Cascade Algorithms in Wavelet Analysis †

Rong-Qing JiaDepartment of Mathematical Sciences

University of AlbertaEdmonton, Canada T6G 2G1

Abstract

In this paper we survey some recent results on cascade algorithms. Let a be a finitelysupported sequence on ZZ. The cascade operator Qa is the the linear operator on Lp(IR)(1 ≤ p ≤ ∞) given by

Qaf :=∑j∈ZZ

a(j)f(2 · − j), f ∈ Lp(IR).

The iteration scheme Qnaf (n = 1, 2, . . .) is called the cascade algorithm associated with a.

The Lp convergence of a cascade algorithm is characterized in terms of the p-norm jointspectral radius of two matrices associated with the corresponding mask. For the specialcase p = 2, convergence of a cascade algorithm is characterized in terms of the spectrumof the transition matrix associated with the mask. Then the basic theory on cascadealgorithms is employed to give a unified treatment of orthogonal wavelets, biorthogonalwavelets, and fundamental refinable functions. Furthermore, we give a comprehensivereview of biorthogonal wavelet bases. Our methods can be used to deal with more com-plicated problems such as biorthogonal wavelet bases on bounded domains. Finally, weextend our study of cascade algorithms to high dimensional spaces.

† Research was supported in part by NSERC Canada under Grants # OGP 121336.

§1. Introduction

In this paper we survey some recent results on cascade algorithms and their applica-tions to wavelet analysis.

As usual, for 1 ≤ p ≤ ∞, Lp(IR) denotes the Banach space of all (complex-valued)measurable functions f on IR such that ‖f‖p <∞, where

‖f‖p :=(∫

IR

|f(x)|p dx)1/p

for 1 ≤ p <∞,

and ‖f‖∞ is the essential supremum of |f | on IR. The Fourier transform of a functionf ∈ L1(IR) is defined by

f(ξ) :=∫

IR

f(x)e−ixξ dx, ξ ∈ IR.

The domain of the Fourier transform can be naturally extended to compactly supporteddistributions.

Let a be a finitely supported sequence on ZZ. The cascade operator Qa is the linearoperator on Lp(IR) (1 ≤ p ≤ ∞) given by

Qaf :=∑j∈ZZ

a(j)f(2 · − j), f ∈ Lp(IR). (1.1)

It is easily seen that Qa is a bounded linear operator on Lp(IR). If φ is a fixed point ofQa, i.e., Qaφ = φ, then φ satisfies the refinement equation

φ =∑j∈ZZ

a(j)φ(2 · − j). (1.2)

Correspondingly, the sequence a is called the refinement mask. Any function satisfyinga refinement equation is called a refinable function.

If a is a mask with∑

j∈ZZ a(j) = 2, then it is known (see [1] and [5]) that there is aunique compactly supported distribution φ satisfying φ(0) = 1 and (1.2). This distributionis said to be the normalized solution to the refinement equation with mask a.

Taking the Fourier transform of both sides of (1.2), we obtain

φ(ξ) = H(ξ/2)φ(ξ/2), ξ ∈ IR,

whereH(ξ) :=

12

∑j∈ZZ

a(j)e−ijξ, ξ ∈ IR. (1.3)

Clearly, H is 2π-periodic and H(0) = 1. Let f be a compactly supported function inLp(IR). Then we also have

Qaf(ξ) = H(ξ/2)f(ξ/2), ξ ∈ IR. (1.4)

1

The iteration scheme Qnaf (n = 1, 2, . . .) is called the cascade algorithm associated

with a. Suppose there exists a function φ in Lp(IR) with φ(0) = 1 such that

limn→∞

‖Qnaf − φ‖p = 0.

Then φ is the normalized solution to the refinement equation (1.2). In what follows wewill show that the initial function f must satisfy the Strang-Fix conditions of order 1 (see[21]), that is,

f(0) = 1 and f(2kπ) = 0 ∀ k ∈ ZZ \ 0.

Let fn := Qnaf , n = 1, 2, . . .. It follows from (1.4) that

fn(2n+1kπ) = fn−1(2nkπ) = · · · = f(2kπ), k ∈ ZZ.

Since the supports of fn are uniformly bounded, we have limn→∞ ‖fn−φ‖1 = 0. Moreover,

|fn(ξ)− φ(ξ)| ≤ ‖fn − φ‖1 ∀ ξ ∈ IR.

Hence, for k ∈ ZZ we have

f(2kπ) = limn→∞

fn(2n+1kπ) = limn→∞

φ(2n+1kπ) =

0 for k ∈ ZZ \ 0,1 for k = 0,

by the Riemann-Lebesgue lemma. Thus, f satisfies the Strang-Fix conditions of order 1.We say that the cascade algorithm associated with a converges in the Lp norm if there

exists a compactly supported function φ ∈ Lp(IR) such that

limn→∞

‖Qnaf − φ‖p = 0

for any compactly supported function f in Lp(IR) satisfying the Strang-Fix conditions oforder 1. The L∞ convergence is often referred to as uniform convergence.

Let u be a compactly supported function in Lp(IR) (1 ≤ p ≤ ∞). Then u satisfies theStrang-Fix conditions of order 1 if and only if∑

j∈ZZ

u(· − j) = 1

(See [20]). Suppose g is a compactly supported function in Lq(IR), where 1/p + 1/q = 1.For h > 0, let Ph be the linear operator defined by

Phφ :=∑j∈ZZ

⟨φ, g(·/h− j)/h

⟩φ(·/h− j), φ ∈ Lp(IR),

where〈f, g〉 :=

∫IR

f(x)g(x) dx, for f ∈ Lp(IR) and g ∈ Lq(IR).

2

Then we have the following result (see, e.g., [12]):

limh→0

‖Phφ− φ‖p = 0, φ ∈ Lp(IR).

In particular, we may choose g to be the characteristic function of the interval [0, 1]. Inthis case, ⟨

φ, g(·/h− j)/h⟩

=1h

∫ (j+1)h

jh

φ(x) dx.

Let `(ZZ) denote the linear space of all sequences on ZZ, and let `0(ZZ) denote the linearspace of all finitely supported sequences on ZZ. For a bounded subset K of IR, let `(K)denote the linear space of all sequences supported on K∩ZZ. For j ∈ ZZ we use δj to denotethe sequence on ZZ given by δj(k) = δjk, k ∈ ZZ, where δjk stands for the Kronecker sign,i.e., δjk = 1 if j = k and δjk = 0 if j 6= k. In particular, we write δ for δ0. We use ∇to denote the difference operator on `(ZZ) given by ∇u = u − u(· − 1) for u ∈ `(ZZ). Theconvolution of two elements u and v in `(ZZ) is defined by

u∗v(j) :=∑k∈ZZ

u(j − k)v(k), j ∈ ZZ,

whenever the above sum makes sense. Moreover, `p(ZZ) denotes the Banach space of all(complex-valued) sequences b on ZZ such that ‖b‖p <∞, where

‖b‖p :=(∑

j∈ZZ

|b(j)|p)1/p

for 1 ≤ p <∞,

and ‖b‖∞ is the supremum of |b| on ZZ.Suppose φ is a compactly supported function in Lp(IR). Then there exists a positive

constant B (see [13]) such that∥∥∥∥∑j∈ZZ

b(j)φ(· − j)∥∥∥∥

p

≤ B‖b‖p ∀ b ∈ `p(ZZ).

We say that the shifts of a function φ in Lp(IR) are stable if there are two positiveconstants A and B such that

A‖b‖p ≤∥∥∥∥∑

j∈ZZ

b(j)φ(· − j)∥∥∥∥

p

≤ B‖b‖p ∀ b ∈ `p(ZZ).

It was proved in [14] that the shifts of a compactly supported function φ ∈ Lp(IR) arestable if and only if, for any ξ ∈ IR, there exists an integer k such that

φ(ξ + 2kπ) 6= 0.

3

The Lp convergence of the cascade algorithm associated with a will be characterizedin terms of the p-norm joint spectral radius of two finite matrices derived from the mask a.

Let A be a finite collection of linear operators on a finite dimensional vector spaceV . A vector norm ‖·‖ on V induces a norm on the linear operators on V as follows. For alinear operator A on V , define

‖A‖ := max‖v‖=1

‖Av‖

.

For a positive integer n we denote by An the Cartesian power of A:

An =(A1, . . . , An) : A1, . . . , An ∈ A

.

Let‖An‖∞ := max

‖A1 · · ·An‖ : (A1, . . . , An) ∈ An

.

Then the uniform joint spectral radius of A is defined to be

ρ∞(A) := limn→∞

‖An‖1/n∞ .

The uniform joint spectral radius was introduced by Rota and Strang in [19]. Daubechiesand Lagarias used the uniform joint spectral radius as a tool in their study of refinementequations (see [6]).

The p-norm joint spectral radius of a finite collection of linear operators was introducedby Jia in [11]. We define, for 1 ≤ p <∞,

‖An‖p :=

( ∑(A1,...,An)∈An

‖A1 · · ·An‖p

)1/p

.

For 1 ≤ p ≤ ∞, the p-norm joint spectral radius of A is defined to be

ρp(A) := limn→∞

‖An‖1/np .

Clearly, ρp(A) is independent of the choice of the vector norm on V .We claim that

limn→∞

‖An‖1/np = inf

n≥1‖An‖1/n

p .

Let ρ := infn≥1 ‖An‖1/np . To justify our claim we observe that, for positive integers r

and s,‖Ar+s‖p ≤ ‖Ar‖p‖As‖p.

For ε > 0, there exists a positive integer s such that ‖As‖1/sp ≤ ρ+ε, i.e., ‖As‖p ≤ (ρ+ε)s.

Let n be a positive integer. Then n can be written as n = ms + r, where m ≥ 0 is aninteger and 0 ≤ r < s. Consequently,

‖An‖p ≤ ‖Ams‖p‖Ar‖p ≤ ‖As‖mp ‖Ar‖p ≤ (ρ+ ε)sm‖Ar‖p.

4

It follows that‖An‖1/n

p ≤ (ρ+ ε)1−r/n‖Ar‖1/np .

Therefore, we havelim sup

n→∞‖An‖1/n

p ≤ ρ+ ε.

On the other hand,lim infn→∞

‖An‖1/np ≥ inf

n≥1‖An‖1/n

p = ρ.

But ε > 0 can be arbitrary. We conclude that limn→∞ ‖An‖1/np = ρ.

If A consists of a single linear operator A, then

ρp(A) = ρ(A),

where ρ(A) denotes the spectral radius of A, which is independent of p. If A consists ofmore than one element, then ρp(A) depends on p in general. By some basic properties of`p spaces we have that, for 1 ≤ p ≤ r ≤ ∞,

(#A)1/r−1/pρp(A) ≤ ρr(A) ≤ ρp(A),

where #A denotes the number of elements in A. Furthermore, it is easily seen from thedefinition of the joint spectral radius that ρ(A) ≤ ρ∞(A) for any element A in A.

Here is an outline of the paper. Section 2 is devoted to investigation of convergenceof cascade algorithms. The Lp convergence of a cascade algorithm will be characterized interms of the p-norm joint spectral radius of two matrices associated with the correspondingmask. For the special case p = 2, convergence of a cascade algorithm can be characterizedin terms of the spectrum of the transition matrix associated with the mask. In Section 3,the basic theory on cascade algorithms developed in Section 2 will be employed to give aunified treatment of orthogonal wavelets, biorthogonal wavelets, and fundamental refinablefunctions. In Section 4, we will provide a comprehensive study of biorthogonal waveletbases. The technique developed in this section can be used to deal with more complicatedproblems such as biorthogonal wavelet bases on bounded domains. Finally, in Section 5,we will extend our study of cascade algorithms to high dimensional spaces. In particular,we will give a characterization of uniform convergence of a cascade algorithm associatedwith a nonnegative mask in terms of stochastic matrices.

§2. Convergence of Cascade Algorithms

This section is devoted to a study of convergence of cascade algorithms. We mainlyfollow the lines of [11]. But some proofs are modified, so that the results in this sectioncan be easily extended to more general situations.

Let a be a finitely supported sequence on ZZ, and let Qa be the cascade operator asdefined in (1.1). We claim that

Qnaf =

∑j∈ZZ

an(j)f(2n· − j), (2.1)

5

where the sequences an are given by

a1 = a and an(j) =∑k∈ZZ

an−1(k)a(j − 2k), j ∈ ZZ, n = 2, 3, . . . . (2.2)

This can be proved by induction on n. Indeed, (2.1) is valid for n = 1. Suppose (2.1) holdstrue for n− 1. Then by the induction hypothesis we have

Qnaf = Qn−1

a (Qaf) =∑k∈ZZ

an−1(k)(Qaf)(2n−1· − k) =∑k∈ZZ

an−1(k)∑j∈ZZ

a(j)f(2n· − 2k− j).

It follows that

Qnaf =

∑j∈ZZ

(∑k∈ZZ

an−1(k)a(j − 2k))f(2n· − j) =

∑j∈ZZ

an(j)f(2n· − j).

This completes the induction procedure.

Theorem 2.1. Let a be a finitely supported sequence on ZZ such that∑

j∈ZZ a(j) = 2,and let Q = Qa be the cascade operator associated with a. Suppose u is a compactlysupported function in Lp(IR) (1 ≤ p ≤ ∞), u satisfies the Strang-Fix conditions of order 1,and the shifts of u are stable. If there exists a function φ ∈ Lp(IR) (a continuous functionφ in the case p = ∞) such that

limn→∞

‖Qnu− φ‖p = 0,

then for any compactly supported function v ∈ Lp(IR) satisfying the Strang-Fix conditionsof order 1 we also have

limn→∞

‖Qnv − φ‖p = 0.

Proof. For n = 0, 1, 2, . . ., let

fn :=∑j∈ZZ

bn(j)u(2n · − j) and gn :=∑j∈ZZ

bn(j)v(2n · − j),

where

bn(j) :=1h

∫ (j+1)h

jh

φ(x) dx with h =12n.

Since both u and v satisfy the Strang-Fix conditions of order 1, we have

limn→∞

‖φ− fn‖p = 0 and limn→∞

‖φ− gn‖p = 0.

Observe that

Qnu =∑j∈ZZ

an(j)u(2n · − j) and Qnv =∑j∈ZZ

an(j)v(2n · − j),

6

where an (n = 1, 2, . . .) are the sequences given in (2.2). Thus, we obtain

Qnu− fn =∑j∈ZZ

[an(j)− bn(j)

]u(2n · − j).

Since the shifts of u are stable, there exists a constant C1 > 0 such that

‖an − bn‖p ≤ C1‖(fn −Qnu)(2−n·)‖p = 2n/pC1‖fn −Qnu‖p.

Furthermore, there exists a constant C2 > 0 such that

‖(gn −Qnv)(2−n·)‖p =∥∥∥∥∑

j∈ZZ

[an(j)− bn(j)

]v(· − j)

∥∥∥∥p

≤ C2‖an − bn‖p.

Combining the above two estimates together, we see that there exists a constant C > 0such that

‖gn −Qnv‖p ≤ C ‖fn −Qnu‖p.

Therefore we have

‖φ−Qnv‖p ≤ ‖φ− gn‖p + ‖gn −Qnv‖p

≤ ‖φ− gn‖p + C‖fn −Qnu‖p

≤ ‖φ− gn‖p + C(‖φ− fn‖p + ‖φ−Qnu‖p

).

But as n→∞, ‖φ− gn‖p → 0, ‖φ− fn‖p → 0, and ‖φ−Qnu‖p → 0; hence we concludethat limn→∞ ‖Qnv − φ‖p = 0.

Let φ be the normalized solution to the refinement equation (1.2). Then Qaφ = φ.Suppose φ ∈ Lp(IR) (φ is continuous in the case p = ∞) and the shifts of φ are stable. InTheorem 2.1 we may choose u to be φ. Consequently, the cascade algorithm associatedwith mask a converges to φ in the Lp norm, provided the shifts of φ are stable.

For ε ∈ ZZ, let Aε be the linear operator on `0(ZZ) given by

Aεv(α) =∑β∈ZZ

a(ε+ 2α− β)v(β), α ∈ ZZ, v ∈ `0(ZZ). (2.3)

Suppose α = ε1 + 2ε2 + · · ·+ 2n−1εn + 2nγ, where ε1, . . . , εn, γ ∈ ZZ. Then

(an∗v)(α) = (Aεn· · ·Aε1v)(γ). (2.4)

This can be proved by induction on n. For n = 1, we have

(a1∗v)(α) =∑β∈ZZ

a(α− β)v(β) =∑β∈ZZ

a(ε1 + 2γ − β)v(β) = (Aε1v)(γ).

7

Hence, (2.4) is true for n = 1. Suppose n > 1 and (2.4) is valid for n−1. By (2.2) we have

(an∗v)(α) =∑β∈ZZ

v(β)an(α− β) =∑β∈ZZ

∑η∈ZZ

v(β)an−1(η)a(α− β − 2η).

Write α = ε1 + 2α1, where α1 := ε2 + · · ·+ 2n−2εn + 2n−1γ. Consequently,

(an∗v)(α) =∑η∈ZZ

∑β∈ZZ

an−1(η)v(β)a(ε1 + 2α1 − β − 2η) =∑η∈ZZ

an−1(η)(Aε1v)(α1 − η).

By the induction hypothesis we obtain

(an∗v)(α) =(an−1∗(Aε1v)

)(α1) = (Aεn

· · ·Aε2)(Aε1v)(γ) = (Aεn· · ·Aε2Aε1v)(γ).

This completes the induction procedure.As a consequence of (2.4), we have the following identity for 1 ≤ p ≤ ∞:

‖an∗v‖p =

( ∑ε1,...,εn∈0,1

‖Aεn · · ·Aε1v‖pp

)1/p

, v ∈ `p(ZZ). (2.5)

Indeed, for 1 ≤ p <∞ we have

‖an∗v‖pp =

∑α∈ZZ

|(an∗v)(α)|p =∑

ε1,...,εn∈0,1

∑γ∈ZZ

∣∣(an∗v)(ε1 + 2ε2 + · · ·+ 2n−1εn + 2nγ)∣∣p.

In light of (2.4), it follows that

‖an∗v‖pp =

∑ε1,...,εn∈0,1

∑γ∈ZZ

∣∣(Aεn· · ·Aε1v)(γ)

∣∣p =∑

ε1,...,εn∈0,1

‖Aεn· · ·Aε1v‖p

p.

This verifies (2.5) for 1 ≤ p <∞. For the case p = ∞ we have

‖an∗v‖∞ = maxε1,...,εn∈0,1

‖Aεn· · ·Aε1v‖∞.

A necessary condition for the Lp convergence of the cascade algorithm associated witha is that a satisfies the basic sum rule:∑

j∈ZZ

a(2j) =∑j∈ZZ

a(2j + 1) = 1. (2.6)

In order to prove this statement we choose f to be the characteristic function of the interval[0, 1) and consider the cascade algorithm Qn

af (n = 1, 2, . . .). If Qnaf converges to some

φ in the Lp norm, then Qna(Qaf) also converges to φ in the Lp norm. Hence, Qaf also

satisfies the Strang-Fix conditions of order 1. Consequently,

0 = Qaf(2π) = H(π)f(π),

8

where H is the trigonometric polynomial given in (1.3). Since f(π) 6= 0, it follows thatH(π) = 0. Therefore, ∑

j∈ZZ

a(j)(−1)j = 0.

This in connection with∑

j∈ZZ a(j) = 2 gives (2.6), as desired.Suppose a is supported on [0, N ], where N is a positive integer. It is easily seen that,

for j ≤ 0 and k ≥ N − 1, `([j, k]) is invariant under both A0 and A1. Let V be thelinear space of those elements v ∈ `([0, N ]) for which

∑k∈ZZ v(k) = 0. We claim that V is

invariant under both A0 and A1 if and only if a satisfies the basic sum rule. Indeed, if asatisfies the basic sum rule, then for ε = 0, 1 and v ∈ `0(ZZ) we have

∑j∈ZZ

Aεv(j) =∑j∈ZZ

∑k∈ZZ

a(ε+ 2j − k)v(k) =∑k∈ZZ

[∑j∈ZZ

a(ε+ 2j − k)]v(k) =

∑k∈ZZ

v(k).

Hence, v ∈ V implies Aεv ∈ V for ε = 0, 1. In other words, V is invariant under both A0

and A1. Conversely, suppose V is invariant under A0. Since ∇δ ∈ V , we have A0(∇δ) ∈ V .Hence, ∑

j∈ZZ

(A0(∇δ)

)(j) = 0, i .e.,

∑j∈ZZ

∑k∈ZZ

a(2j − k)∇δ(k) = 0.

It follows that ∑j∈ZZ

a(2j)−∑j∈ZZ

a(2j − 1) = 0.

This show that a satisfies the basic sum rule.

Theorem 2.2. The cascade algorithm associated with a converges in the Lp norm if andonly if the following two conditions are satisfied:(a) V is invariant under both A0 and A1;(b) ρp(A0|V , A1|V ) < 21/p.

Proof. Suppose the cascade algorithm associated with a converges to φ in Lp(IR). Thena satisfies the basic sum rule. Hence, V is invariant under both A0 and A1.

Let f be the hat function given by f(x) = max1 − |x|, 0, x ∈ IR. Let fn := Qnaf ,

n = 1, 2, . . .. Then we have ‖fn − φ‖p → 0 as n→∞. By (2.1) we have

fn =∑j∈ZZ

an(j)f(2n· − j), n = 1, 2, . . . ,

where an (n = 1, 2, . . .) are the sequences given by (2.2). It follows that

fn − fn(· − 1/2n) =∑j∈ZZ

(∇an)(j)f(2n· − j).

Since f is the hat function, there exists a positive constant C such that

2−n/p‖∇an‖p ≤ C‖fn − fn(· − 1/2n)‖p.

9

We observe that

‖fn − fn(· − 1/2n)‖p ≤ ‖fn − φ‖p + ‖φ− φ(· − 1/2n)‖p + ‖φ(· − 1/2n)− fn(· − 1/2n)‖p.

Consequently,lim

n→∞‖fn − fn(· − 1/2n)‖p = 0.

Hence, we obtainlim

n→∞2−n/p‖∇an‖p = 0. (2.7)

Let A := A0|V , A1|V . Fix a vector norm on V . Then there exist two positiveconstants C1 and C2 such that

C1‖∇an‖p ≤ ‖An‖p ≤ C2‖∇an‖p, 1 ≤ p ≤ ∞. (2.8)

In order to prove (2.8) we observe that ∇δj : j = 0, . . . , N − 1 is a basis for V . Hence,there exist two positive constants C3 and C4 such that

C3

N−1∑j=0

‖B(∇δj)‖p ≤ ‖B‖ ≤ C4

N−1∑j=0

‖B(∇δj)‖p

for every linear operator B on V . Recall that

‖An‖pp =

∑ε1,...,εn∈0,1

‖(Aεn· · ·Aε1)|V ‖p, 1 ≤ p <∞.

Moreover, by (2.5) we get∑ε1,...,εn∈0,1

∥∥(Aεn · · ·Aε1)(∇δj)∥∥p

p= ‖an∗(∇δj)‖p

p = ‖(∇an)∗δj‖pp = ‖∇an‖p

p ∀ j ∈ ZZ.

For the case p = ∞, we have ‖An‖∞ = maxε1,...,εn∈0,1 ‖(Aεn· · ·Aε1)|V ‖∞ and

maxε1,...,εn∈0,1

∥∥(Aεn· · ·Aε1)(∇δj)

∥∥∞ = ‖an∗(∇δj)‖∞ = ‖∇an‖∞ ∀ j ∈ ZZ.

Hence, (2.8) is valid for 1 ≤ p ≤ ∞.Combining (2.8) and (2.7) together, we obtain

limn→∞

2−n/p‖An‖p = 0.

Recall thatinfn≥1

‖An‖1/np = lim

n→∞‖An‖1/n

p = ρp(A0|V , A1|V ).

Consequently,

2−n/p‖An‖p =(2−1/p‖An‖1/n

p

)n ≥ (2−1/pρp(A0|V , A1|V ))n.

10

The above estimates tell us that

limn→∞

(2−1/pρp(A0|V , A1|V )

)n = 0.

Therefore, 2−1/pρp(A0|V , A1|V ) < 1. This completes the proof of the necessity part.Let us establish the sufficiency part of the theorem. Suppose f is a compactly sup-

ported function in Lp(IR) (f is continuous in the case p = ∞) satisfying the Strang-Fixconditions of order 1. Observe that Qn+1

a f − Qnaf = Qn

ag, where g := Qaf − f . Forx ∈ [0, 1), let vx(j) := g(x + j), j ∈ ZZ. We claim that

∑j∈ZZ vx(j) = 0 for almost every

x ∈ [0, 1). Since f satisfies the Strang-Fix conditions of order 1, we have∑

j∈ZZ f(x+j) = 1for almost every x ∈ ZZ. By condition (a), the mask a satisfies the basic sum rule. Hence,for almost every x ∈ IR,∑

j∈ZZ

Qaf(x+ j) =∑j∈ZZ

∑k∈ZZ

a(k)f(2x+ 2j − k)

=∑k∈ZZ

[∑j∈ZZ

a(k + 2j)]f(2x− k) =

∑k∈ZZ

f(2x− k) = 1.

Consequently,∑j∈ZZ

vx(j) =∑j∈ZZ

Qaf(x+ j)−∑j∈ZZ

f(x+ j) = 0 for a.e. x ∈ IR.

This justifies our claim.By (2.1) we have

Qnag(2

−nx) =∑j∈ZZ

an(j)g(x− j), x ∈ IR.

Hence, for 1 ≤ p <∞, we obtain

‖Qnag‖p

p =∫

IR

∣∣Qnag(y)

∣∣p dy = 2−n

∫IR

∣∣Qnag(2

−nx)∣∣p dx

= 2−n

∫IR

∣∣∣∣∑j∈ZZ

an(j)g(x− j)∣∣∣∣p dx = 2−n

∑k∈ZZ

∫[0,1)+k

∣∣∣∣∑j∈ZZ

an(j)g(x− j)∣∣∣∣p dx

= 2−n

∫[0,1)

∑k∈ZZ

∣∣∣∣∑j∈ZZ

an(j)g(x+ k − j)∣∣∣∣p dx = 2−n

∫[0,1)

‖an∗vx‖pp dx.

It follows from (2.5) that

‖an∗v‖p ≤ ‖An‖p‖v‖p ∀ v ∈ `p(ZZ).

By condition (b), ρ := ρp(A0|V , A1|V ) < 21/p. Hence, ρ < η21/p for some η, 0 < η < 1.Thus, there exists a positive constant C such that ‖An‖p ≤ C(η21/p)n for all n ∈ IN.Consequently,

‖an∗vx‖p ≤ C(η21/p)n‖vx‖p.

11

Note that∫[0,1)

‖vx‖pp dx = ‖g‖p

p. Therefore,

‖Qnag‖p

p ≤ 2−nCp(η21/p)pn‖g‖pp = Cpηnp‖g‖p

p.

It follows that‖Qn+1

a f −Qnaf‖p = ‖Qn

ag‖p ≤ Cηn‖g‖p.

Similarly, for the case p = ∞, we have

‖Qn+1a f −Qn

af‖∞ = ‖Qnag‖∞ ≤ Cηn‖g‖∞.

Since 0 < η < 1, the above estimates tell us that (Qnaf)n=1,2,... converges in Lp(IR). In

other words, the cascade algorithm associated with a converges in the Lp norm.

In what follows, we use ω(f, h)p to denote the Lp modulus of continuity of f :

ω(f, h)p := sup|t|≤h

‖f − f(· − t)‖p, h > 0.

For 1 ≤ p ≤ ∞ and 0 < µ ≤ 1, we use Lip(µ,Lp(IR)) to denote the linear space of allfunctions f in Lp(IR) for which

supt>0

ω(f, t)p/tµ <∞.

In particular, if p = ∞, then Lip(µ,L∞(IR)) is the Lipschitz space Lipµ (see [8, p. 51]).Suppose the cascade algorithm associated with mask a converges to φ in Lp(IR). By

Theorem 2.2 we have ρp(A0|V , A1|V ) < 21/p. Hence, there exists some η, 0 < η < 1, suchthat ρp(A0|V , A1|V ) < η21/p. Consequently, there exists a positive constant C1 such that‖An‖p ≤ C1(η21/p)n ∀n ∈ IN. This in connection with (2.8) gives ‖∇an‖p ≤ C2(η21/p)n,where C2 is a constant independent of n. Note that

φ = Qnaφ =

∑j∈ZZ

an(j)φ(2n · − j).

It follows thatφ− φ(· − 1/2n) =

∑j∈ZZ

(∇an)(j)φ(2n · − j).

Therefore, we obtain

‖φ− φ(· − 1/2n)‖p ≤ C32−n/p‖∇an‖p ≤ C2C3ηn ∀n ∈ IN,

where C3 is a constant independent of n. Let µ := − log2 η > 0. Then η = 2−µ. Suppose

h =12n

+d1

2n+1+

d2

2n+2+ · · · ,

12

where d1, d2, . . . ∈ 0, 1. By what has been proved, we have

‖φ− φ(· − h)‖p ≤ C2C3

∞∑m=n

2−mµ ≤ Chµ,

where C is a positive constant independent of h. This shows that φ ∈ Lip(µ,Lp(IR)) forsome µ > 0, whenever the corresponding cascade algorithm converges in Lp(IR).

In the rest of this section we will discuss the L2 convergence of cascade algorithms.This problem was investigated in terms of subdivision operators by Villemoes [23], andby Goodman, Micchelli, and Ward [9]. More generally, Zhou [24] showed that the 2-normjoint spectral radius of a finite collection of square matrices is equal to the spectral radiusof a certain finite matrix derived from the given matrices.

Suppose a ∈ `0(ZZ). For n = 1, 2, . . ., let an be the sequences given in (2.2). Let Ta bethe transition operator on `0(ZZ) given by

Tav(j) :=∑k∈ZZ

a(2j − k)v(k), j ∈ ZZ, v ∈ `0(ZZ).

Clearly, Ta is the same as the linear operator A0 defined in (2.3). Hence, by (2.4) we havethe following formula:

Tna v(j) = (an∗v)(2nj) =

∑k∈ZZ

an(2nj − k)v(k), j ∈ ZZ, n ∈ IN. (2.9)

For two functions f and g in L2(IR), we define

f g(x) :=∫

IR

f(x+ y)g(y) dy, x ∈ IR.

Evidently, f g is continuous and ‖f g‖∞ ≤ ‖f‖2‖g‖2. Similarly, for two sequences uand v in `2(ZZ), we define

u v(j) :=∑k∈ZZ

u(j + k)v(k), j ∈ ZZ.

Also, ‖u v‖∞ ≤ ‖u‖2‖v‖2.Let b := a a/2. We claim that

limn→∞

‖an∗v‖1/n2 =

√2ρ(Tb|W ) , v ∈ `0(ZZ), (2.10)

where W is the minimal invariant subspace of the transition operator Tb generated byw := v v.

For n = 1, 2, . . ., let bn be the sequences given by

b1 = b and bn(j) =∑k∈ZZ

bn−1(k)b(2j − k), j ∈ ZZ.

13

Let vn := an∗v and wn := vn vn. It can be easily proved that wn = 2nbn∗w. Hence, itfollows from (2.9) that

2nTnb w(j) = wn(2nj), j ∈ ZZ.

Since wn = vn vn, we have

2n‖Tnb w‖∞ ≤ ‖wn‖∞ ≤ ‖vn‖2

2.

On the other hand,

2nTnb w(0) = wn(0) =

∑k∈ZZ

vn(k)vn(k) = ‖vn‖22.

Consequently,‖vn‖2

2 = 2n‖Tnb w‖∞.

Therefore,

limn→∞

‖an∗v‖2/n2 = lim

n→∞‖vn‖2/n

2 = limn→∞

(2n‖Tn

b w‖∞)1/n = 2ρ(Tb|W ).

This verifies (2.10).

Theorem 2.3. Let a be a sequence supported on [0, N ] and satisfying∑

j∈ZZ a(j) = 2.Then the cascade algorithm associated with a converges in the L2 norm if and only if thefollowing two conditions are satisfied:(a) a satisfies the basic sum rule;(b) ρ(Tb|W ) < 1, where W := w ∈ `([−N,N ]) :

∑j∈ZZ w(j) = 0.

Proof. Suppose conditions (a) and (b) are satisfied. Let v := ∇δ. Then w := v∗v belongsto W . In light of (2.10) we have

limn→∞

‖∇an‖1/n2 = lim

n→∞‖an∗(∇δ)‖1/n

2 ≤√

2ρ(Tb|W ) <√

2.

Let A := A0|V , A1|V , where V := v ∈ `([0, N ]) :∑

j∈ZZ v(j) = 0. It follows from (2.8)that

ρ2(A0|V , A1|V ) = limn→∞

‖An‖1/n2 = lim

n→∞‖∇an‖1/n

2 .

Hence, ρ2(A0|V , A1|V ) < 21/2. Therefore, the cascade algorithm associated with a con-verges in L2(IR), by Theorem 2.2.

Conversely, suppose the cascade algorithm associated with a converges to φ in L2(IR).Then a satisfies the basic sum rule. Consequently, b satisfies the basic sum rule. Let φ0

be the characteristic function of the interval [0, 1), and let φn := Qnaφ0, n = 1, 2, . . .. Then

‖φn − φ‖2 → 0 as n → ∞. Moreover, let gn := φn φn (n = 0, 1, . . .) and g := φ φ. Itis easily seen that gn = Qn

b g0, n = 1, 2, . . .. We have

‖gn − g‖∞ = ‖φn φn − φ φ‖∞≤ ‖φn (φn − φ)‖∞ + ‖(φn − φ) φ‖∞≤ (‖φn‖2 + ‖φ‖2)‖φn − φ‖2.

Hence, Qnb g0 converges to g in L∞(IR). Therefore, we must have ρ(Tb|W ) < 1.

We remark that condition (b) holds true if and only if 1 is a simple eigenvalue ofthe transition matrix (b(2j − k))−N≤j,k≤N and all the other eigenvalues are less than 1 inmodulus.

14

§3. Applications to Wavelet Analysis

In this section, the basic theory on cascade algorithms developed in the last sectionwill be used to give a unified treatment of orthogonal wavelets, biorthogonal wavelets, andfundamental refinable functions.

To begin with, we discuss orthogonal refinable functions. It was Daubechies who firstconstructed smooth orthogonal wavelets with compact support (see [4]). The followingtheorem gives a characterization for a refinable function to be orthogonal.

Theorem 3.1. Let φ be the normalized solution of the refinement equation (1.2), andlet H be the trigonometric polynomial given in (1.3). Then φ(· − j) : j ∈ ZZ forms anorthonormal system in L2(IR) if and only if the following two conditions are satisfied:(a) |H(ξ)|2 + |H(ξ + π)|2 = 1 for all ξ ∈ IR;(b) The cascade algorithm associated with a converges in the L2 norm.

Proof. Let us first establish the necessity part of the theorem. Suppose the shifts of φare orthonormal. Then

∑k∈ZZ |φ(ξ+2πk)|2 = 1 for all ξ ∈ IR. Since φ(ξ) = H(ξ/2)φ(ξ/2),

ξ ∈ IR, we deduce that

1 =∑k∈ZZ

|φ(ξ + 2πk)|2 =∑k∈ZZ

∣∣H(ξ/2 + πk)φ(ξ/2 + πk)∣∣2

= |H(ξ/2)|2∑k∈ZZ

|φ(ξ/2 + 2πk)|2 + |H(ξ/2 + π)|2∑k∈ZZ

|φ(ξ/2 + π + 2πk)|2

= |H(ξ/2)|2 + |H(ξ/2 + π)|2.

This verifies condition (a). Moreover, if the shifts of φ are orthonormal, then the shifts ofφ are stable. Hence, the cascade algorithm associated with a converges in the L2 norm,by Theorem 2.1.

For the sufficiency part we observe that condition (a) implies invariance of orthonor-mality under the cascade operator Qa. Suppose f is a compactly supported function inL2(IR) and the shifts of f are orthornormal. We wish to show that the shifts of Qaf arealso orthonormal. For this purpose it suffices to show that

∑k∈ZZ |(Qaf ) (ξ + 2πk)|2 = 1

for all ξ ∈ IR. Since (Qaf ) (ξ) = H(ξ/2)f(ξ/2), ξ ∈ IR, we have∑k∈ZZ

|(Qaf ) (ξ + 2πk)|2 =∑k∈ZZ

∣∣H(ξ/2 + πk)f(ξ/2 + πk)∣∣2.

The right of the above equation is equal to

|H(ξ/2)|2∑k∈ZZ

|f(ξ/2 + 2πk)|2 + |H(ξ/2 + π)|2∑k∈ZZ

|f(ξ/2 + π + 2πk)|2.

But∑

k∈ZZ |f(ξ/2 + 2πk)|2 =∑

k∈ZZ |f(ξ/2 + π + 2πk)|2 = 1. Hence,∑k∈ZZ

∣∣H(ξ/2 + πk)f(ξ/2 + πk)∣∣2 = |H(ξ/2)|2 + |H(ξ/2 + π)|2 = 1,

15

where condition (a) has been used to derive the last equality. This shows that the shiftsof Qaf are orthonormal.

Let φ0 be the characteristic function of the interval [0, 1), and let φn := Qnaφ0 for

n = 1, 2, . . .. Clearly, the shifts of φ0 are orthonormal. By what has been proved, theshifts of φn are also orthonormal. In other words,

〈φn, φn(· − j)〉 = δ0j , j ∈ ZZ.

Since ‖φn − φ‖2 → 0 as n→∞, we obtain

〈φ, φ(· − j)〉 = limn→∞

〈φn, φn(· − j)〉 = δ0j , j ∈ ZZ.

Consequently, the shifts of φ are orthonormal.

Suppose the mask a is supported on [0, N ]. Then the sequence b := a a/2 issupported on [−N,N ]. As we mentioned at the end of Section 2, the cascade algorithmassociated with a converges in the L2 norm if and only if 1 is a simple eigenvalue ofthe transition matrix (b(2j − k))−N≤j,k≤N and all the other eigenvalues are less than 1 inmodulus. Thus, Theorem 3.1 recovers a result of Lawton [16] on orthogonality of a refinablefunction in terms of the eigenvalue condition of the corresponding transition matrix.

Next, we investigate biorthogonal refinable functions. Let φ and φ be two compactlysupported functions in L2(IR). We say that the shifts of φ and φ are biorthogonal if

〈φ(· − j), φ(· − k)〉 = δjk, j, k ∈ ZZ.

By the Poisson summation formula, the shifts of φ and φ are biorthogonal if and only if

∑k∈ZZ

φ(ξ + 2kπ)ˆφ(ξ + 2kπ) = 1 ∀ ξ ∈ IR.

Suppose φ and φ are refinable:

φ =∑j∈ZZ

a(j)φ(2 · − j) and φ =∑j∈ZZ

b(j)φ(2 · − j), (3.1)

where a and b are finitely supported sequences on ZZ. By taking the Fourier transform ofboth sides of the refinement equations in (3.1), we obtain

φ(ξ) = H(ξ/2)φ(ξ/2) and ˆφ(ξ) = G(ξ/2)ˆφ(ξ/2), ξ ∈ IR,

whereH(ξ) :=

12

∑j∈ZZ

a(j)e−ijξ and G(ξ) :=12

∑j∈ZZ

b(j)e−ijξ, ξ ∈ IR. (3.2)

16

If the shifts of φ and φ are biorthogonal, then for ξ ∈ IR,

1 =∑k∈ZZ

φ(2ξ + 2kπ) ˆφ(2ξ + 2kπ)

=∑k∈ZZ

H(ξ + kπ)φ(ξ + kπ)G(ξ + kπ) ˆφ(ξ + kπ)

=∑

ε=0,1

H(ξ + επ)G(ξ + επ)∑k∈ZZ

φ(ξ + επ + 2kπ) ˆφ(ξ + επ + 2kπ)

= H(ξ)G(ξ) +H(ξ + π)G(ξ + π).

Theorem 3.2. The shifts of φ and φ are biorthogonal if and only if the following twoconditions are satisfied:(a) H(ξ)G(ξ) +H(ξ + π)G(ξ + π) = 1 for ξ ∈ IR;(b) The cascade algorithms associated with a and b converge in L2(IR).

Proof. We have proved necessity of condition (a). If the shifts of φ and φ are biorthogonal,then they are stable; hence, the cascade algorithms associated with a and b converge inL2(IR), by Theorem 2.1.

In order to establish sufficiency of conditions (a) and (b), we consider the cascadeoperators Qa and Qb given by

Qaf :=∑j∈ZZ

a(j)f(2 · − j) and Qbg :=∑j∈ZZ

b(j)g(2 · − j),

where f and g are compactly supported functions in L2(IR). If condition (a) holds true,and if the shifts of f and g are biorthogonal, then the shifts of Qaf and Qbg are alsobiorthogonal. Indeed, for ξ ∈ IR, the expression∑

k∈ZZ

Qaf(ξ + 2kπ) Qbg(ξ + 2kπ)

is equal to ∑ε=0,1

H(ξ + επ)G(ξ + επ)∑k∈ZZ

f(ξ + επ + 2kπ) g(ξ + επ + 2kπ).

Using biorthgonality of the shifts of f and g, we obtain∑k∈ZZ

Qaf(ξ + 2kπ) Qbg(ξ + 2kπ) = H(ξ)G(ξ) +H(ξ + π)G(ξ + π) = 1.

This justifies our claim.

17

Choose f0 and g0 to be the characteristic function of the interval [0, 1). Clearly, theshifts of f0 and g0 are biorthogonal. For n = 1, 2, . . ., let fn := Qn

af0 and gn := Qnb g0. By

what has been proved, the shifts of fn and gn are biorthogonal. In other words,

〈fn(· − j), gn(· − k)〉 = δjk, j, k ∈ ZZ.

By condition (b), ‖fn − φ‖2 → 0 and ‖gn − φ‖2 → 0 as n→∞. Hence, we obtain

〈φ(· − j), φ(· − k)〉 = limn→∞

〈fn(· − j), gn(· − k)〉 = δjk, j, k ∈ ZZ.

This shows that the shifts of φ and φ are biorthogonal.

In a slightly different form, Theorem 3.2 was established by Cohen and Daubechies[2]. But their proof relies on the fact that a trigonometric polynomial of one variable hasonly a finite number of zeros in the interval [0, 2π]. By contrast, the proof given here canbe easily extended to the multivariate case (see Section 5).

Finally, let us study refinable fundamental functions. A continuous function on IR issaid to be fundamental if

φ(0) = 1 and φ(j) = 0 ∀ j ∈ ZZ \ 0.

Suppose in addition that φ is compactly supported and satisfies the refinement equation

φ(x) =∑j∈ZZ

a(j)φ(2x− j), x ∈ IR.

Setting x = k in the above equation, we obtain φ(k) = a(2k). Hence,

a(0) = 1 and a(2k) = 0 ∀ k ∈ ZZ \ 0.

If a finitely supported sequence a satisfies the above conditions and∑

j∈ZZ a(2j + 1) = 1,then we say that a is interpolatory. Thus, in order for a continuous refinable functionwith compact support to be fundamental, the corresponding mask must be interpolatory.In general, this condition is not sufficient. For example, let φ(x) := max1 − |x|/3, 0,x ∈ IR. Then

φ(x) =12φ(2x+ 3) + φ(2x) +

12φ(2x− 3), x ∈ IR.

Hence, φ is refinable and the corresponding mask a is interpolatory. But φ is not funda-mental.

Theorem 3.3. If a is an interpolatory mask and the cascade algorithm associated witha converges uniformly, then the limit function φ is fundamental. Conversely, if φ is acontinuous fundamental function on IR with compact support and φ satisfies the refinement

18

equation with mask a, then a is interpolatory and the cascade algorithm associated witha converges uniformly.

Proof. If a is interpolatory and f is a continuous fundamental function on IR, then Qafis also fundamental. Indeed, for k ∈ ZZ we have

(Qaf)(k) =∑j∈ZZ

a(j)f(2k − j) = a(2k) = δ(k), k ∈ ZZ.

Let φn := Qnaφ0, where φ0 is the hat function given by φ0(x) := max1 − |x|, 0, x ∈ IR.

Note that φ0 is fundamental. Hence, for each n = 1, 2, . . ., φn is fundamental. If thecascade algorithm associated with a converges uniformly, then there exists a continuousfunction φ such that ‖φn − φ‖∞ → 0 as n→∞. It follows that

φ(k) = limn→∞

φn(k) = δ(k) ∀ k ∈ ZZ.

This shows that φ is fundamental.Conversely, suppose that φ is a continuous fundamental function on IR with compact

support and φ satisfies the refinement equation with mask a. Then a is interpolatory.Moreover, the shifts of φ are stable. Consequently, the cascade algorithm associated witha converges uniformly, by Theorem 2.1.

In [7] Deslauriers and Dubuc constructed a family of refinable fundamental functionswith increasing smoothness, which are related to the orthogonal refinable functions con-structed by Daubechies [4].

§4. Biorthogonal Wavelet Bases

Let φ and φ be two compactly supported functions in L2(IR). Suppose φ and φ satisfythe refinement equations in (3.1) with masks a and b, respectively. Let H and G be thetrigonometric polynomials given in (3.2). Suppose the shifts of φ and φ are biorthogonal.By Theorem 3.2, this is the case if and only if the cascade algorithms associated with aand b converge in the L2 norm and H(ξ)G(ξ) +H(ξ + π)G(ξ + π) = 1 for all ξ ∈ IR. It iseasily seen that the latter is equivalent to the following relation:∑

j∈ZZ

a(2k + j)b(j) = 2δ0k ∀ k ∈ ZZ. (4.1)

Since the cascade algorithms associated with a and b converge in the L2 norm, the masksa and b satisfy the basic sum rule. Moreover, there exists some µ > 0 such that φ and φbelong to Lip(µ,L2(IR)).

For µ > 0, we denote by Hµ(IR) the Sobolev space of all functions f ∈ L2(IR) suchthat ∫

IR

|f(ξ)|2(1 + |ξ|µ)2 dξ <∞.

For 0 < µ ≤ 1, it is easily seen that Hµ(IR) ⊆ Lip(µ,L2(IR)).

19

Let

ψ :=∑j∈ZZ

(−1)j b(1− j)φ(2 · − j) and ψ :=∑j∈ZZ

(−1)j a(1− j) φ(2 · − j). (4.2)

For j, k ∈ ZZ, let

ψjk :=√

2jψ(2j · − k) and ψjk :=

√2

jψ(2j · − k). (4.3)

It was proved by Cohen and Daubechies [2] (also see [3]) that (ψjk)j,k∈ZZ and (ψjk)j,k∈ZZ

are Riesz bases for L2(IR). In this section we shall give a self-contained proof for thisimportant result. The technique developed in this section can be used to deal with morecomplicated problems such as biorthogonal wavelet bases on bounded domains.

A sequence (fn)n=1,2,... in a Hilbert space H is called a Riesz sequence if there existtwo positive constants A and B such that

A

( ∞∑n=1

|cn|2)1/2

≤

∥∥∥∥∥∞∑

n=1

cnfn

∥∥∥∥∥ ≤ B

( ∞∑n=1

|cn|2)1/2

for all sequences (cn)n=1,2,... in `2. A Riesz sequence (fn)n=1,2,... is called a Riesz basisif additionally the linear span of fn : n = 1, 2, . . . is dense in H.

The inner product of two elements f and g in H is denoted by 〈f, g〉. A sequence(gn)n=1,2,... in H is called a Bessel sequence if there exists a positive constant B suchthat

∞∑n=1

∣∣〈f, gn〉∣∣2 ≤ B‖f‖2 ∀ f ∈ H.

We observe that ψ lies in Hµ(IR) for some µ > 0. Moreover, since b satisfies the basicsum rule, we have

∫IRψ(x) dx = 0. The following theorem tells us that (ψjk)j,k∈ZZ is a

Bessel sequence in L2(IR).

Theorem 4.1. Assume ψ ∈ Hµ(IR) for some µ > 0. If ψ furthermore is compactlysupported and

∫IRψ(x) dx = 0, then there is a positive constant B such that∑

j,k∈ZZ

∣∣〈f, ψjk〉∣∣2 ≤ B‖f‖2

2 ∀ f ∈ L2(IR),

where ψjk(x) := 2j/2ψ(2jx− k), x ∈ IR. In other words, (ψjk)j,k∈ZZ is a Bessel sequence inL2(IR).

This theorem was established by Villemoes [22]. Its proof will be given at the end ofthis section.

We claim that the sequences (ψjk)j,k∈ZZ and (ψjk)j,k∈ZZ are biorthogonal, that is,

〈ψjk, ψrs〉 = δjrδks ∀ j, k, r, s ∈ ZZ. (4.4)

20

To justify our claim we first verify (4.4) for the case j = r = 0. In this case we have

〈ψ0k, ψ0s〉 =⟨∑

α∈ZZ

(−1)α b(1− α)φ(2 · − 2k − α),∑β∈ZZ

(−1)β a(1− β) φ(2 · − 2s− β)⟩.

Note that

〈φ(2 · − 2k − α), φ(2 · − 2s− β)〉 =

1/2 if 2k + α = 2s+ β,0 otherwise.

It follows that

〈ψ0k, ψ0s〉 =12

∑α∈ZZ

a(2(s− k) + 1− α

)b(1− α) = δks,

where (4.1) has been used to derive the second equality. For j, k ∈ ZZ, let

φjk :=√

2jφ(2j · − k) and φjk :=

√2

jφ(2j · − k).

Then we have

〈ψ0k, φ0s〉 =⟨∑

α∈ZZ

(−1)αb(1− α)φ(2 · − 2k − α),∑β∈ZZ

b(β) φ(2 · − 2s− β)⟩

=12

∑α∈ZZ

(−1)αb(1− α) b(2k − 2s+ α) = 0.

Similarly, 〈φ0k, ψ0s〉 = 0. By using a simple change of variables we obtain

〈ψjk, ψjs〉 = δks, 〈ψjk, φjs〉 = 0, and 〈φjk, ψjs〉 = 0.

In particular, (4.4) is valid for j = r. For j > r, ψrs is a linear combination of φjn (n ∈ ZZ).Hence, 〈ψjk, ψrs〉 = 0 for j > r. Similarly, 〈ψjk, ψrs〉 = 0 for j < r. This verifies (4.4) forthe case j 6= r.

Thus, (ψjk)j,k∈ZZ and (ψjk)j,k∈ZZ are biorthogonal Bessel sequences in L2(IR). Thefollowing theorem tells us that they are Riesz sequences.

Theorem 4.2. Let (fn)n=1,2,... and (gn)n=1,2,... be two Bessel sequences in a Hilbert spaceH. If, in addition, they are biorthogonal, i.e., 〈fm, gn〉 = δmn, then both (fn)n=1,2,... and(gn)n=1,2,... are Riesz sequences.

Proof. Let (cn)n=1,2,... be a sequence in `2. For N ∈ IN, let uN :=∑N

n=1 cnfn. Then wehave

〈uN , uN 〉 =N∑

n=1

cn〈fn, uN 〉.

21

By the Schwarz inequality we deduce from the above equation that

‖uN‖4 ≤

(N∑

n=1

|cn|2)(

N∑n=1

∣∣〈fn, uN 〉∣∣2). (4.5)

But the sequence (fn)n=1,2,... is a Bessel sequence. Hence, there exists a constant A suchthat

∞∑n=1

∣∣〈fn, uN 〉∣∣2 ≤ A‖uN‖2, (4.6)

where A is independent of N . Combining (4.5) and (4.6) together, we obtain

‖uN‖4 ≤ A‖uN‖2N∑

n=1

|cn|2.

It follows that

‖uN‖2 ≤ AN∑

n=1

|cn|2. (4.7)

Since (cn)n=1,2,... is a sequence in `2, this shows that (uN )N=1,2,... converges to someelement u in H. Letting N →∞ in (4.7), we obtain

‖u‖2 ≤ A∞∑

n=1

|cn|2. (4.8)

We may write u as∑∞

n=1 cnfn. Clearly, biorthogonality of (fn)n=1,2,... and (gn)n=1,2,...

implies cn = 〈u, gn〉. Since (gn)n=1,2,... is a Bessel sequence, there exists a constant B suchthat

∞∑n=1

|cn|2 =∞∑

n=1

∣∣〈u, gn〉∣∣2 ≤ B‖u‖2. (4.9)

By (4.8) and (4.9) we conclude that (fn)n=1,2,... is a Riesz sequence. Similarly, (gn)n=1,2,...

is also a Riesz sequence.

We are in a position to establish the main result of this section.

Theorem 4.3. Let φ and φ be two compactly supported functions in L2(IR) such that theshifts of φ and φ are biorthogonal. Suppose φ and φ are the normalized solutions to therefinement equations in (3.1) with masks a and b, respectively. For j, k ∈ ZZ, let (ψjk)j,k∈ZZ

and (ψjk)j,k∈ZZ be defined as in (4.2) and (4.3). Then (ψjk)j,k∈ZZ is a Riesz basis for L2(IR).Moreover,

f =∑

j,k∈ZZ

〈f, ψjk〉ψjk, f ∈ L2(IR),

22

where the series converges in the L2 norm.

Proof. For j ∈ ZZ, let Pj be the linear operator on L2(IR) given by

Pjf :=∑k∈ZZ

〈f, φjk〉φjk, f ∈ L2(IR),

where φjk :=√

2jφ(2j · − k) and φjk :=

√2

jφ(2j · − k). Note that ‖Pj‖ = ‖P0‖ for all

j ∈ ZZ. By the hypothesis of the theorem, both φ and φ satisfy the Strang-Fix conditionsof order 1. Hence,

∑k∈ZZ φ(· − k) = 1 and

∫IRφ(x) dx = 1. Consequently, Pj preserves

constants, i.e., Pj1 = 1. Therefore, ‖Pnf − f‖2 → 0 as n → ∞. Moreover, ‖Pnf‖2 → 0as n → −∞. For this, one can first prove that limn→−∞ ‖Png‖2 = 0 for any continuousfunction g with compact support. For general f ∈ L2(IR) we argue as follows. Given ε > 0,there exists a compactly supported continuous function g such that ‖f − g‖2 ≤ ε/‖P0‖.It follows that ‖Pn(f − g)‖2 ≤ ε for all n. There exists a positive integer N such that‖Png‖2 ≤ ε for all n ≤ −N . Consequently, ‖Pnf‖2 ≤ 2ε for all n ≤ −N . This shows that‖Pnf‖2 → 0 as n→ −∞.

We claim thatP1f − P0f =

∑k∈ZZ

〈f, ψ0k〉ψ0k. (4.10)

In order to verify (4.10) we express ψ0k and ψ0k as follows:

ψ(x− k) =∑r∈ZZ

(−1)rb(1− r)φ(2x− 2k − r) =∑r∈ZZ

(−1)rb(1− r + 2k)φ(2x− r),

ψ(x− k) =∑s∈ZZ

(−1)sa(1− s)φ(2x− 2k − s) =∑s∈ZZ

(−1)sa(1− s+ 2k)φ(2x− s).

Hence,∑k∈ZZ

〈f, ψ0k〉ψ0k =∑r∈ZZ

∑s∈ZZ

∑k∈ZZ

(−1)r+sa(1− s+ 2k)b(1− r + 2k)〈f, φ(2 · − s)〉φ(2 · − r).

Moreover,φ(x− k) =

∑r∈ZZ

a(r)φ(2x− 2k − r) =∑r∈ZZ

a(r − 2k)φ(2x− r),

φ(x− k) =∑s∈ZZ

b(s)φ(2x− 2k − s) =∑s∈ZZ

b(s− 2k)φ(2x− s).

Hence, ∑k∈ZZ

〈f, φ0k〉φ0k =∑r∈ZZ

∑s∈ZZ

∑k∈ZZ

a(r − 2k)b(s− 2k)〈f, φ(2 · − s)〉φ(2 · − r).

Thus, (4.10) will be established if we can prove∑k∈ZZ

(−1)r+sa(1− s+ 2k)b(1− r + 2k) +∑k∈ZZ

a(r − 2k)b(s− 2k) = 2δrs. (4.11)

23

The proof of (4.11) will be divided into two cases: r + s is odd and r + s is even. If r + sis odd, letting k → (r + s− 1)/2− k in the second sum of (4.11), we obtain∑

k∈ZZ

a(r − 2k)b(s− 2k) =∑k∈ZZ

a(1− s+ 2k)b(1− r + 2k).

This verifies (4.11) for the case when r+s is odd. If r+s is even, letting k → (r+s)/2−kin the first sum of (4.11), we obtain∑

k∈ZZ

a(1− s+ 2k)b(1− r + 2k) =∑k∈ZZ

a(r + 1− 2k)b(s+ 1− 2k).

Consequently, the left-hand side of (4.11) is equal to∑k∈ZZ

a(r + 1− 2k)b(s+ 1− 2k) +∑k∈ZZ

a(r − 2k)b(s− 2k) =∑j∈ZZ

a(r + j)b(s+ j) = 2δrs,

where (4.1) has been used to derive the last equality. This verifies (4.11) for the case whenr + s is even.

Let n be a positive integer. It follows from (4.10) that

Pn+1f − P−nf =n∑

j=−n

∑k∈ZZ

〈f, ψjk〉ψjk, f ∈ L2(IR).

But ‖(Pn+1f − P−nf) − f‖2 → 0 as n → ∞. Hence, the linear span of ψjk : j, k ∈ ZZis dense in L2(IR). By Theorems 4.1 and 4.2, (ψjk)j,k∈ZZ is a Riesz sequence. Therefore,(ψjk)j,k∈ZZ is a Riesz basis for L2(IR). Moreover, by what has been proved, we have

f = limn→∞

(Pn+1f − P−nf) = limn→∞

n∑j=−n

∑k∈ZZ

〈f, ψjk〉ψjk =∑

j,k∈ZZ

〈f, ψjk〉ψjk, f ∈ L2(IR),

where the series converges in the L2 norm. This completes the proof of the theorem.

It remains to prove Theorem 4.1. Its proof requires two auxiliary lemmas. The fol-lowing lemma extends [22, Lemma 3.1] from the case p = 2 to the general case 1 ≤ p ≤ ∞.

Lemma 4.1. Suppose f ∈ Lip (µ,Lp(IR)) for some µ > 0 and g ∈ Lq(IR), 1/p+ 1/q = 1.Assume that g has compact support and

∫IRg(x) dx = 0. Let

bn(k) :=⟨2−n/pf(2−n ·), g(· − k)

⟩, k ∈ ZZ, n = 0, 1, 2, . . . .

Then ‖bn‖p ≤ C2−nµ, where C is a constant depending only on g, f and µ.

Proof. Let fn(x) := 2−n/pf(2−nx), x ∈ IR. By Theorem 2.4 in [8, Chap. 6] there exists aconstant C1 and a function un ∈ Lp(IR) such that u′n ∈ Lp(IR),

‖fn − un‖p ≤ C1ω(fn, 1)p and ‖u′n‖p ≤ C1ω(fn, 1)p, (4.12)

24

where the constant C1 is independent of n. For t ∈ IR we have

‖fn − fn(· − t)‖p = ‖2−n/pf(2−n ·)− 2−n/pf(2−n · − 2−nt)‖p = ‖f(·)− f(· − 2−nt)‖p.

Since f lies in Lip (µ,Lp(IR)), there exists a constant C2 such that

‖f(·)− f(· − 2−nt)‖p ≤ C2(2−n|t|)µ.

Hence,ω(fn, 1)p = sup

|t|≤1

‖fn − fn(· − t)‖p ≤ C22−nµ. (4.13)

We may write bn(k) = cn(k) + dn(k), where

cn(k) := 〈fn − un, g(· − k)〉 and dn(k) := 〈un, g(· − k)〉, k ∈ ZZ.

Since g ∈ Lq(IR) is compactly supported, by [13, Theorem 3.1], there exists a constant C3

(independent of n) such that

‖cn‖p ≤ C3‖fn − un‖p. (4.14)

Let h(x) :=∫ x

−∞ g(y) dy. Since∫IRg(y) dy = 0, h is compactly supported. Moreover,

h′ = g. Consequently,

dn(k) = 〈un, h′(· − k)〉 = −〈u′n, h(· − k)〉.

Hence, there exists a constant C4 (independent of n) such that

‖dn‖p ≤ C4‖u′n‖p. (4.15).

Combining the estimates (4.12)–(4.15) together, we see that there exists a constant C suchthat ‖bn‖p ≤ C2−nµ for all n ∈ IN0 := IN∪0. This completes the proof of the lemma.

Lemma 4.2. Suppose a ∈ `2(ZZ) and w is a sequence supported on [−K2n,K2n], whereK ∈ IN and n ∈ IN0. Let

u(k) :=∑j∈ZZ

a(2nk − j)w(j) and v(k) :=∑j∈ZZ

w(k − 2nj)a(j), k ∈ ZZ.

Then‖u‖2 ≤

√2K + 1 ‖a‖2‖w‖2 and ‖v‖2 ≤

√2K + 1 ‖a‖2‖w‖2.

Proof. Since w is supported on [−K2n,K2n], by the Schwarz inequality we have

|u(k)|2 ≤2nK∑

j=−2nK

∣∣a(2nk − j)∣∣2 2nK∑

j=−2nK

|w(j)|2, k ∈ ZZ.

25

It follows that ∑k∈ZZ

|u(k)|2 ≤∑k∈ZZ

2nK∑j=−2nK

|a(2nk − j)|2∑j∈ZZ

|w(j)|2.

But ∑k∈ZZ

2nK∑j=−2nK

|a(2nk − j)|2 ≤ (2K + 1)∑k∈ZZ

|a(k)|2.

Hence, ‖u‖2 ≤√

2K + 1‖a‖2‖w‖2.In order to prove the second inequality we write k = 2nr + s, where 0 ≤ s < 2n and

r ∈ ZZ. Fix s for the time being. Since w is supported on [−K2n,K2n], w(2nr+s−2nj) 6= 0only if −K2n ≤ 2nr + s − 2nj ≤ K2n, or −K + r + 2−ns ≤ j ≤ K + r + 2−ns. By theSchwarz inequality we have

|v(2nr+s)|2 ≤∑

−K+r+2−ns≤j≤K+r+2−ns

|w(2nr+s−2nj)|2∑

−K+r+2−ns≤j≤K+r+2−ns

|a(j)|2.

It follows that

∑r∈ZZ

|v(2nr + s)|2 ≤∑j∈ZZ

|w(s− 2nj)|2[∑

r∈ZZ

∑−K+r+2−ns≤j≤K+r+2−ns

|a(j)|2].

≤∑j∈ZZ

|w(s− 2nj)|2 (2K + 1)∑j∈ZZ

|a(j)|2.

Hence,

‖v‖22 =

2n−1∑s=0

∑r∈ZZ

|v(2nr + s)|2 ≤ (2K + 1)‖a‖22

2n−1∑s=0

∑j∈ZZ

|w(s− 2nj)|2.

This shows that ‖v‖2 ≤√

2K + 1 ‖a‖2‖w‖2.

Proof of Theorem 4.1 Let h be the orthonormal Haar wavelet, defined by h(x) = 1for 0 ≤ x < 1/2, h(x) = −1 for 1/2 ≤ x < 1, and h(x) = 0 elsewhere. Defininghjk(x) = 2j/2h(2jx − k), j, k ∈ ZZ, we see that (hjk)j,k∈ZZ is an orthonormal basis forL2(IR). Decompose f =

∑j,k∈ZZ aj(k)hjk and then compute bj(k) := 〈f, ψjk〉, j, k ∈ ZZ.

We have

〈hrs, ψjk〉 =∫

IR

2r/2h(2rx− s) 2j/2ψ(2jx− k) dx =∫

IR

2−n/2h(2−nx− s)ψ(x− k) dx,

where n := j − r. For n ∈ IN0, define

wn1(k) :=∫

IR

2−n/2h(2−nx)ψ(x− k) dx, k ∈ ZZ.

26

Then for j ≥ r we have〈hrs, ψjk〉 = wn1(k − 2ns).

Similarly, for n ∈ IN0 define

wn2(k) :=∫

IR

h(x+ k) 2−n/2ψ(2−nx) dx, k ∈ ZZ.

Then for j < r we have〈hrs, ψjk〉 = wn2(2nk − s),

where n := r − j. Hence, for j, k ∈ ZZ we have

bj(k) = 〈f, ψjk〉 =∑

r,s∈ZZ

ar(s)〈hrs, ψjk〉

=∞∑

n=0

∑s∈ZZ

aj−n(s)wn1(k − 2ns) +∞∑

n=1

∑s∈ZZ

aj+n(s)wn2(2nk − s).

Note that there exists a positive integer K such that both wn1 and wn2 are supported on[−K2n,K2n]. By Lemma 4.2, there exists a positive constant C1 such that

‖bj‖2 ≤ C1

[ ∞∑n=0

‖aj−n‖2‖wn1‖2 +∞∑

n=1

‖aj+n‖2‖wn2‖2

].

Note that ψ ∈ Hµ(IR) ⊆ Lip(µ,L2(IR)), where µ > 0. Hence, by Lemma 4.1 we obtain

‖wn1‖2 ≤ C22−nµ and ‖wn2‖2 ≤ C22−nµ,

where C2 is a constant independent of n. Consequently,

‖bj‖2 ≤ C1C2

[ ∞∑n=0

‖aj−n‖22−nµ +∞∑

n=1

‖aj+n‖22−nµ

].

It follows that ∑j∈ZZ

‖bj‖22 ≤ (C1C2)2σ2

∑j∈ZZ

‖aj‖22,

where σ :=∑

n∈ZZ 2−µ|n| <∞. But∑

j∈ZZ ‖aj‖22 = ‖f‖2

2. Therefore, there exists a positiveconstant B such that∑

j,k∈ZZ

∣∣〈f, ψjk〉∣∣2 =

∑j∈ZZ

‖bj‖22 ≤ B‖f‖2

2 ∀ f ∈ L2(IR).

This completes the proof of Theorem 4.1.

27

§5. Multivariate Refinable Functions

In this section we extend our study of cascade algorithms to high dimensional spaces.Several results on characterization of convergence of cascade algorithms are given. Inparticular, uniform convergence of a cascade algorithm associated with a nonnegative maskis characterized in terms of stochastic matrices. Finally, cascade algorithms are used toinvestigate multivariate biorthogonal and fundamental refinable functions. These resultsare stated without proof. For detailed proofs the reader is referred to relevant papers.

Let us consider the following refinement equation

φ =∑

α∈ZZs

a(α)φ(M · − α),

where a is a finitely supported sequence on ZZs, called the refinement mask, and M is ans× s integer matrix such that limn→∞M−n = 0, called a dilation matrix.

If∑

α∈ZZs a(α) = m := |detM |, then there exists a unique compactly supporteddistribution φ with φ(0) = 1 satisfying the above refinement equation. We say that φ isthe normalized solution of the refinement equation.

Let Qa be the cascade operator defined by

Qaf :=∑

α∈ZZs

a(α)f(M · − α), f ∈ Lp(IRs).

Suppose the normalized solution φ lies in Lp(IRs). If limn→∞ ‖Qnaf − φ‖p = 0, then f

must satisfy the Strang-Fix conditions of order 1. In other words,∑α∈ZZs

f(· − α) = 1.

We say that the cascade algorithm associated with a converges in the Lp norm if thereexists a compactly supported function φ in Lp(IRs) such that

limn→∞

‖Qnaf − φ‖p = 0

for any compactly supported function f ∈ Lp(IRs) satisfying the Strang-Fix conditions oforder 1.

Theorems 5.1 and 5.2, established by Han and Jia [10], are generalizations of Theorems2.1 and 2.2.

Theorem 5.1. Suppose u is a compactly supported function in Lp(IRs) (1 ≤ p ≤ ∞), usatisfies the Strang-Fix conditions of order 1, and the shifts of u are stable. If there existsa function φ ∈ Lp(IRs) (a continuous function in the case p = ∞) such that

limn→∞

‖Qnau− φ‖p = 0,

then for any compactly supported function v ∈ Lp(IRs) satisfying the Strang-Fix conditionsof order 1 we also have

limn→∞

‖Qnav − φ‖p = 0.

28

Consequently, if the normalized solution φ of the refinement equation lies in Lp(IRs) (φ isa continuous function in the case p = ∞), and if the shifts of φ are stable, then the cascadealgorithm associated with mask a converges in the Lp norm.

If the cascade algorithm associated with a converges in the Lp norm (1 ≤ p ≤ ∞),then a satisfies the basic sum rule:∑

α∈ZZs

a(γ +Mα) = 1 ∀ γ ∈ ZZs.

Let E be a complete set of representatives of the distinct cosets of the quotient groupZZs/MZZs. We assume that E contains 0. Thus, each element α ∈ ZZs can be uniquelyrepresented as ε+Mγ, where ε ∈ E and γ ∈ ZZs.

By `0(ZZs) we denote the linear space of all finitely supported sequences on ZZs. Thesupport of an element a ∈ `0(ZZs) is defined by suppa := α ∈ ZZs : a(α) 6= 0. For afinite subset K of ZZs, we denote by `(K) the linear subspace of `0(ZZs) consisting of allsequences supported on K.

For ε ∈ E, let Aε be the linear operator on `0(ZZs) given by

Aεv(α) =∑

β∈ZZs

a(ε+Mα− β)v(β), v ∈ `0(ZZs), α ∈ ZZs.

Let K := ZZs ∩ (∑∞

n=1M−nH), where H := (suppa − E) ∪ M([0, 1]s). Then `(K) is

invariant under Aε for each ε ∈ E. Moreover, if a satisfies the basic sum rule, then

V :=v ∈ `(K) :

∑α∈ZZs

v(α) = 0

is invariant under Aε for each ε ∈ E.

Theorem 5.2. The cascade algorithm associated with mask a and dilation matrix Mconverges in the Lp norm (1 ≤ p ≤ ∞) if and only if a satisfies the basic sum rule andρp

(Aε|V : ε ∈ E

)< m1/p, where m := |detM |.

The following theorem on L2 convergence of cascade algorithms was obtained inde-pendently by Lawton, Lee, and Shen [17], and by Han and Jia [10].

Theorem 5.3. For a given mask a and a dilation matrix M , let b be the sequence givenby b(α) :=

∑β∈ZZs a(α+β)a(β)/m, α ∈ ZZs, and let K := ZZs∩(

∑∞n=1M

−n(suppb)). Thenthe cascade algorithm associated with mask a and dilation matrix M converges in the L2

norm if and only if the following two conditions are satisfied:(a) a satisfies the basic sum rule;(b) The matrix B := (b(Mα − β))α,β∈K has 1 as its simple eigenvalue and all the other

eigenvalues are less than 1 in modulus.

The uniform convergence of a cascade algorithm associated with a nonnegative maskcan be characterized in terms of stochastic matrices. A square matrix (aij)1≤i,j≤r is calleda column-stochastic matrix if aij ≥ 0 for all i, j = 1, . . . , r and

∑ri=1 aij = 1 for all

j = 1, . . . , r. A column-stochastic matrix A is called scrambling if each pair of columnsof A have positive entries in some common row. The following theorem was establishedby Jia and Zhou [15].

29

Theorem 5.4. Let a be a nonnegative mask. Then the cascade algorithm associated withmask a converges uniformly if and only if(a) a satisfies the basic sum rule, and(b) there is a positive integer k such that for every k-tuple (ε1, . . . , εk) ∈ Ek, the finite

matrix((Aεk

· · ·Aε1)(α, β))α,β∈K

is scrambling.

Let us discuss applications of cascade algorithms to biorthogonal refinable functionsand wavelets. Given two compactly supported functions φ and φ in L2(IRs), we say thatthe shifts of φ and φ are biorthogonal if

〈φ(· − α), φ(· − β)〉 = δαβ ∀α, β ∈ ZZs.

Suppose φ and φ are refinable:

φ =∑

α∈ZZs

a(α)φ(M · − α) and φ =∑

α∈ZZs

b(α)φ(M · − α),

where a and b are finitely supported sequences on ZZs such that∑α∈ZZs

a(α) =∑

α∈ZZs

b(α) = m = |detM |.

LetH(ξ) :=

1m

∑α∈ZZs

a(α)e−iα·ξ and G(ξ) :=1m

∑α∈ZZs

b(α)e−iα·ξ, ξ ∈ IRs.

Let Ω be a complete set of representatives of the distinct cosets of the quotient groupZZs/MT ZZs. We assume that Ω contains 0.

Theorem 5.5. The shifts of φ and φ are biorthogonal if and only if the following twoconditions are satisfied:(a)

∑ω∈ΩH

(ξ + (MT )−12πω

)G(ξ + (MT )−12πω

)= 1 for all ξ ∈ IR;

(b) The cascade algorithms associated with a and b converge in L2(IRs).

By Theorem 5.3, the L2 convergence of the cascade algorithms associated with a andb can be characterized in terms of the eigenvalue condition on the corresponding transitionmatrices. Thus, Theorem 5.5 recovers the results of Long and Chen [18] on biorthogonalwavelet bases.

Finally, we mention some useful facts about multivariate fundamental refinable func-tions. A continuous function φ on IRs is said to be fundamental if φ(0) = 1 and φ(α) = 0for all α ∈ ZZs \ 0. A mask a is called interpolatory (with respect to the lattice MZZs) if

a(0) = 1 and a(Mα) = 0 ∀α ∈ ZZs \ 0.

Theorem 5.6. If a is an interpolatory mask and the cascade algorithm associated with aconverges uniformly, then the limit function is fundamental. Conversely, if φ is a continuousfundamental function on IRs with compact support and φ satisfies the refinement equationwith mask a, then the cascade algorithm associated with a converges uniformly.

By using Theorem 5.4, it can be proved that for any dilation matrix M , there existsan interpolatory nonnegative mask a such that the normalized solution of the refinementequation is fundamental.

30

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