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L. Levaggi – A. Tabacco

WAVELETS ON THE INTERVAL AND RELATED TOPICS

Abstract. We use an abstract framework to obtain a multilevel decomposition of a variety of functional spaces, using biorthogonal wavelet bases satisfying homo-geneous boundary conditions on the unit interval.

1. Introduction

Wavelet bases and, more generally, multilevel decompositions of functional spaces are often presented as a powerful tool in many different areas of theoretical and numerical analysis. The construction of biorthogonal wavelet bases and decompositions of Lp( ), 1< p<+∞, is now well understood (see [17, 11, 15] and also [5]).

We aim at considering a similar problem on the interval(0,1). Many authors have studied this subject, but, to our knowledge, none of them gives a unitary and general approach. For example, the papers [3, 12, 13, 10, 22] deal only with orthogonal multiresolution analyses; in [2] the authors study biorthogonal wavelet systems, but they do not obtain polynomial exactness for the dual spaces and fundamental inequalities such as the generalized Jackson and Bernstein ones. Dahmen, Kunoth and Urban ([16]) get all such properties, but they address specifically the construction arising from (biorthogonal) B-splines on the real line. While their construction of the scaling functions is similar to ours, the wavelets are defined in a completely different way using the concept of stable completions (see [8]). Finally, we recall the work of Chiavassa and Liandrat ([9]), which is concerned with the characterization of spaces of functions satisfying homogeneous boundary value conditions (such as the Sobolev spaces H₀s(0,1) ₌ Bs_22,0(0,1), see (2) for a definition) with orthogonal wavelet systems.

In this paper, we shall build multilevel decompositions of functional spaces (such as scales of Besov and Sobolev spaces) of functions defined on the unit interval and, possibly, subject to homogeneous boundary value conditions. Moreover, we do not work necessarily in an Hilbertian setting (as all the papers cited above do), but more generally we deal with subspaces of Lp(0,1), 1< p<+∞.

Working on subintervals I of the real line, we cannot find, in general, a unique function whose dilates and translates form a multilevel decomposition of Lp(I). Indeed, the main differ-ence is the lack of translation invariance. We shall divide our construction in two steps. Firstly we shall deal with the half-line(0,+∞); secondly, in a suitable way, we shall glue together two such constructions (on(0,+∞)and(−∞,1)) to get the final result on the unit interval(0,1).

The outline of the paper is as follows. In Section 2, we recall some basic facts about abstract multilevel decompositions and characterization of subspaces. In Section 3 and 5, we construct scaling function and wavelet spaces for the half-line. In Section 4, we prove all the needed properties, such as the Jackson- and Bernstein-type inequalities, to get the characterization of Besov spaces. To get a similar result for spaces of functions satisfying homogeneous boundary value conditions (result contained in Section 7), in Section 6 we study the boundary values of

the scaling functions and wavelets. In Section 8, we collect all our results to obtain a multilevel decomposition of spaces of functions defined on the unit interval(0,1). Finally, in Section 9, we describe in detail how our construction applies to the B-spline case.

We set here some basic notation that will be useful in the sequel.

Throughout the paper, C will denote a strictly positive constant, which may take different values in different places.

Given two functions Ni : V → +(i=1,2)defined on a set V , we shall use the notation N1(v) N2(v), if there exists a constant C>0 such that N1(v)≤C N2(v), for allv∈V . We say that N₁(v)_≍N₂(v), if N₁(v) N₂(v)and N₂(v) N₁(v).

Moreover, for any x ∈ we will indicate by⌈x⌉(or⌊x⌋) the smallest (largest) integer greater (less) than or equal to x.

Letdenote either or the half-line(0,+∞)or the unit interval(0,1). In this paper, we will use the Besov spaces Bs_pq()as defined in [23] (see also [21]) and the Sobolev spaces Ws,p()(= Bs_pp(), unless p 6= 2 and s ∈ ✁), and H

s_{() = B}s

22(). For the sake of completeness, we recall the definition of Besov space Bs_pq(). Forv∈Lp(), 1< p<+∞, let us denote by1r_hthe difference of order r_∈✁ \ {0}and step h∈ , defined as

1r_hv(x)= r

X

j=0 (−1)r+j

r j

v(x+ j h), ∀x∈rh = {x∈: x+r h∈}.

For t>0, letω(pr)be the modulus of smoothness of order r defined as ω(pr)(v,t)= sup

|h|≤t

1r_hv_Lp_( rh) .

For s>0 and 1< p,q<+∞, we sayv∈ Bs_pq()whenever the semi-norm |v_|Bs

pq()=

Z +∞

0

h

t−sω(pr)(v,t)

iq dt t

1/q

is finite, provided r is any integer>s (different values of r give equivalent semi-norm). Bs_pq() is a Banach space endowed with the norm

kvkBs

pq():= kvkLp()+ |v|Bspq().

Moreover, we recall the following real interpolation result: Lp(),Bs1

pq()s2 s1,q=

Bs2

pq() (1)

where 1<p,q<+∞and 0<s2<s1.

Finally, we will be interested in spaces of functions satisfying homogeneous boundary value conditions. To this end, let C∞₀ ()denote the space of C∞()-functions whose support is a compact subset of; hereis either(0,+∞)or(0,1), i.e., a proper subset of . We consider the Besov spaces Bs_pq,0()(s_≥0, 1< p,q<+∞) defined as the completion of C∞₀ ()in Bs_pq(). One can show that, if s ≤ 1p, Bspq,0() = Bspq(), while, if s > 1p, Bspq,0()is strictly contained in Bs_pq(). In this second case one also has

Bs_pq,0()=

(

v∈Bs_pq(): d j_v

d xj =0 on∂if 0≤ j<s− 1 p

)

. (2)

Moreover, for s≥0, 1< p,q<+∞, let

Bs_pq,00()=nv∈Bs_pq( ): suppv⊆

o

; (3)

then, if s− 1p∈/✁,

Bs_pq,00()= Bs_pq,0() . (4)

For these spaces, we have the real interpolation result:

Lp(),Bs1

pq,0()

s2/s1,p=

Bs2

pq,00() , (5)

where 1<p,q<+∞and 0<s₂<s₁. In particular, if 1< p<+∞, 0<s₂<s₁, we have

Lp(),Ws1,p

0 ()

s2/s1,p=

    

Ws2,p

00 () if s2− 1 p∈✁, Bs2

pq,0() if s2∈✁, Ws2,p

0 () if s2,s2− 1 p ∈/✁, (6)

where W₀s,p(), W₀₀s,p()are defined similarly to the Besov spaces (2) and (3). We will also consider negative values of s. For s < 0, 1 < p,q < _+∞, let us denote by p′ and q′ the conjugate index of p and q respectively (i.e. 1_p+ _p1′ = 1_q+_q1′ =1). Then, we set

Bs_pq()₌B−_p′s_q′_,0()

′

.

2. Multilevel decompositions

We will recall how to define an abstract multilevel decomposition of a separable Banach space V , with norm denoted byk·k, and how to obtain, using Jackson- and Bernstein-type inequalities, characterization of subspaces of V . For more details and proofs we refer, among the others, to [5, 6, 15].

2.1. Abstract setting

Let{Vj}j∈ (✁ = ✂or✁ = {j ∈ ✂: j ≥ j0 ∈ ✂}) be a family of closed subspaces of V such that Vj⊂Vj+1,∀j∈✁. For all j ∈✁, let Pj : V →Vjbe a continuous linear operator satisfying the following properties:

kPjk✄

(V,Vj)≤C (independent of j ),

(7)

Pjv=v , ∀v∈Vj, (8)

Pj◦Pj+1=Pj. (9)

Observe that (7) and (8) imply

kv−Pjvk ≤C inf u∈Vj

kv−uk, ∀v∈V,

where C is a constant independent of j . Through Pj, we define another set of operators Qj : V _→Vj+1by

and the detail spaces

Wj :=Im Qj, ∀j∈✁ . If✁ is bounded from below, it will be convenient to set Pj

0−1 = 0 and Vj0−1 = {0}; thus,

Qj0−1 = Pj0 and Wj0−1 = Vj0. In this case, let us also set = ✁ ∪ {j0−1}, otherwise = ✁. Thanks to the assumptions (7)-(8)-(9), every Qj is a continuous linear projection on Wj, the sequence{Qj}j∈ is uniformly bounded in✁(V,Vj+1)and each space Wj is a complement space of Vjin Vj+1, i.e.,

Vj+1=Vj⊕Wj. (10)

By iterating the decomposition (10), we get for any two integers j₁, j_2∈ such that j₁< j₂

V_{j2 =}V_j1⊕

 

j2−1

M

j=j1

W_j

 , (11)

so that every element in Vj2 can be viewed as a rough approximation of itself on a coarse level plus a sum of refinement details. Making some more assumptions on the operators Pj, we obtain a similar result for anyv_∈V . In fact, if

Pjv→v as j→ +∞, (12)

and

Pj0−1=0 if✁ = {j∈✂: j ≥ j0∈✂}, Pjv→0 as j→ −∞ if✁ =✂,

(13) then

V =

+∞ M

j=inf✂ Wj, (14)

and

v= X j≥inf✂

Qjv , ∀v∈V. (15)

The decomposition (15) is said to be q-stable, 1<q<∞, if

kvk ≍



 X

j≥inf✂ kQjvkq

 

1/q

, ∀v∈V. (16)

For each j∈✁, let us fix a basis for the subspaces Vj 8j =

n

ϕj k: k∈ ˘ ✄ j

o

, (17)

and for the subspaces W_j

9j =

n

ψj k: k∈ ˆ ✄ j

o

, (18)

with ˘ ✄

j and ˆ ✄

We can represent the operators Pj and Qjin the form Pjv=

X

k∈ ˘j

˘

vj kϕj k, Qjv=

X

k∈ ˆj

ˆ

vj kψj k, ∀v∈V. (19)

Thus, if (12) and (13) hold, (15) can be rewritten as v= X

j≥inf✂

X

k∈ ˆj

ˆ

vj kψj k, ∀v∈V. (20)

The bases chosen for the spaces Vj (and Wj) are called uniformly p-stable if, for a certain 1< p<∞,

Vj =

X

k∈ ˘j

αkϕj k:{αk}_{k∈ ˘}

j ∈ℓ

p

and

X

k∈ ˘j

αkϕj k

≍

{αk}_{k∈ ˘}

j

ℓp , ∀ {αk} ∈ℓ

p_,

the constants involved in the definition of_≍being independent of j .

If the multilevel decomposition is q-stable, the bases (18) of each Wj are uniformly p-stable, (12) and (13) hold, we can further transform (16) as

kv_{k ≍}

   

X

j≥inf✂

 

X

k∈ ˆj

| ˆvj k|p

  

q/p

  

1/q

, _∀v_∈V.

2.2. Characterization of intermediate spaces

Let us consider a Banach space Z_⊂V , whose norm will be denoted by_{k · k}Z. We assume that there exists a semi-norm| · |Z in Z such that

kv_kZ ≍ kvk + |v|Z, ∀v∈Z. (21)

In addition, we assume that

Vj ⊂Z, ∀j∈✁. (22)

Thus, Z is included in V with continuous embedding.

We recall that the real interpolation method ([4]) allows us to define a family of intermediate spaces Zα_q, with 0< α <1 and 1<q<∞, such that

Z ⊂Zα2

q2 ⊂ Z

α1

q1 ⊂V, 0< α1< α2<1, 1<q1,q2<∞, with continuous inclusion. The space Z_qαis defined as

Z_qα₌(V,Z)α,q=

v_∈V :_|v_|q_α,q :=

Z ∞

0

[t−αK(v,t)]qdt t <∞

where

K(v,t)= inf

z∈Z{kv−zk +t|z|Z}, v∈V, t>0. Z_qαis equipped with the normkvkα,q =

kvkq+ |v|qα,q

1/q

. We note that one can replace the semi-norm|v|α,qby an equivalent, discrete version as follows:

|v|α,q≍



X

j∈

bαq jK(v,b−j)q

 

1/q

, ∀v∈V,

where b is any real number>1.

We can characterize the space Z_qαin terms of the multilevel decomposition introduced in the previous Subsection (see [5, 6] and also [20]). This general result is based on two inequalities classically known as Bernstein and Jackson inequalities. In our framework, these inequalities read as follows: there exists a constant b>1 such that the Bernstein inequality

|v|Z bjkvk, ∀v∈Vj, ∀j∈✁ (23)

and the Jackson inequality

kv−Pjvk b−j|v|Z, ∀v∈Z, ∀j∈✁ (24)

hold. The Bernstein inequality is also known as an inverse inequality, since it allows the stronger normkvkZ to be bounded by the weaker normkvk, providedv∈Vj. The Jackson inequality is an approximation result, which yields the rate of decay of the approximation error by Pjfor an element belonging to Z . Note that, if we assume Z to be dense in V , then the Jackson inequality implies the consistency condition (12). The following characterization theorem holds.

THEOREM1. Let_{Vj,Pj}j∈ be a family as described in Section 2.1. Let Z be a subspace

of V satisfying the hypotheses (21) and (22). If the Bernstein and Jackson inequalities (23) and (24) hold, then for all 0< α <1, 1<q<∞one has

Z_qα=

 

v∈V :

X

j≥inf✂

bαq jkQjvkq<+∞

  

with

|v|α,q≍



 X

j≥inf✂

bαq jkQjvkq

 

1/q

, ∀v∈ Z_qα.

If in addition both (12) and (13) are satisfied, the bases (18) of each W_j ( j _∈✁) are uniformly p-stable (for suitable 1< p<∞) and the multilevel decomposition (15) is q-stable, then the following representation of the norm of Z_qαholds:

kv_kα,q≍

   

X

j∈

(1₊bαq j)

 

X

k∈ ˆj

| ˆvj k|p

  

q/p

  

1/q

or

kvkα,q≍ kPj0vk +

   

X

j∈

bαq j

 

X

k∈ ˆj

| ˆvj k|p

  

q/p

  

1/q

, ∀v∈Zα_q,

if✁ = {j∈✂: j ≥ j0∈✂}.

It is possible to prove a similar result for dual topological spaces (see e.g. [5]). Indeed, let V be reflexive and_hf, v_idenote the dual pairing between the topological dual space V′and V . For each j∈✁, letPej : V

′_→V′_{be the adjoint operator of P}

jandVej =ImPej ⊂V′. It is possible to show that the family{eVj,Pej}satisfies the same abstract assumptions of{Vj,Pj}. Moreover, let 1<q<∞and let q′be the conjugate index of q1_q+ 1

q′ =1

. Set Z_q−α ₌Zα_q′

′

, for 0< α <1, then one has:

THEOREM2. Under the same assumptions of Theorem 1 and with the notations of this Section, we have

Z_q−α₌

 

f ∈ Z

′_: X

j≥j0

b−αq j_kQejfkq_V′<+∞

  ,

with

kf_kZ−α

q ≍

ePj0f

V′+



X

j≥j0

b−αq j_kQejvkq_V′

 

1/q

, _∀f _∈Z−_qα.

2.3. Biorthogonal decomposition in

The abstract setting can be applied to construct a biorthogonal system of compactly supported wavelets on the real line and a biorthogonal decomposition of Lp( )(1<p<_∞). We will be very brief and we refer to [5, 6, 11, 17] for proofs and more details.

We suppose to have a couple of dual compactly supported scaling functionsϕ∈Lp( )and

e

ϕ∈ Lp′( )

1

p+ p1′ =1

satisfying the following conditions. There exists a couple of finite real filters h = {hn}n1

n=n0,h˜ =

_˜

hn n˜1

n= ˜n0 with n0,n˜0 ≤ 0 and n1,n˜1 ≥ 0, so thatϕandeϕ satisfy the refinement equations:

ϕ(x)=√2 n1

X

n=n0

hnϕ(2x−n) , eϕ(x)=√2

˜

n1

X

n= ˜n0 ˜

hneϕ(2x−n) , (25)

and

suppϕ=[n0,n1], suppeϕ=

˜ n0,n˜1

.

From now on, we will only describe the primal setting, the parallel˜construction following by analogy; it will be understood that there p has to be replaced by the conjugate index p′.

Setting, as usual, for j,k ∈ ✂,ϕj k(x) = 2

j/p_ϕ(2j_{x−k), we have the biorthogonality} relations

hϕj k,eϕj k′i =

Z

Thus8j = {ϕj k: k∈✂}are uniformly p-stable bases for the spaces

Vj =Vj( )=spanLp{ϕ_{j k}: k∈✂} =

  

X

k∈

αkϕj k:{αk}k∈ ∈ℓp   ,

and, for anyv∈Vj, we can write v=X

k∈

αkϕj k=

X

k∈

˘

vj kϕj k with v˘j k= hv,eϕj ki and

kvkLp_{( )}≍ 

X

k∈

| ˘vj k|p

 

1/p . (26)

We have

Vj ⊂Vj+1,

\

j∈

Vj = {0},

[

j∈

Vj =Lp( ) .

We suppose there exists an integer L_≥1 so that, locally, the polynomials of degree up to L₋1 (we will indicate this set✁L−1) are contained in Vj. It is not difficult to show that L must satisfy the relation

L≤n1−n0 (27)

(see [6], equation (3.2)). We set

ψ (x)=√2 1_X−_en0

n=1−_en1

gnϕ(2x−n) , with gn=(−1)neh1−n, andψj k(x)=2j/pψ (2jx−k),∀j,k∈✂. Thenhψj k,ψe

j′_k′i =δ_{j j}′δ_kk′,∀j,j′,k,k′∈✂. The wavelet spaces

Wj =

  

X

k∈

αkψj k:{αk}k∈ ∈ℓp  

, ∀j∈✂,

satisfy Lp( )_{= ⊕j}∈ Wj. For anyv∈Lp( ), this implies the expansion v= X

j,k∈

ˆ

vj kψj k, withvˆj k= hv,ψ˜j ki ; (28)

in addition, if p=2,

kv_kL2_{( )}≍



X

j,k∈

| ˆvj k|2

 

1/2

, _∀v_∈L2( ) . (29)

Next, let us recall that ifϕ_∈Bs0

pq( )(s0>0, 1<p,q<+∞) then the Bernstein and Jackson inequalities hold for every space Z=Bs_pq( )with 0≤s<min(s0,L). Indeed, we have

|v|Bs

pq( ) 2

j s

kvkLp_{( )}, ∀v∈V_j,∀j∈✂ (30)

and

kv−PjvkLp_{( )} 2−j s|v|_Bs

pq( ), ∀v∈B

s

pq( ) ,∀j∈✂. (31)

Thus, taking into account (1), we can apply the characterization Theorems 1 and 2 to the Besov space Z .

3. Scaling function spaces for the half-line

Starting from a biorthogonal decomposition on as described in Subsection 2.3, we aim to the construction of dual scaling function spaces Vj +

andVej +

which will form a multilevel decomposition of Lp +and of Lp′ +, respectively. For simplicity, we will work on the scale j = 0 and again we will not explicitly describe the dual˜construction. Without loss of generality, we shall suppose L ≤eL andn˜0≤n0≤0≤n1 ≤ ˜n1so that suppϕ⊆suppeϕ; if this is not the case, it is enough to exchange the role of the primal and the dual spaces.

From now on, we will append a suffix to all the functions defined on the real line. Note that if k_{≥ −}n₀,ϕ_0khave support contained in [0,_+∞). More precisely

suppϕ_0k=[n0+k,n1+k]. Let us fix a nonnegative integerδand set k₀∗= −n0+δ; observe that

k∗₀₌minnk_∈✂: suppϕ

0k⊆[δ,+∞)

o

.

Let us define

V(+)₌spannϕ_0k[0,+∞): k_≥k₀∗o_; (32)

this space will be identified in a natural way with a subspace of V0( )and will not be modified by the subsequent construction. To obtain the right scaling space V0 +

for the half-line, we will add to the basis

n

ϕ_0k[0,+∞): k≥k₀∗

o

of V(+)a finite number of new functions. These functions will be constructed so that the property of reproduction of polynomials is maintained. In fact we know that for any polynomial p∈✁L−1and every fixed x∈ ,

p(x)₌X k∈

˘

p_0kϕ_0k(x) .

So, if_{pα:α=0, . . . ,L−1}is a basis for✁

L−1, for every x≥0, we have pα(x)=

X

k≥−n1+1

c_αkϕ_0k(x)

= k∗

0−1

X

k=−n1+1

cαkϕ0k(x)+

X

k≥k∗

0

cαkϕ0k(x) , (33)

where

cαk :=(p˘α)0k=

Z

pα(y)eϕ(y−k)d y, α=0, . . . ,L−1.

Since the second sum in (33) is a linear combination of elements of V(+), in order to locally generate all polynomials of degree≤ L−1 on the half-line, we will add to this space all the linear combinations of the functions

φα(x)= k∗

0−1

X

k=−n1+1

cαkϕ0k(x) , α=0, . . . ,L−1. (35)

REMARK1. Let{pα|α=0, . . . ,L−1}and{pα⋆|α=0, . . . ,L−1}be two bases of✁L−1. Let us denote byφαandφα⋆the functions defined by the previous argument and by M the matrix

of the change of basis of✁L−1, then one can prove that φ_α⋆ ₌

L_X−1

β=0 Mαβφβ

for everyα.

PROPOSITION1. The functionsφα,α=0, . . . ,L−1 are linearly independent.

Proof. Ifδis strictly positive, the linear independence of the boundary functions is obvious. Indeed, on [0, δ], they coincide with linearly independent polynomials. Ifδ₌0 let us observe that the functionsϕ_0k[0,+∞)involved in (35) have staggered support, i.e., suppϕ_0k[0,+∞) = [0,n1+k], thus they are linearly independent. To obtain the linear independence of the functions φα it is sufficient to prove that the matrix

C=(cαk)_αk=−=0,... ,n1+L1−,... ,1−n0−1

induces an injective transformation. Thanks to Remark 1, we can choose any polynomial bases to prove the maximality of rank(C); if pα(x)=xαfor anyα, one has

cαk= α X β=0

α β

kβMeα−β,

whereMei =

R

xieϕ(x)d x is the i -th moment ofeϕon . Letvbe a vector in L such that CTv₌0, then the polynomial of degree L₋1

L_X−1

β=0 L_X−1

α=β

α β

vαMeα−βxβ

has n1−n0−1 distinct zeros; thus, recalling the relation (27), it is identically zero. This means Mv=0 where M =(Mi j)is an upper triangular matrix with Mi j = j_i

_e

Mj−i if j ≥i . M is non-singular, in fact det(M)=(R eϕ(x)d x)L 6=0 and the proof is complete.

The building blocks of our multiresolution analysis on(0,+∞)will be the border functions (35) and the basis elements of V(+). Using Proposition 1 and the linear independence on of the functionsϕ_0k, one can easily check that

PROPOSITION2. The functionsφα,α=0, . . . ,L−1 andϕ_0k

_[0,+∞), k≥k₀∗, are linearly independent.

Thus, it is natural to define

V0 +

=span{φα:α=0, . . . ,L−1} ⊕V(+).

(36)

We rename the functions in the following way ϕ_0k=

(

φk if k=0, . . . ,L−1, ϕ_0,k∗

0+k−L

if k≥L. (37)

Observe that

V(+)₌span_{ϕ0k: k≥L}. (38)

We study now the biorthogonality of the dual generators of V0 +

andVe0 +

. Setting k∗₌maxk∗₀,k˜∗_{0 =}max₋n₀₊δ,− ˜n_{0+ ˜}δ ,

(39)

let us observe that_{ϕ_0k : k_≥k∗_}and_{eϕ_0k: k_≥k∗_}are already biorthogonal. In order to get a pair of dual systems using our “blocks” we have therefore to match the dimensions of the spaces spanned by

{φα :α=0, . . . ,L−1} ∪ n

ϕ_0k: k=k₀∗, . . . ,k∗−1

o

and by

e

φβ :β=0, . . . ,eL−1 ∪ n

e

ϕ_0k: k_{= ˜}k∗₀, . . . ,k∗₋1

o

.

This requirement can be translated into an explicit relation betweenδandδ˜; indeed, we must have L−k₀∗=eL− ˜k₀∗, i.e.,

˜

δ−δ=eL−L+ ˜n0−n0. (40)

SinceeL≥L, we get k∗= − ˜n0+ ˜δ.

REMARK2. The two parametersδandδ˜have been introduced exactly because we want the equality of the cardinality of the sets previously indicated. On the other hand, we want to choose them as small as possible in order to minimize the perturbation due to the boundary. Thus, it will be natural to fix one, betweenδandδ˜, equal to zero and determine the other one from the relation (40). In particular, ifeL₋L_{+ ˜}n0−n0≥0, we setδ=0 andδ˜=eL−L+ ˜n0−n0; in this caseδ <˜ eL; whereas ifeL−L+ ˜n0−n0<0, we chooseδ˜=0 andδ=L−eL+n0− ˜n0. In analogy with the previous case, we will suppose

0≤δ <L. (41)

Let us define the spaces V₀BandeV₀Bspanned by the so called boundary scaling functions as

V₀B:=spanϕ0k: k=0, . . . ,eL−1 , Ve0B=span

e

ϕ0k: k=0, . . . ,eL−1 , (42)

and the spaces V₀I andeV₀Ispanned by the interior scaling functions as V₀I :=spanϕ0k: k≥eL , eV0I :=span

e

ϕ0k: k≥eL . (43)

Thus we have

V0 +

=V₀B⊕V₀I, eV0 +

=eV₀B⊕Ve₀I. (44)

Note that, if L< eL, the subspace V₀I is strictly contained in V(+)(see (38)). In other words, some of the functions in V(+)(which are scaling functions on supported in [0,+∞)) are thought as boundary scaling functions, i.e., are included in V₀B. As we already observed, the basis functions of V₀IandVe₀I are biorthogonal. Recalling (35) and (37), if 0≤k≤eL−1 there exist coefficientsαkmsuch thatϕ0k = Pm<k∗α_kmϕ

0m; thus, if l≥eL, due to the position of the supports, we have

hϕ0k,eϕ0li =

X

m<k∗

αkm

Z

ϕ_0meϕ

0,k∗_+l−eLd x=0,

i.e., V₀B _⊂eV₀I⊥. The only functions that we have to modify in order to obtain biorthogonal systems are the border ones. The problem is to find a basis of V₀B, sayη0k: k=0, . . . ,eL−1 , and one ofVe₀B, sayη˜0l: l=0, . . . ,eL−1 , such that

hη0k,η˜0li =δkl, k,l=0, . . . ,eL−1.

Settingη0k=PemL−=10dkmϕ0mandη˜0k=PemL−=10d˜kmeϕ0m, and calling X the Gramian matrix of components

Xkl= hϕ0k,eϕ0li, k,l=0, . . . ,eL−1, (45)

this is equivalent to the problem of finding twoeL×eL real matrices, say D= (dkm)andeD= ˜

dkm

, satisfying

D XDeT =I. (46)

A necessary and sufficient condition for (46) to have solutions is clearly the non-singularity of X , or equivalently V₀B ∩Ve₀B

⊥

= {0}. If this is the case, there exist infinitely many couples which satisfy equation (46); indeed if we chooseD non-singular then it is sufficient toe set D = XDeT

−1

. We know at present of no general result establishing the invertibility of X , although it can be proved, e.g., for orthogonal systems and for systems arising from B-spline functions (see Section 9 and also [16]). From now on we will assume this condition is verified and we suppose (renaming if necessary) that_{ϕ0k}k≥0and{eϕ0l}l≥0are dual biorthogonal bases.

Let us prove that the functionsϕ0k, k≥0, form a p-stable basis of V0 +

. PROPOSITION3. We have

V0 +

=

 

v=

X

k≥0

αkϕ0k:{αk}k∈ ∈ℓp   

with

kvkLp₍ +₎≍ k{α_k}_k∈ k_ℓp, ∀v∈V₀ +.

Proof. Letvbe any function in V0 +

; by (44) it can be written asv=vB+vI withvB =

PeL−1

k=0αkϕ0k andvI = Pk≥eLαkϕ0k. The sequence{αk}k≥eL is p-summable thanks to the “inclusion” of V₀I in V0( )and by (26), so the first part of the Proposition is proved.

To show (47), let us set K =max|suppϕ0k|: k=0, . . . ,eL−1 and note that

kvBkLpp_(0,K)≍ eL_X−1 k=0

|αk|p kvk_Lpp₍ +₎.

(48)

Indeed, since for any N∈✁ \ {0}, the application x=(x0, . . . ,xN)∈ N+17−→

N

X

n=0 xnϕ0n

Lp_(0,K)

,

defines a norm on N+1 (for a proof see, e.g., [5], Proposition 6.1), and every two norms on a finite dimensional space are equivalent, the first equivalence is proven. The second inequality follows from

|αk|p =

Z

suppϕ˜0k

v(x)ϕ_˜0k(x)d x

p

≤

Z

suppϕ˜0k

|v(x)|pd x·

Z

suppϕ˜0k

| ˜ϕ0k(x)|p

′

d x

!p/p′

kvk_Lpp₍ +₎.

Thus, by (48) and the p-stability ofϕ0kon the line (26), we have kvkp_Lp₍ +₎= kvB+vIk

p

Lp₍ +₎ kvBk

p

Lp_(0,K)+ kvIk_Lpp₍ +₎≍

X

k≥0 |αk|p.

On the other hand, we have

kvIkLp_(0,K)= kv−v_Bk_Lp_(0,K)≤ kvk_Lp_(0,K)+ kv_Bk_Lp_(0,K) kvk_Lp₍ +₎

and so

kvIkp_Lp₍ +₎ kvIk

p

Lp_(0,K)+ kvIkpLp_([K,+∞))

= kvIkp_Lp_(0,K)+ kvk

p

Lp_([K,+∞)) kvk

p Lp₍ +₎.

Then

X

k≥0 |αk|p=

eL_X−1 k=0

|αk|p+

X

k≥eL

|αk|p≍ kvBkLpp_(0,K)+ kvIk_Lpp₍ +₎ kvk

p Lp₍ +₎,

and the result is completely proven.

Similarly, it is possible to show that the dual basis{eϕ0k}k≥0ofeV0 +

is a p′-stable basis. Let us introduce the isometries Tj : Lp +

→ Lp +and eTj : Lp

′ ₊

→ Lp′ + defined as

(Tjf)(x)=2j/pf(2jx) , eTjf

(x)=2j/p′f(2jx) . (49)

and setϕj k=Tjϕ0k,eϕj k=eTjeϕ0k,∀j,k≥0. We define the j -th level scaling function spaces as

Vj +

:=Tj V0 +

, eVj +

:=eTj Ve0 +

. (50)

Let us now show that these are families of refinable spaces. PROPOSITION4. For any j ∈ ✁, one has the inclusions Vj

+

⊂ Vj+1 +

and

e

Vj +

⊂eVj+1 +

.

Proof. As before, we will only prove the result for the primal setting. By (50), we can restrict ourselves to j=0 and show that V0 +

⊂V1 +

. For k≥L, rewriting (25), one has

ϕ0k(x)=ϕ_0,k∗

0+k−L(x)=2

1 2−1p

X

m

h_m−2(k∗

0+k−L)

T1ϕ0m

(x) .

As the filter{hn}is finite, we see that the first non vanishing term in the sum corresponds to ϕ_0,k∗

0+δ+2(k−L)

, which belongs to V(+)for any k_≥L, so that the function on the left hand side belongs to V₁ +.

Suppose now k < L; without loss of generality we can choose pα(x)= xα for anyα, so

that, by (33), (34) and (35), one has, for x_≥0,

21/p(2x)k ₌ 21/p

 

φk(2x)+

X

m≥k∗

0

ckmϕ0m(2x)

  

= ϕ1k(x)+

X

m≥k∗

0

ckm

T1ϕ0m

(x) .

Again, using (35),

ϕ0k = 2−(k+1/p)

  ϕ1k+

X

m≥k∗

0

ckmϕ1m

 

− X

m≥k∗

0

ckmϕ0m

= 2−(k+1/p)

 ϕ1k+

X

m≥L c_k,k∗

0+m−Lϕ1m



₋ X

m≥L c_k,k∗

0+m−Lϕ0m, which completes the proof.

REMARK3. Note that, choosing the basis of monomials for✁L−1, the refinement equation for the modified border functions in V₀Btakes the form

φα=2−(α+1/p)T1φα+ X

k≥k∗

0

H_αkϕ1k (51)

where

Hαk =cαk2−(α+1/p)−2

1 2−

1

p

X

l≥k∗

0

cαlhk−2l, (52)

4. Projection operators

Following the guiding lines of the abstract setting, we will define a sequence of continuous linear operators Pj : Lp +

→Vjfor j∈✁, satisfying (7), (8) and (9).

By (50), it is obvious that the definition of P0gives naturally the complete sequence, by posing Pj =Tj◦P0◦T_j−1, where Tj is the isometry defined in (49). Forv∈Lp +

, let us set

P0v=

X

k≥0 ˘

v0kϕ0k, with v˘0k=

Z

v(x)eϕ0k(x)d x.

We will first prove that P0is a well-defined and continuous operator. PROPOSITION5. We have

kP0vk_Lpp₍ +₎≍

X

k≥0

| ˘v0k|p kvk_Lpp₍ +₎, ∀v∈L

p +

.

Proof. Let us write

P₀v₌

eL_X−1 k=0 ˘

v_0kϕ_0k+X k≥eL ˘ v_0kϕ_0k.

Observe that, by the H¨older inequality,

eL_X−1 k=0

| ˘v0k|p≤ kvkp_Lp₍ +₎

  e

L_X−1

k=0 keϕ0kkp

Lp′

( +₎



₌Ckvk_Lpp₍ +₎.

Thus, by the p-stability property on the line (26),

X

k≥0

| ˘v0k|p kvk_Lpp₍ +

).

This implies P0v∈V0 +

and the result follows by (47).

Since Tj is an isometry for any j , we immediately get (7). Equality (8), follows by the biorthogonality of the systems. Equality (9) is a consequence of the inclusion Vj +

⊂ Vj+1 +

, proven in Proposition 4. Similarly, one can define a sequence of dual operators,

e

Pj : Lp

′ ₊

→Vej +

, satisfying the same properties of the primal sequence. 4.1. Jackson and Bernstein inequalities

The main property of the original decomposition on the real line we have inherited, is the way polynomials are reconstructed through basis functions. This is what we call the approximation property, and it is fundamental for the characterization of functional spaces. In this section we will exploit it to prove Bernstein- and Jackson-type inequalities on the half-line and then apply the general characterization results of the abstract setting (Theorems 1 and 2).

As in Section 2.2, we consider a Banach subspace Z of Lp + and suppose that the scaling functionϕbelongs to Z . In the following, Z will be the Besov space Bs0

pq +, with s0>0 and 1< p,q<+∞.

PROPOSITION6. For any 0≤s≤s0, the Bernstein-type inequality |v|Bs

pq( +) 2

j s_kvk

Lp₍ +_{) ,} ∀v∈V_j +

, ∀j∈✁, (53)

holds.

Proof. Applying the definition of the operator Tj, it is easy to see that |Tjv|Bs

pq( +)=2

j s_|v| Bs

pq( +) , ∀v∈B

s pq +

, ∀j∈✁; (54)

so, by (50), it is enough to prove the inequality for j= 0. Proceeding as in Proposition 3, we choosev∈ V0 +

and writev=vB+vI. Let us estimate separately the semi-norms of the two terms. We have

|vB|p_Bs pq( +)=

e

L_X−1

k=0 ˘ v_0kϕ_0k

p

Bs pq( +)

≤

 

eL_X−1 k=0

| ˘v_{0k| |}ϕ_0k|Bs pq( +)

 

p _e

L_X−1

k=0 | ˘v_0k|p,

the constants depending on the semi-norms of the basis functions and the equivalence of norms in eL. Observe thatvI is an element of V0= V0( ); using the Bernstein inequality (30) and the p-stability on the line (26), we get

|vI|Bs

pq( +) kvIkLp( )≍ 

X

k≥eL | ˘v_0k|p

 

1/p .

Thus, by (47), |v|_Bps

pq( +)

|vB|p_Bs

pq( +)+ |vI|

p Bs

pq( +)

k{ ˘v0k}k∈ kℓpp kvkLp₍ +_{) .}

Next, we prove the generalized Jackson inequality, following the same ideas used in show-ing the analogous property (31) on the real line (see [5]).

PROPOSITION7. For each 0≤s<min(s0,L), we have kv−PjvkLp₍ +₎ 2−j s|v|_Bs

pq( +) , ∀v∈B

s pq +

, ∀j∈✁. (55)

Proof. As before (see (54)), it is enough to prove that kv₋P0vkLp₍ +₎ |v|_Bs

pq( +) , ∀v∈B

s pq +

.

Let us divide the half-line into unitary intervals, +_{= ∪l}≥0Ilwhere Il =[l,l+1], and estimate kv−P0vkLp_(I

l). Recalling that polynomials up to degree L−1 are locally reconstructed through

the basis of V₀ +, it is easy to see that for any q_∈✁

L−1there existsvqin V0 +

such that vq=q on Il. Then, using (8), we have

kv−P0vkLp_(I

l) = kv−q+P0q−P0vkLp(Il)

≤ kv−qkLp_(I

Moreover, from the compactness of the supports of the basis functions, setting Rl = {k∈ ✁ : suppϕ0k∩Il 6= ∅}and Jl= ∪k∈Rlsuppeϕ0k, for any f ∈L

p +_{, one gets}

kP₀f_kLpp_(I l) =

Z

Il

X

k∈Rl

˘ f_0kϕ_0k(x)

p d x

≤

max k∈Rl

kfkLp_(suppeϕ

0k)keϕ0kkLp′

( +₎

pZ

Il X

k∈Rl

|ϕ0k(x)|

p

d x kfkp_Lp_(J

l).

Thus

kv−P0vkLp_(I

l) kv−qkLp(Jl), ∀q∈✁L−1. Taking the infimum over all q∈✁L−1, we end up with

kv−P0vkLp₍ +₎

X

l≥0 inf q∈ L−1

kv−qkLp_(J l).

(56)

A local version of Whitney’s Theorem (see [21]) for a given interval of the real line J , states that, for anyv∈Lp(J),

inf q∈ L−1

kv₋q_kLp_(J) w(L)(v,J) ,

(57) where

w(L)(v,J)₌

"

1 2|J|

Z |J| −|J|

dh

Z

J(Lh)|

1L_hv_|pd x

#1/p

and J(s)= {x ∈ J : x+s ∈ J}. Observing that h∗= |Jl|(independent of l) and using (57), we have

kv−P0vkp_Lp₍ +₎

X

l≥0

w(L)(v,Jl)

p

= 1

2h∗

Z h∗

−h∗dh

X

l≥0

Z

Jl(Lh)

|1_hLv|pd x

sup

|h|≤h∗

Z

+|1

L hv|pd x

= ω(_pL)(v,h∗)p ,

whereω(pL)is the modulus of smoothness (see [21]). To conclude the proof, it is enough to observe that, for any 1<q<∞,

ω(pL)(v,h∗)



X

j∈

2j sq

ω(pL)(v,2−j)

q

 

1/q ≍ |v|Bs

pq( +) .

Since Bs_pq +is dense in Lp +, the following property immediately follows. COROLLARY1. The union∪j∈ Vj +

5. Wavelet function spaces for the half-line

We now have all the tools to build the detail spaces and the wavelets on the half-line. Recalling the abstract construction, we start from level j₌0 and look for a complement space W0 + such that V1 +

=V0 +

⊕W0 +

(note that the sum is not, in general, orthogonal) and

W0 +

⊥Ve0. To this end, let us consider the basis functions of V1 +

and let us write them as a sum of a function of V0 +

and a function which will be an element of W0 +

. Since we have based our construction on the existence of a multilevel decomposition on the real line, we report two equations that will be largely used in the sequel (see, e.g., [6]):

ϕ_1k = 2 1

p−

1 2



 X

˜

n0≤k−2m≤ ˜n1 ˜

hk−2mϕ0m+

X

1−n1≤k−2m≤1−n0 ˜

gk−2mψ0m



,

(58)

ψ_0m = 2 1 2−

1

p

1+2m_X− ˜n0

l=1+2m− ˜n1

gl−2mϕ1l. (59)

Interior wavelets. Since V₀ +contains the subspace V(+)defined in (32), V₁ +contains the subspace T1V(+)=

ϕ_1k[0,+∞): k≥k₀∗ . Considering equation (59), let us determine the integer m such that all the indices l in the sum are greater or equal to k₀∗. This is equivalent to 2m≥k∗₀+ ˜n1−1, so we set

m≥

_k∗

0+ ˜n1−1 2

=: m∗₀. (60)

Since, (see (2.9) in [11]),

X

n∈

˜

hnhn−2k=δ0k, ∀k∈✂, (61)

it is easy to see thatn_˜1−n₀is always odd; thus, by (39), m∗₀= n˜1−n0−1

2 +

δ 2

. (62)

Let us set

W₀I :=span

n

ψ_0m[0,+∞): m≥m∗₀

o

; (63)

we observe that W₀Ican be identified with a subspace of W₀( ), thus it is orthogonal toVe₀and W₀I ⊆W0 +

. The functions

ψ0m:=ψ0m

[0,+∞)

are called interior wavelets.

Border wavelets. Let us now call W₀Ba generic supplementary space of W₀I in W0 +

and set V₁B₌T1V₀B.

Proof. Let K >0 be an integer such that on [K,+∞)all non-vanishing wavelets and scaling functions are interior ones. Then

V₁B⊕spanϕ_1k: k₀∗≤k<−n0+2K =

h

V₀B_⊕spanϕ_0k: k₀∗_≤k<−n₀₊K i_⊕hW₀B_⊕spanψ_0m: m∗_0≤m_≤K₋1 i, since, by (58), the first interior wavelet used to generateϕ_1,−n

0+2K isψ0K. Then the result easily follows.

To build W₀B we need some functions that, added to W₀I, will generate both V₁B and the interior scaling functions that cannot be obtained in (58) using V(+))and W₀I. Thanks to (51), we only have to consider the problem of generating interior scaling functions. Let us now look for the functionsϕ_1kgenerated byϕ_0m, for m_≥k₀∗, and byψ_0m, with m_≥ m∗₀. Let us work separately on the two sums of (58):

(a) we must have m≥(k− ˜n1)/2. Imposing m≥k∗₀and seeking for integer solutions, we get

k− ˜n1 2

≥k₀∗= −n0+δ; (64)

(b) similarly, we obtain m≥(n0−1+k)/2. Again, we want m≥m∗₀, so we must have

n₀₋1₊k 2

≥m∗₀.

Using (62), this means

n0−1+k 2

≥ −n0+ ˜n1−1

2 +

δ 2

. (65)

Since (64) and (65) have to be both satisfied, we obtain the following condition k≥ −2n0+ ˜n1+2δ−1=2k∗0+ ˜n1−1. (66)

Indeed, this can be seen considering all possible situations. For instance, if n0andδare even, thenn˜1is odd and

l δ 2

m

= δ₂. If k satisfies both (64) and (65), so does k−1; thus we can look for the least k as an even integer. In this case

l

k− ˜n1

2

m

= k− ˜n1

2 +

1 2and

l

n0−1+k

2

m

= n0−1+k

2 +

1 2, and (66) easily follows. The other cases are dealt with similarly. Let us set

k=2k∗₀+ ˜n1−1, (67)

so that

spannϕ_1k[0,+∞): k_≥ko_⊆V(+)_⊕W₀I. We are left with the problem of generating some functions of V1 +

, preciselyϕ_1kwith k₀∗≤ k < k. Observe that we have to generate k−k∗₀functions using a space of dimension m∗₀ =

k−k∗

0

2

. In fact one can show that one out of twoϕ_1k, for k₌k∗₀, . . . ,k₋1, depends on the previous ones through elements of level zero.

PROPOSITION9. Let e=0 ifδis even, e=1 ifδis odd. For all 1≤m≤m∗₀−e, one has ϕ

1,k−2m

[0,+∞)∈Sm⊕V I 0 ⊕W0I, with

Sm:=span

n

ϕ 1,k−2l+1

[0,+∞): 1≤l≤m o

.

Proof. Let us setn_˜1−n_{0 =}2r₊1 with r >0 (recall thatn_˜1−n₀is odd). Indeed, it is not difficult to see that for r=0 there is nothing to prove. Observe that, by (58), for any l∈✂, we have

ϕ

1,k−2l=2

1 p− 1 2     X

n≥k∗

0−l ˜

h_k−2l−2nϕ_0n+ X n≥m∗

0+ j δ 2 k −l ˜

g_k−2l−2nψ_0n

    =2 1 p− 1 2 ˜ h_n˜

1−1ϕ0,k∗

0−l+ ˜

h_n˜

1−3ϕ0,k∗

0−l+1+. . . + ˜g−n0ψ

0,m∗

0+

j

δ

2

k

−l+ ˜g−n0−2ψ0,m∗

0+

j

δ

2

k

−l+1+. . .

(68)

and

ϕ

1,k−2l+1=2

1 p− 1 2     X

n≥k∗

0−l ˜

h_{k−2l+1−2n}ϕ_0n+ X n≥m∗

0+ j δ 2 k −l ˜

g_{k−2l+1−2n}ψ_0n

    =2 1

p−12

˜ hn˜1ϕ0,k∗

0−l+ ˜

hn˜1−2ϕ0,k∗

0−l+1+

. . . + ˜g−n0+1ψ

0,m∗

0+

j

δ

2

k

−l+ ˜g−n0−1ψ0,m∗

0+

j

δ

2

k

−l+1+. . .

. (69)

Let us prove the stated result by induction on m. For m=1 we consider the linear combination hn0ϕ_1,k−2+hn0+1ϕ_1,k−1

=2 1 p− 1 2   

hn₀h˜n˜1−1+hn0+1h˜n˜1

ϕ_0,k∗

0−1+

X

n≥k∗

0

c1nϕ0n

+ hn0g˜−n0+hn0+1g˜−n0+1

ψ 0,m∗

0+

j

δ

2

k −1+

X

n≥m∗

0+

j

δ

2

kd1nψ0n    ,

for some coefficients c1nand d1n. Writing (61) with k= n0− ˜n₂1+16=0 and

X

n∈

˜

gnhn−2k=0 (see (3.29) in [11]) with k_{= −}n₀, we have

Thus we have

hn0ϕ_1,k−2

[0,+∞)+hn0+1ϕ_1,k−1

[0,+∞)∈V I 0 ⊕W0I, and the result follows because hn06=0.

Set now 1<m≤m∗₀−e. As before, we choose a certain linear combination of the scaling functionsϕ

1,k−2landϕ1,k−2l+1with 1≤ l ≤ m. Then we use (68), (69) to represent them through functions of level 0. More precisely

hn₀ϕ

1,k−2m+hn0+1ϕ_1,k−2m+1+. . .+hn0+2m−2ϕ_1,k−2+hn0+2m−1ϕ_1,k−1 =hn₀h˜n˜1−1+hn0+1h˜n˜1

ϕ_0,k∗

0−m

+hn₀h˜n˜1−3+hn0+1h˜n˜1−2+hn0+2h˜n˜1−1+hn0+3h˜n˜1

ϕ_0,k∗

0−m+1 +. . .

+hn₀h˜n˜1−2m+1+hn0+1h˜n˜1−2m+2+. . .+hn0+2m−1h˜n˜1

ϕ_0,k∗

0−1

+ X

n≥k∗

0

cmnϕ0n

+ hn₀g˜−n0+hn0+1g˜−n0+1

ψ 0,m∗

0+

_δ

2

−m

+ hn₀g˜−n0−2+hn0+1g˜−n0−1+hn0+2g˜−n0+hn0+3g˜−n0+1

ψ 0,m∗

0+

_δ

2

−m+1 +. . .

+ hn0g˜−n0−2m+2+hn0+1g˜−n0−2m+1+. . .+hn0+2m−2g˜−n0+1

ψ 0,m∗

0+

_δ

2

−1

+ X

n≥m∗

0+

_δ

2

dmnψ0n,

(70)

for some cmn and dmn. The coefficients of the functionsϕ_0,k∗

0−m

, . . . , ϕ_0,k∗

0−1

can be written as

X

n∈

hnh˜n−2k=δ0k, (71)

with k= n0− ˜n1+1

2 , . . . ,

n0− ˜n1+1

2 +m−1= −r,· · ·,−r+m−1, respectively. Similarly, the coefficients of the functionsψ

0,m∗

0+

_δ

2

−m, . . . , ψ0,m∗

0+

_δ

2

−1can be written as

X

n∈

hn−2kg˜n=0,

with k= −n0,· · ·,−n0−m+1, respectively. Observe now that m≤m∗₀−e=r+

_δ

2

. If m_≤r , all indices k in (71) are negative, so

hn₀ϕ

1,k−2m+hn0+1ϕ_1,k−2m+1

[0,+∞)∈

−hn0+2ϕ_1,k−2m+2+hn0+3ϕ_1,k−2m+3+. . . + hn0+2m−2ϕ1,k−2+hn0+2m−1ϕ1,k−1

[0,+∞)+V I 0 ⊕W0I

and the result is proven by induction since hn0 6=0. If m>r (i.e.,δ≥2), we get

hn₀ϕ

1,k−2m+hn0+1ϕ1,k−2m+1

[0,+∞)∈

−hn0+2ϕ_1,k−2m+2+hn0+3ϕ_1,k−2m+3+. . . + hn0+2m−2ϕ1,k−2+hn0+2m−1ϕ1,k−1

[0,+∞)

+2

1

p−12

ϕ_0,k∗

0−m+r

[0,+∞)+V I 0 ⊕W0I. Therefore, by the induction hypothesis, we only have to prove that

ϕ_0,k∗

0−m+r

[0,+∞)∈Sm⊕V I 0 ⊕W0I. We immediately get m₋r_≤m∗₀₋e₋r ₌δ₂, so we show that

ϕ_0,k∗

0−l

[0,+∞)∈Sl, 1≤l≤

δ 2 , (72)

by induction on l. If l=1, from (69) we have ϕ_0,k∗

0−1

[0,+∞) = 2 1 2− 1 p ˜

hn˜1ϕ1,k−1

[0,+∞)+ X

n≥k∗

0

c1nϕ0n

[0,+∞)

+ X

n≥m∗

0+

_δ

2

−1

d_1nψ_0n[0,+∞)∈S_1⊕V₀I_⊕W₀I

(for some c1nand d1n). If l>1, using induction, we similarly get ϕ_0,k∗

0−l

[0,+∞) = 2

1 2−1p

˜

hn˜1ϕ_1,k−2l+1

[0,+∞)+ X

n≥k∗

0−l+1

cl,nϕ0n

[0,+∞)

+ X

n≥m∗

0+

_δ

2

−l

dl,nψ0n

[0,+∞)∈Sl⊕V I 0 ⊕W0I

(again cln and dln are fixed coefficients). Thus we have proven (72), and this completes the proof.

Using this result, setting ψ_0,m∗

0−l:=ϕ1,k−2l+1

[0,+∞)−P0

ϕ 1,k−2l+1

[0,+∞)

, l=1, . . . ,m∗₀, (73)

and

W₀B= {ψ0m|m=0, . . . ,m∗0−1}, (74)

we get

W0 +

=W₀B⊕W₀I, (75)

with W₀Ias defined in (63). With the same process we can build

e

W0 +

=We₀B⊕We₀I.

As for the scaling functions, we must find a couple of biorthogonal bases for the spaces W0 + andWe0 +

. To fix notations, we suppose thatme∗₀≤m∗₀(otherwise we only have to exchange

e

m∗₀and m∗₀in what follows). From the biorthogonality properties on the real line we have hψ0m,ψe0ni =δmn, ∀m,n≥m∗0.

Note, however, that the modified wavelets we have defined are no longer orthogonal to the inte-rior ones. In fact, from definition (73), it follows that

hψ0m,ψe0ni =

ϕ 1,k−2(m∗

0−m)+1

,ψe0n

, m=0, . . . ,m∗₀−1, n≥m∗₀. Using the refinement equation for wavelets on the real line, we easily show that

hψ0m,ψe0ni =0, n=0, . . . ,me∗0−1 if m≥

&

˜

k_{+ ˜}n1−1 2

'

=: m∗

hψ0m,ψe0ni =0, m=0, . . . ,m∗0−1 ifn≥

&

k+n1−1 2

'

=:me∗. (76)

Observing that m∗≥em∗, it is sufficient to find two m∗×m∗matrices E=(emr)andeE=(ens)˜ such that

*_m∗₋₁

X

r=0

emrψ0r, m∗₋₁

X

s=0 ˜ ensψe0s

+

=δmn, ∀m,n=0, . . . ,m∗−1.

Calling Y the m∗×m∗matrix of components Ymn = hψ0m,ψe0ni, this condition is equivalent to E YEeT = I.

Again, it is enough to prove that the matrix Y is non-singular. In fact this follows from the assumed invertibility of the matrix X (defined in (45)); since

det Y 6=0 iff W0 +

∩We0 +

⊥

= {0}, we immediately get the result observing that

W₀ +_∩We₀ +⊥_⊂V₁ +_∩Ve₁⊥_{= {}0_}. Moreover

W0 +

⊂eV₀⊥; indeed, for anyv∈Lp +, we have

hv−P0v,eϕ0ki = ˘v0k−

X

l≥0 ˘

v0lhϕ0l,eϕ0ki =0. Finally, for any j_∈✁, we set

Wj +

=TjW0 +

and Wej +

=eTjWe0 +

; (77)

settingψj m=Tjψ0m,ψej m=eTjψe0mfor every j,m∈✁, it is easy to check that the biorthog-onality relations

hψj m,ψej′_ni =δ_{j j}′δmn, ∀j,j′,m,n≥0,

hold. Moreover, with a proof similar to the one of Proposition 3, one has PROPOSITION10. The bases9j = {ψj m: m ∈ ✁}of Wj

+_{, are uniformly p-stable}

bases and the bases9ej =

_e

ψj m : m ∈ ✁ ofWej

+_{, are uniformly p}′_{-stable bases for all}

j≥0.

Moreover, let us state a characterization theorem for Besov spaces based on the biorthogonal multilevel decomposition Vj +,Vej_j≥0as described in Section 3. With the notation of the Introduction, for 1< p,q<+∞, let us set

Xs_pq:=

(

Bs_pq + if s≥0,

B−_p′s_q′

+′ _{if s<0,}

and denote by +the space of distributions. THEOREM3. Letϕ∈ Bs0

pq +

for some s0> 0, 1 < p,q <+∞. For all s ∈ S := (₋min(s₀,L),min(s₀,L))_{\ {}0_}, the following characterization holds:

Xs_pq=

            

n

v∈Lp +:P_j≥02sq j P_k≥0|vj k|p

q/p

<+∞o if s>0,

n

v∈ +:v∈Xs_pq, for some s∈S and

P

j≥02sq j

P

k≥0|vj k|p

q/p

<+∞o if s<0, where

vj k=

ˆ

vj k= hv,ψej ki if s>0, ˆ

evj k= hv, ψj ki if s<0. In addition, for all s∈S andv∈ Xs_pq, we have

|v|Xs pq ≍

 

X

j≥0 2sq j



X

k≥0 |vj k|p

 

q/p

 

1/q . (78)

Finally, if p ₌ q ₌ 2, the characterization and the norm equivalence hold for all index s _∈ (−min(s0,L),min(s0,L)).

Proof. It is sufficient to apply Theorems 1 and 2 (since the Bernste and Jackson- type in-equalities have been proven in Proposition 6 and 7, respectively) and remember the interpolation result (1).

REMARK4. It is possible to obtain a characterization result using the same representation of a functionv∈ Xs_pq for both positive and negative s. Indeed, ifϕ∈ Bs0

pq( ),eϕ∈ Bspq˜0( ), given any s∈ (−min(s˜0,eL),min(s0,L))\ {0}andv=Pj,k≥0vˆj kψj k ∈ Xspq, we have the same norm equivalence as in (78) withv_ˆj kinstead ofvj k(see, e.g., [14]). We also observe that, in general,s˜0<s0and thus Theorem 3 gives characterization for a larger interval.

It is easy to see that

suppϕ(l)= − l 2 , l 2

=: [n0,n1], and thatϕ(l)satisfies the refinement equation

ϕ(l)(x)₌ n1 X k=n0

21−l

l k+₂l

ϕ(l)(2x₋k) .

It has been shown in [11] that for any integerl˜≥l such that l+ ˜l is even, there exists a function e

ϕl,˜l

∈ L2( )that satisfies the conditions in Section 2.3 with L = l andeL = ˜l. This is a compactly supported function such that

suppeϕl,l˜ = − l 2 − ˜l₊1,

l 2

+ ˜l₋1

=: [n_˜0,n_˜1].

Let us consider a fixed couple l,l˜and drop, for a moment, the superscripts. Substituting in (40) we must have

˜

δ0−δ0=1−l. (96)

For l= ˜l=1, we obtain the orthogonal Haar basis and we can obviously chooseδ˜0=δ0=0. In all the other cases, observing thatn_{˜1− ˜}n₀₋1₌2l˜₊l₋3_{≥ ˜}l, we can chooseδ˜0=0 and δ0=l−1, so k∗0=

_l 2

+l−1 andk˜₀∗=2l

+ ˜l−1. Considering now (93), we must have ˜

δ1−δ1=1−l,

and again we can setδ˜1=0 andδ1=l−1. Recalling the definition (94), we must fix a coarsest level j0such that

j0≥ l

log₂

l+2l˜−2 m

. For j≥ j0, we have (see Section 8)

Vj(0,1) = span n

φ(_{j k}0): k=0, . . . ,l−1 o

⊕

⊕ span

ϕ_{j k}: k=l+

l 2

−1, . . . ,2j−l− l 2 +1 ⊕ ⊕ spannφ(_{j k}1): k₌0, . . . ,l₋1o.

From the definition, it is easy to see that, for any integer p,

ϕ(2 p)(−x)=ϕ(2 p)(x) , ϕ(2 p+1)(1−x)=ϕ(2 p+1)(x) , ∀x∈ , and these properties are in fact maintained by the dual functions, i.e.,

e

ϕ(2 p,l˜)(₋x)₌ϕe(2 p,l˜)(x) , eϕ(2 p+1,l˜)(1₋x)₌eϕ(2 p+1,l˜)(x) , _∀x_∈ , for any suitable integerl. These symmetry features can be used to generate boundary functions.˜ In fact, if we choose p(α1)(y)= p(α0)(−y), it is straightforward to see that

As we said before, in this setting we can prove that the matrix we get imposing the biorthogonal-ity conditions is non-singular. To simplify matters, we carry out the proof for the left boundary point only, the other one being treated analogously. To this end, let us set

ϕl, ˜ l

0k =

  

φ_0k(0) k=0, . . . ,l−1, ϕ(l)

0,l₂+k−1 k=l, . . . ,l˜−1, e

ϕl, ˜ l

0k = eφ

(0)

0k k=0, . . . ,l˜−1, and

X l,l˜=Dϕl, ˜ l 0α ,eϕ

l,l˜ 0β

E

α,β=0,... ,l−˜ 1 .

We prove now a stronger proposition about the non-singularity of X l,l˜. As the proof is done using induction, when necessary we will keep track of the dependence of the variables on the parameters l andl. Let I˜ _{= {}i₁, . . . ,ik}and J = {j1, . . . ,jk}be two selections of row and column indices, respectively; suppose k≥2, in6=imfor every n6=m and jn+1= jn+1. Let us denote by X_IJthe corresponding k×k submatrix of X l,l˜.

PROPOSITION12. For all l,l˜ ≥ 1 such that l+ ˜l is even, every submatrix Xj1,... ,jk

i1,... ,ik of

X(l,l)˜ is non-singular.

Proof. Using biorthogonality on the line and the definition of the modified functions on the boundary, it is easy to see that, ifl˜≥l,

ϕl,

˜ l 0α ,eϕ

l,l˜ 0β

=

ϕl,

˜ l 0α , (·)

β

for allα,β₌0, . . . ,l˜₋1 and any couple l,l˜.

We will prove the result by induction on l. First, let us compute the matrix arising from the Haar functionϕ₌χ[0,1), that is the B-spline of order l=1 (observe that in this case we do not

have boundary scaling functions of the form (35)). Choosingδ0,δ˜0=0 one has

ϕ l, ˜ l 0α ,eϕ

l,l˜ 0β

=

Z α+1

α

xβd x= 1 β+1

h

(α+1)β+1−αβ+1 i

.

Setting Pβ(x)= x β+1

β+1 we have, for any oddl,˜

X 1,l˜₌ Pβ(α+1)−Pβ(α)_{α,β=0,...l−}˜ ₁.

If, starting from the second, we add to any row the preceding one and multiply the j -th column by j , we obtain the Vandermonde matrix V =(Vi j)= (ij), for i , j =1, . . . ,l. It is easy to˜ prove that V has the property stated in the proposition, which in turn means that X(1,l)˜ has it as well.

Secondly, let us suppose that the proposition is true for all matrices X(n,m)with n+m even and n<l and observe that

ϕ l,

˜ l 0α , (·)

β = −

d d xϕ

l,l˜ 0α ,Pβ

.

Using the relations (see [18]) d d xϕ

(l)_(x)=ϕ(l−1)_(x+r

l−1)−ϕ(l−1)(x+rl−1−1) (where rs=s mod 2),

cl, ˜ l

α,k −c l,l˜

α,k−1=αc (l−1,l+˜ 1) α−1,k−rl−1 (see (34)) and the definition of boundary functions, we get

           d d xϕ

l,l˜

00 =ϕ

(l−1,l+˜ 1) 0l d

d xϕ l,l˜

0α =αϕ

(l−1,l+˜ 1)

0,α−1 −c

l,l˜

α,k∗₋₁ϕ (l−1,l+˜ 1)

0α α=1, . . . ,l−1,

d d xϕ

l,l˜

0α =ϕ

(l−1,l+˜ 1)

0,α−1 −ϕ

(l−1,l+˜ 1)

0α α=l, . . . ,l˜−1.

We can now express the elements of B ₌ X l,l˜in terms of the components of A₌ X(l₋ 1,l˜+1)as follows:

Bαβ=

1 β+1

    

−Al,β+1 α=0,

−αA_α−1,β+1+c l,˜l

α,k∗₋₁Al,β+1 α=1, . . . ,l−1, −Aα−1,β+1+Aα,β+1 α=l, . . . ,l˜−1.

With elementary operations on the rows and columns of B, we can transform it in the matrix obtained deleting the last two rows and the first and last column of A. These operations never switch columns and do not affect the singularity of the minors of B, so by induction the proof is complete.

Finally, with the notation of Section 8, we have m∗₀=m#₀−1= l+ ˜l

2 +

l₊r

2 −1, me ∗ 0=em

#

0−1=

l_{+ ˜}l

2 +r−1, and

m∗=m#−1=l+r+ ˜l−2+ &

˜ l−1

2 '

, me∗=me#−1=2l+r−2+ &

˜ l−1

2 '

,

where r is 1 if l is odd, zero otherwise. Then it is easy to see thatem∗≤m∗for all couples l,l˜. Therefore, one has

Wj(0,1) = span n

ψ(_{j m}0): m=0, . . . ,m∗−1 o

⊕

⊕ span

n

ψj m: m=m∗, . . . ,2j−m∗ o

⊕

⊕ span

n

ψ(_{j m}1): m=1, . . . ,m∗−1 o

.

Since the border wavelets have the same supports at the two edges, we only have to compute their lengths once. From the definition (74), substituting the values of n₀,n˜₀and n₁,n˜₁it is easy to see that

max m=0,... ,m∗₋₁

suppψ(_{j m}0) =2−j

2l˜+ 3l

2 +r−3

To obtain a lower limit for j0it is enough to require 2−j0

2l˜+3l2 +r−3

≤ 12. In this way we obtain

j0≥

log₂

2l˜+3l 2 +r−3

+1

. (97)

Acknowledgments. The authors would like to thank Claudio Canuto for many clarifying and

useful discussions. This research has been partially supported by CNR ST/74 Progetto Strategico Modelli e Metodi per la Matematica e l’Ingegneria.

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AMS Subject Classification: 42C15.

Laura LEVAGGI

Dipartimento di Matematica, Universit`a di Genova via Dodecaneso 35

16146 Genova Anita TABACCO

Dipartimento di Matematica, Politecnico di Torino corso Duca degli Abruzzi 24

10129 Torino