), this time involving derivatives: D(k)ϕ(ξ)−D(k)ϕ(ξ+η) = ∞ L=1 ∈(Z+)k sup i=L 1 2|| L p=1 p−1 j=1 m(ε j) (2−j(ξ+η))[m(ε p) (2−pξ)−m(ε p) (2−p(ξ+η))] × L j=p+1 m(ε j)(2−jξ)ϕ(2−Lξ) + L j=1 m(ε j)(2−j(ξ+η))[ϕ(2−Lξ)−ϕ(2−L(ξ+η))] .
Connectivity in the set of α-localized wavelets on R323 To simplify the notation, we will write (∆ηF)(ξ)≡F(ξ)−F(ξ+η). Now, using the previous equality with ϕand ϕn, and subtracting both quantities, we obtain D(k)(ϕn−ϕ)(ξ)−D(k)(ϕn−ϕ)(ξ+η) = ∞ L=1 sup i=L 1 2|| L p=1 p−1 q=1 q−1 j=1 m(ε j) (2−j(ξ+η))[(mn−m)(ε q) (2−q(ξ+η))] × p−1 j=q+1 mn(ε j) (2−j(ξ+η))[(∆ηmn(ε p) (2−p·))(ξ)] L j=p+1 mn(ε j) (2−jξ)ϕn(2−Lξ) + p−1 j=1 m(ε j) (2−j(ξ+η))[(∆η(mn−m)(ε p) (2−p·))(ξ)] L j=p+1 mn(ε j) (2−jξ)ϕn(2−L ξ) + L q=p+1 p−1 j=1 m(ε j)(2−j(ξ+η))[(∆ηm(ε p)(2−p·))(ξ)] q−1 j=p+1 m(ε j)(2−jξ) ×[(mn−m)(ε q)(2−qξ)] L j=q+1 mn(ε j)(2−jξ)ϕn(2−Lξ) + p−1 j=1 m(ε j) (2−j(ξ+η))[(∆ηm(ε p) (2−p·))(ξ)] L j=p+1 m(ε j) (2−jξ)(ϕn−ϕ)(2−Lξ) + L q=1 q−1 j=1 m(ε j)(2−j(ξ+η))[(mn−m)(ε q)(2−q(ξ+η))] L j=q+1 mn (ε j)(2−j(ξ+η)) ×[(∆ηϕn(2−L·))(ξ)] + L j=1 m(ε j)(2−j(ξ+η))[(∆η(ϕn−ϕ)(2−L·))(ξ)] = ∞ L=1 =(L,... ,L) + ∞ L=1 =(L,... ,L) =[A1+···+A6]+[B1+···+B6]=A+B.
324 G. Garrig´ os The estimates needed to show that ωε(A)→0, when n→∞, are essentially the same as in the previous cases: we isolate the factor that contains the difference mn−m ∞(or, more generally, mn−m Hα(T)), while we control the rest of the integral &R&R... dξ dη |η|1+2εby periodizing in ξ, and with the increments ∆ηG, for different choices of G=mn(ε j),m (ε j),ϕn,... To control the “tales of the products”, one can use the following lemma, whose simple proof can be found in Chapter 4 of [7]. Lemma 3.24. Let Hbe the normed space (modulo null-functions) defined by H='f:R→Cmeasurable: f 2 ∗= ess-supξ∈R k∈Z |f(ξ+2kπ)|2<∞(. Let τbea2π-periodic measurable function and let P=Pτbe the operator defined at f∈Hby (Pf)(ξ)=τ(ξ/2)f(ξ/2),ξ∈R. Then, for every f∈H Pf ∗≤ess-supξ∈T[|τ(ξ)|2+|τ(ξ+π)|2]1 2 f ∗. After a change of variable, and periodization, the case A1becomes: ωε (A1)≤ ∞ L=1 Lk 2L L p=1 (p−1) m k α mn−m α mn k α2−p(ε−1 2) P m (ε p+1) n ...P m (ε L) nϕn ∗ ×RT |(∆ηm(ε p) n)(ξ)|2dη |η|1+2ε1 2 ≤C ∞ L=1 Lk+2 2L/22k/2 ϕn ∗ mn−m α→0,as n→∞. One deals similarly with A2,A3and A5. In the case A4, we use Lemma 3.13 to obtain ωε(A4)≤C ϕn−ϕ ∗→0,as n→∞, while for A6we have ωε(A6)≤Cωε(ϕn−ϕ)→0,as n→∞, the convergence to 0 following now from (3.18) in case 1 above.
Connectivity in the set of α-localized wavelets on R325 The calculations for ωε(B)→0 follow the same pattern, the main difference with Abeing that the kth derivatives have to be treated with ·L2(T)-norms rather than ·∞. The cases B1,B5, and B6contain the essential features of these modifications, and are presented below. The remaining cases can be easily derived from these. When dealing with B1, we are led to the following equality, in which we have separated the terms 1 ≤p≤L−1 and p=L: ωε(B1)≤ ∞ L=1 1 2L L−1 p=1 (p−1) mn−m ∞2−p(ε−1 2) ×RR |(∆ηmn)(ξ)|2|mn(k)(2−L+pξ)ϕn(2−L+pξ)|2dξ dη |η|1+2ε1 2 + ∞ L=1 L−1 2L mn−m ∞2−L(ε−1 2) ϕn ∗ ×RT |(∆ηm(k) n)(ξ)|2dη |η|1+2ε1 2 . The second summand is easily treated. For the first, we use the Sobolev Embedding Theorem to choose a θ∈(0,1 2) such that 0 <ε+θ<1, and (3.25) |mn(ξ)−mn(ξ+η)|≤C mn α|η|ε+θ,ξ∈R,|η|≤1. Then the double integral in the second line of the previous inequality can be estimated by: RR |(∆ηmn)(ξ)|2|mn(k)(2−L+pξ)ϕn(2−L+pξ)|2dξ dη |η|1+2ε ≤C2L−pR |mn(k)(ξ)|2|ϕn(ξ)|2dξ|η|>1 4dη |η|1+2ε+|η|≤1 dη |η|1−2θ ≤C2L−p ϕn 2 ∗ mn(k) 2 L2(T). Thus, ωε(B1)≤C ∞ L=1 L2 2L/2 mn−m ∞+C ∞ L=1 L 2L/2 mn−m ∞→0, as napproaches ∞.
326 G. Garrig´ os To deal with B5, we use a Lipschitz condition that is a little stronger than (3.25): (3.26) ϕ(·)−ϕn(·+η) ∗≤'C|η|1 2 ϕn H1(R),when 0 <ε<1 2 C|η|γ ϕn H1 2+γ(R),when 1 2≤ε<1, for a suitable γ∈(ε, 1) (see, e.g., Chapter 4 of [7]). Now, no matter what εis, we have ωε(B5)≤ ∞ L=1 1 2L L−1 q=1 2−L(ε−1 2) mn−m ∞ ×RR |mn(k)(ξ)|2|(∆ηϕn)(ξ)|2dξ dη |η|1+2ε1 2 + ∞ L=1 1 2L2−L(ε−1 2) RR |(mn−m)(k) (ξ)|2|(∆ηϕn)(ξ)|2dξ dη |η|1+2ε1 2 . Separating the integrals over ηin two parts: &|η|≤1+&|η|>1, using (3.26), and periodizing in ξ, we obtain: ωε(B5)≤C mn−m Hα(T)→0,as n→∞. Finally, for B6, we replace ϕby ϕn−ϕin the Lipschitz condition (3.26), and use the same argument as for B5, obtaining, when 0 <ε<1 2, ωε(B6)≤C ϕn−ϕ H1(R)→0,as n→∞, the convergence to 0 following from case 2 above. This completes the proof of (3.23) when 0 <ε<1 2. When 1 2≤ε<1, we have, by the same kind of argument, ωε(B6)≤C ϕn−ϕ H1 2+γ(R)→0,as n→∞, where the convergence to 0 now follows from the fact that 1 <1 2+γ<3 2 and the previous case. This completes the proof of (3.23) and, with it, establishes Theorem 3.8.
Connectivity in the set of α-localized wavelets on R327 Remark 3.27. The map M:m→ M(m)≡∞ 1m(2−j·), is welldefined, and takes values in Hα(R), whenever m∈F α(see, Chapter 4 of [7]), although, as we saw in Example 3.12, it is not continuous in this case. It is not too hard to modify the proof we just presented to show that, when mn→min Fα, then ϕn→ϕin the local Sobolev space Hα loc(R), where ϕn,ϕare the images of mn,munder M. That is, if α=k+ε,k∈Z+∪{0},0≤ε<1, for every compact set K⊂R,we have lim n→∞{ϕn−ϕL2(K)+ϕ(k) n−ϕ(k)L2(K)+ωε,K (ϕ(k) n−ϕ(k))}=0, where ωε,K(g)= KK |g(x+h)−g(x)|2dx dh |h|1+2ε1 2 ,if 0 <ε<1 0,if ε=0. In particular, when α>k+1 2, the Sobolev Embedding Theorem implies that ϕ(h) n→ϕ(h), uniformly on compact sets of R,0≤h≤k, extending a result of Bonami, Durand and Weiss (Proposition 2.4 of [1]). A careful proof of these facts appears in section 3.3 of [3]. 4. Connectivity in the space Wα In subsection 2.3 we have studied two important objects in the theory of α-localized MRA’s: the low-pass filters, and the scaling functions. We have seen that the sets Eαand Sα, consisting, respectively, of all these elements, are topologically equivalent and arcwise connected. In this section, we shall prove similar properties for wavelets arising from α-localized MRA’s. First, we recall that these are precisely the wavelets belonging to the set Wα(theorems of Herv´e and Lemari´e) and, later, we analyze the “phases” associated with these wavelets to obtain the connectivity result in Theorem 1.3. We will also see the relation between the “center of mass” of a wavelet and the homotopy degree of an associated “phase”. 4.1. Wavelets in Wαand α-localized MRA’s. It is well-known that, whenever {Vj}j∈Zis an α-localized MRA, with scaling function ϕ, and associated low-pass filter m, then one can produce a wavelet ψ0by letting (4.1) ψ0(ξ)=e−iξ/2m(ξ/2+π)ϕ(ξ/2),ξ∈R
334 G. Garrig´ os When we move to the case 1 2<α<∞, the situation changes dramatically. To be more precise, suppose we have a function ν∈Hα(T), unimodular, and with, say, homotopy degree 0. Consider, then, the following functional equation: (5.3) ν(2ξ)=µ(2ξ)µ(ξ)µ(ξ+π),ξ∈T, where µis the unknown, to be in the space M(0) β, for the largest possible β(we would like β=α). If we could solve this equation, then, by letting ˜m(ξ)=µ(2ξ)µ(ξ)m(ξ),ξ∈R, we obtain a “nice” representation for the wavelet ψin (4.3), since: ψ(ξ)=ce −iξ 2µ(ξ)µ(ξ/2+π)µ(ξ/2) m(ξ/2+π) ∞ j=2 m(2−jξ) =ce −iξ 2˜m(ξ/2+π) ∞ j=2 ˜m(2−jξ). If the solution µbelongs to the Sobolev space Hβ(T), then the new filter ˜mis in Eβ, and we have a partial answer to our question. Moreover, if we could take β=α, then we would have a natural extension of Proposition 5.1 to the α-localized situation. Unfortunately, this case is not attainable when α<∞, the most we can say being contained in the following result: Theorem 5.4. Let 1 2<α≤∞and ν∈M (0) αand write ν(ξ)= eiθ(ξ), for some real-valued θ(ξ)=∈Zθeiξ ∈Hα(T). Consider the functional equation in (5.3). Then, (i) If 1<α≤∞, (5.3) has a unique solution µ(ξ)=eiγ(ξ)∈ M(0) 1given by γ(ξ)=∈Zγeiξ, where γ0=−θ0and γ= ∞ j=0 2jθ2j,=0. Moreover, in this case µ∈∩ β<αHβ(T).If α=∞, there are examples for which the solution µ/∈Hα(T). (ii) If 1 2<α≤1, (5.3) has infinitely many solutions µ∈∩ β<αM(0) β. There are examples for which none of the solutions belong to Hα(T). As a corollary we obtain the following extension of Proposition 5.1:
Connectivity in the set of α-localized wavelets on R335 Corollary 5.5. Let 1 2<α≤∞,k∈Zand ψ∈W (k) α. Then, there exists a low-pass filter m∈∩ β<αEβsuch that (5.2) holds for some unimodular constant c∈C. Furthermore, if (c, m)and (c',m ')are two solutions to (5.2) such that m, m'∈H1(T), then, there exists an integer M∈Zsuch that c'=(−1)Mcand m'(ξ)=eiMξm(ξ). To prove the theorem we rewrite the functional equation (5.3) in terms of the real-valued functions θand γ, and their Fourier coefficients. In this setting, it is easy to check that (5.3) is equivalent to solve the difference equation: (5.6) θ=γ−2γ2,=0 θ0=−γ0+2Kπ, for some fixed constant K∈Z(that we will assume equal to 0), and with the additional assumption γ=γ−,∈Z(so that γis real-valued). By iteration in (5.6), we obtain the following sequence as candidate to solve the equation: γ=∞ j=0 2jθ2j,if =0 −θ0,if =0. Assume now that α>1. Then since θ∈Hα(T) (and, hence, θ∈o(||α), as ||→∞), it follows that the series ∞ j=0 2jθ2jconverges absolutely for every = 0, providing {γ}a solution to the difference equation (5.6). We claim, in addition, that γbelongs to Hβ(T), for every β<α. Indeed, =0 |γ|2||2β≤ =0 ∞ j=0 2j|θ2j| 2 ||2β ≤ =0 ∞ j=0 1 22j(α−1) ∞ j=0 22jα|θ2j|2 ||2β =22(α−1) 22(α−1) −1 ∞ p=0 m∈2Z+1 ∞ j=0 (2j+p|m|)2α|θ2j+pm|21 |2pm|2(α−β) ≤4α−1 4α−1−1 ∞ p=0 s∈Z |s|2α|θs|21 22(α−β)p ≤C(α, β) s∈Z |s|2α|θs|2.
336 G. Garrig´ os This shows the existence part of the theorem (when α>1). For the uniqueness note that, if we assume {˜γ}∈H1(T), then γ=o(||), as ||→∞, and iterating the difference equation (5.6) we obtain, for =0, (5.7) ˜γ= n j=0 2jθ2j+2 n+1˜γ2n+1→ ∞ j=0 2jθ2j,as n→∞, showing that ˜γ=γ. With respect to the cases 1 2<α≤1, note that when θ∈H1(T) the series ∞ j=0 2jθ2jmight be divergent (see Example 5.9 below). To construct a solution to the difference equation (5.6) we iterate “backwards”, assuming a priori that γ2+1 =0,∈Z. Then, we define the rest of the coefficients γby: (5.8) γ= −θ0,if =0 0,if ∈2Z+1 −n j=1 2−jθ2n−jp,if =2 np, n ≥1,p∈2Z+1. An easy computation shows that γ=θ+2γ2,∈Z. We claim, further, that γ∈Hβ(T), for every β<α. Indeed, =0 ||2β|γ|2= ∞ n=1 p∈2Z+1 |2np|2β n j=1 2−jθ2n−jp 2 ≤ ∞ n=1 p∈2Z+1 |2np|2β n j=1 2−2j(1−α) n j=1 2−2jα|θ2n−jp|2 ≤ ∞ n=1 p∈2Z+1 n j=1 22nβ|p|2βn2−2jα|θ2n−jp|2 = ∞ n=1 p∈2Z+1 n j=1 |2n−jp|2α|θ2n−jp|21 |p|2(α−β) n 22n(α−β) ≤ ∞ n=1 p∈2Z+1 n−1 s=0 |2sp|2α|θ2sp|2n 22n(α−β) = ∞ s=0 p∈2Z+1 |2sp|2α|θ2sp|2 ∞ n=s+1 n 22n(α−β) ≤Cα,β r∈Z |r|2α|θr|2<∞.
Connectivity in the set of α-localized wavelets on R337 The non-uniqueness of the solutions to (5.6), in this case, can be guessed from the freedom to choose the odd coefficients γ2+1. A more constructive proof is the following. Consider the lacunary sequence γ=2−j,if ||=2 j,j≥0 0,otherwise. Note that ∈Z|γ|2(1+||2)β<∞, for all β<1, while γ=2γ2,=0, and γ0= 0. We have then obtained a non-trivial solution in ∩β<1Hβ(T) to the difference equation (5.6) when θ≡0. In fact, multiplying {γ}by any real constant we obtain infinitely many others. From here it follows immediately that (5.3) has infinitely many solutions µ∈∩ β<αM(0) β, when α≤1. We now turn to the “sharpness” of our results. In the example below we give a choice of θ∈Hα(T) for which the solutions to (5.6) do not belong to Hα(T). Example 5.9. We define the following lacunary sequence {θ}∈Z: θ='1 2α2jj,if ||=2 2j,j≥1 0,otherwise. Note that ∈Z|θ|2||2α=j≥11 j2<∞. Suppose α≥1. Then, if we expect the solutions γto (5.6) to belong to the space Hα(T), we must have that γ=o(||), as ||→∞, and the same type of argument as in (5.7) would imply that γ=∞ j=0 2jθ2j. In the case α= 1, this is already a contradiction since γ1=∞ j=0 2jθ2j=∞ r=1 1 r=∞. For the case α>1, we have ∞ =1 |γ|2||2α= ∞ m=1 22mα|γ2m|2= ∞ m=1 22mα ∞ j=0 2jθ2j+m 2 = ∞ m=1 22(α−1)m ∞ j=m 2jθ2j 2 ≥ ∞ m=2 22(α−1)m ∞ r=[log2m]+1 22r1 2α2rr 2 ≥C ∞ m=2 22(α−1)m 2m 2αm log2m 2 =C ∞ m=2 1 (log2m)2=∞, which is also a contradiction. This completes the proof of part (i) in Theorem 5.4, and the case α= 1 in part (ii).
338 G. Garrig´ os Suppose now that, with the same sequence {θ}, we let instead 1 2< α<1, and assume there is a solution γ∈Hα(T) to the difference equation (5.6). Then, by iteration, we must have that γ2n=1 2nγ1− n j=1 1 2jθ2n−j,n≥1. Now, if γ1= 0, then ∞ =1 ||2α|γ|2≥ ∞ n=3 22nα n j=1 2−jθ2n−j 2 = ∞ n=3 1 22n(1−α) [log2(n−1)] r=1 22r1 2α2rr 2 ≥C ∞ n=3 1 22n(1−α) 2(1−α)n log2n 2 =C ∞ n=3 1 (log2n)2=∞, which is a contradiction. On the other hand, if γ1= 0, then ∞ n=3 22nα|γ2n|2= ∞ n=3 1 22n(1−α)γ1− n j=1 2n−jθ2n−j 2 ≥ ∞ n=3 1 22n(1−α) |γ1|2+ n j=1 2n−jθ2n−j 2 −2|γ1| n j=1 2n−jθ2n−j ≥|γ1|2 ∞ n=3 1 22n(1−α) +C ∞ n=3 1 (log2n)2−2|γ1|) C ∞ n=3 1 2n(1−α)log2n=+∞, where in the last inequality we have used that n j=1 2n−jθ2n−j≃2n(1−α) log2n. But this is also a contradiction. Thus, we have completed the second part of (ii), and established Theorem 5.4. Final comments. Before ending, we would like to mention the work done by other authors related to the connectivity of wavelets since the article [1] appeared in the literature. X. Dai, D. Larson and G. Weiss, together with their collaborators, have studied this question from a slightly different point of view. They considered MRA wavelets, but with no localization or smoothness assumption on them, and obtained connecting
Connectivity in the set of α-localized wavelets on R339 paths continuous only in the topology of L2(R). The treatment of filters and phases differs from ours, and no homotopy impediments appear in this case. A complete exposition of their results can be found in [12]. The paper [10] contains the first general connectivity result involving non-MRA wavelets. However, the question of whether the set of all wavelets is connected still remains open. References 1. A. Bonami, S. Durand and G. Weiss, Wavelets obtained by continuous deformations of the Haar wavelet, Rev. Mat. Iberoamericana 12(1) (1996), 1–18. 2. A. Cohen,“Ondelettes et traitement num´erique du signal,” Rech. Math. Appl. 25, 1992. 3. G. Garrig´ os, The characterization of wavelets and related functions and the connectivity of α-localized wavelets on R, Ph. D. Thesis, Washington University (1998). (Available at http://www.math.wustl.edu/∼gustavo). 4. L. Herv´ e,M´ethodes d’op´erateurs quasi-compacts en analyse multir´esolution, applications `a la construction de bases d’ondelettes et `a l’interpolation, Th`ese, Laboratoire de Probabilit´es, Universit´ede Rennes-I (1992). 5. E. Hern´ andez and G. L. Weiss,“A First Course on Wavelets,” CRC Press, 1996. 6. E. Hern´ andez, X. Wang and G. Weiss, Smoothing Minimally Supported Frequency Wavelets. Part I, J. Fourier Anal. Appl. 2(4) (1996), 329–340. 7. J. P. Kahane and P. G. Lemari´ e-Rieusset,“Fourier Series and Wavelets,” Gordon & Breach Publishers, 1995. 8. P. G. Lemari´ e-Rieusset, Projecteurs invariants, matrices de dilatation, ondelettes et analyses multi-r´esolution, Rev. Mat. Iberoamericana 10(2) (1994), 283–347. 9. P. G. Lemari´ e-Rieusset, Fonctions d’´echelle interpolantes, polynˆomes de Bernstein et ondelettes non stationnaires, Rev. Mat. Iberoamericana 13(1) (1997), 91–188. 10. D. M. Speegle, The S-elementary wavelets are path-connected, Proc. Amer. Math. Soc. 127(1) (1999), 223–233. 11. L. F. Villemoes, Energy moments in time and frequency for twoscale difference equation solutions and wavelets, SIAM J. Math. Anal. 23(6) (1992), 1519–1543.
340 G. Garrig´ os 12. The Wutam Consortium, Basic Properties of Wavelets, J. Fourier Anal. Appl. 4(4) (1998), 575–594. Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi, 24 10129 Torino ITALY e-mail: gusta[email protected] Primera versi´o rebuda el 27 d’octubre de 1998, darrera versi´o rebuda el 14 de gener de 1999