123 L.Levaggi–A.TabaccoWAVELETSONTHEINTERVALANDRELATEDTOPICS Rend.Sem.Mat.Univ.Pol.TorinoVol.57,2(1999)

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 L^p( ), 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 H₀^s(0,1) = B^s_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 L^p(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 L^p(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

123

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 N_i : 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 bydxe(orbxc) 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 B^s_pq()as defined in [23] (see also [21]) and the Sobolev spaces W^s,p()(= B^s_pp(), unless p 6= 2 and s ∈ ), and H^s() = B₂₂^s (). For the sake of completeness, we recall the definition of Besov space B^s_pq(). Forv∈L^p(), 1< p<+∞, let us denote by1^r_hthe difference of order r∈ \ {0}and step h∈ , defined as

1^r_hv(x)= Xr

j=0 (−1)^r+j

r j

v(x+ j h), ∀x∈_rh = {x∈: x+r h∈}.

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

|h|≤t 1^r_hv

L^p(_rh) .

For s>0 and 1< p,q<+∞, we sayv∈ B^s_pq()whenever the semi-norm

|v|B^s_pq()= Z +∞

0 h

t^−sω^(r)_p (v,t)iq dt t

1/q

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

kvkB^s_pq():= kvkL^p()+ |v|B^s_pq(). Moreover, we recall the following real interpolation result:

L^p(),B^s_pq¹()

s2

s1,q=B^s_pq²() (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 B^s_pq,0()(s≥0, 1< p,q<+∞) defined as the completion of C^∞₀ ()in B^s_pq(). One can show that, if s ≤ ¹_p, B^s_pq,0() = B^s_pq(), while, if s > ¹_p, B^s_pq,0()is strictly contained in B^s_pq(). In this second case one also has

B^s_pq,0()= (

v∈B^s_pq(): d^jv

d x^j =0 on∂if 0≤ j<s− 1 p )

. (2)

Moreover, for s≥0, 1< p,q<+∞, let B^s_pq,00()=n

v∈B^s_pq( ): suppv⊆ o

; (3)

then, if s− ¹_p∈/,

B^s_pq,00()= B^s_pq,0() . (4)

For these spaces, we have the real interpolation result:

L^p(),B^s_pq,0¹ ()

s₂/s₁,p=B^s_pq,00² () , (5)

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

L^p(),W₀^s1,p() s₂/s₁,p=







W₀₀^s2,p() if s₂− ¹_p∈, B^s_pq,0² () if s₂∈, W₀^s2,p() if s₂,s₂− ¹_p ∈/, (6)

where W₀^s,p(), W₀₀^s,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 p⁰ and q⁰ the conjugate index of p and q respectively (i.e. ¹_p+ _p¹⁰ = ¹_q+_q¹⁰ =1). Then, we set

B^s_pq()=

B^−s_p0q⁰,0()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{V_j}j∈ ( = or = {j ∈ : j ≥ j₀ ∈ }) be a family of closed subspaces of V such that V_j⊂V_j+1,∀j∈. For all j ∈, let P_j : V →V_jbe a continuous linear operator satisfying the following properties:

kP_jk(V,Vj)≤C (independent of j ), (7)

P_jv=v , ∀v∈V_j, (8)

P_j◦P_j+1=P_j. (9)

Observe that (7) and (8) imply

kv−P_jvk ≤C inf

u∈V_jkv−uk, ∀v∈V,

where C is a constant independent of j . Through P_j, we define another set of operators Q_j : V →V_j+1by

Q_jv=P_j+1v−P_jv , ∀v∈V, ∀j∈ ,

and the detail spaces

W_j :=Im Q_j, ∀j∈ .

If is bounded from below, it will be convenient to set P_j0−1 = 0 and V_j0−1 = {0}; thus, Q_j0−1 = P_j0 and W_j0−1 = V_j0. In this case, let us also set = ∪ {j₀−1}, otherwise

= . Thanks to the assumptions (7)-(8)-(9), every Q_j is a continuous linear projection on W_j, the sequence{Q_j}j∈ is uniformly bounded in(V,V_j+1)and each space W_j is a complement space of V_jin Vj+1, i.e.,

V_j+1=V_j⊕W_j. (10)

By iterating the decomposition (10), we get for any two integers j₁, j₂∈ such that j₁< j₂ V_j2 =V_j1⊕



 j2−1 M

j=j₁ W_j



, (11)

so that every element in V_j2 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 P_j, we obtain a similar result for anyv∈V . In fact, if

P_jv→v as j→ +∞, (12)

and

P_j0−1=0 if = {j∈: j ≥ j₀∈}, P_jv→0 as j→ −∞ if =,

(13) then

V = +∞M

j=inf W_j, (14)

and

v= X

j≥inf

Q_jv , ∀v∈V. (15)

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

kvk



 X j≥inf

kQ_jvk^q



 1/q

, ∀v∈V. (16)

For each j∈, let us fix a basis for the subspaces V_j 8_j =n

ϕ_{j k}: k∈ ˘j o

, (17)

and for the subspaces W_j

9_j =n

ψ_{j k}: k∈ ˆj o (18) ,

with˘_j andˆ_jsuitable sets of indices.

We can represent the operators P_j and Q_jin the form P_jv= X

k∈ ˘_j

˘

v_{j k}ϕ_{j k}, Q_jv= X k∈ ˆ_j

ˆ

v_{j k}ψ_{j k}, ∀v∈V. (19)

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

v= X

j≥inf

X

k∈ ˆ_j ˆ

v_{j k}ψ_{j k}, ∀v∈V. (20)

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

V_j = X k∈ ˘_j

α_kϕ_{j k}:{α_k}_{k∈ ˘}

j ∈`^p

and

X

k∈ ˘_j α_kϕ_{j k}

{α_k}_{k∈ ˘}

j

`<sup>p</sup> , ∀ {α<sub>k</sub>} ∈`^p,

the constants involved in the definition ofbeing independent of j .

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

kvk





 X

j≥inf



X k∈ ˆ_j

| ˆv_{j 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 byk · kZ. We assume that there exists a semi-norm| · |Z in Z such that

kvkZ kvk + |v|Z, ∀v∈Z. (21)

In addition, we assume that

V_j ⊂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 Z^α_q, with 0< α <1 and 1<q<∞, such that

Z ⊂Z^α_q2²⊂ Z^α_q1¹ ⊂V, 0< α₁< α₂<1, 1<q₁,q₂<∞, with continuous inclusion. The space Z_q^αis defined as

Z_q^α=(V,Z)_α,q=

v∈V :|v|^qα,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_q^αis equipped with the normkvkα,q =

kvk^q+ |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∈

b^{αq j}K(v,b^−j)^q



 1/q

, ∀v∈V,

where b is any real number>1.

We can characterize the space Z_q^αin 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 b^jkvk, ∀v∈V_j, ∀j∈ (23)

and the Jackson inequality

kv−P_jvk 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∈V_j. The Jackson inequality is an approximation result, which yields the rate of decay of the approximation error by P_jfor 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{V_j,P_j}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_q^α=



v∈V : X j≥inf

b^{αq j}kQ_jvk^q<+∞





with

|v|α,q



 X j≥inf

b^{αq j}kQ_jvk^q



 1/q

, ∀v∈ Z_q^α.

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_q^αholds:

kvkα,q





 X

j∈

(1+b^{αq j})



X k∈ ˆ_j

| ˆv_{j k}|^p



 q/p



 1/q

, ∀v∈Z^α_q, if =,

or

kvkα,q kP_j0vk +





 X

j∈ b^{αq j}



X k∈ ˆ_j

| ˆv_{j k}|^p



 q/p



 1/q

, ∀v∈Z^α_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 andhf, videnote the dual pairing between the topological dual space V⁰and V . For each j∈, letPe_j : V⁰→V⁰be the adjoint operator of P_jandVe_j =ImPe_j ⊂V⁰. It is possible to show that the family{eV_j,Pe_j}satisfies the same abstract assumptions of{V_j,P_j}. Moreover, let 1<q<∞and let q⁰be the conjugate index of q

1

q+_q¹⁰ =1

. Set Z_q^−α = Z^α_q0

0 , 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 ∈ Z⁰: X j≥j₀

b^{−αq j}kQe_jfk^q_V⁰<+∞



,

with

kfk_Zq^−α eP_j0f

V⁰+



X j≥j₀

b^{−αq j}kQe_jvk^q_V⁰



 1/q

, ∀f ∈Z^−α_q .

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 L^p( )(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∈L^p( )and e

ϕ∈ L^p0( ) 1

p+ _p¹⁰ =1

satisfying the following conditions. There exists a couple of finite real filters h = {hn}ⁿn=n¹ ₀,h˜ = h˜n 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

n˜1

X

n= ˜n₀

h˜neϕ(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 p⁰.

Setting, as usual, for j,k ∈ ,ϕ_{j k}(x) = 2^j/pϕ(2^jx−k), we have the biorthogonality relations

hϕ_{j k},eϕ_{j k}⁰i = Z

ϕ_{j k}(x)eϕ_{j k}⁰(x)d x=δ_kk⁰, ∀j,k,k⁰∈.

Thus8_j = {ϕ_{j k}: k∈}are uniformly p-stable bases for the spaces V_j =V_j( )=span_Lp{ϕ_{j k}: k∈} =



 X

k∈

α_kϕ_{j k}:{α_k}k∈ ∈`^p



,

and, for anyv∈V_j, we can write

v=X

k∈

α_kϕ_{j k}=X k∈

˘

v_{j k}ϕ_{j k} with v˘_{j k}= hv,eϕ_{j k}i and

kvkL^p( )



X k∈

| ˘v_{j k}|^p



 1/p

. (26)

We have

V_j ⊂V_j+1, \ j∈

V_j = {0}, [ j∈

V_j =L^p( ) .

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 V_j. 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−Xen0

n=1−en1

gnϕ(2x−n) , with gn=(−1)ⁿeh_1−n,

andψ_{j k}(x)=2^j/pψ (2^jx−k),∀j,k∈. Thenhψ_{j k},ψe_j⁰_k⁰i =δ_{j j}⁰δ_kk⁰,∀j,j⁰,k,k⁰∈. The wavelet spaces

W_j =



 X

k∈

α_kψ_{j k}:{α_k}k∈ ∈`^p



, ∀j∈, satisfy L^p( )= ⊕j∈ W_j. For anyv∈L^p( ), this implies the expansion

v= X

j,k∈

ˆ

v_{j k}ψ_{j k}, withvˆ_{j k}= hv,ψ˜_{j k}i ; (28)

in addition, if p=2,

kvkL²( )



X j,k∈

| ˆv_{j k}|²



 1/2

, ∀v∈L²( ) . (29)

Next, let us recall that ifϕ∈B^s_pq⁰( )(s₀>0, 1<p,q<+∞) then the Bernstein and Jackson inequalities hold for every space Z=B^s_pq( )with 0≤s<min(s₀,L). Indeed, we have

|v|B^s_pq( ) 2^{j s}kvkL^p( ), ∀v∈V_j,∀j∈ (30)

and

kv−P_jvkL^p( ) 2^{−j s}|v|B^s_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 V_j ⁺

andVe_j ⁺

which will form a multilevel decomposition of L^{p +}

and of L^{p0 +}

, 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=minn

k∈: suppϕ_0k⊆[δ,+∞) o

.

Let us define

V⁽⁺⁾=spann ϕ_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{p_α:α=0, . . . ,L−1}is a basis for_L−1, for every x≥0, we have p_α(x)= X

k≥−n₁+1

c_αkϕ_0k(x)

= k₀^∗−1

X

k=−n₁+1

c_αkϕ_0k(x)+ X k≥k₀^∗

c_αkϕ_0k(x) , (33)

where

c_αk :=(p˘α)_0k= Z

pα(y)eϕ(y−k)d y, α=0, . . . ,L−1. (34)

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

φ_α(x)= k^∗₀−1

X

k=−n₁+1

c_αkϕ_0k(x) , α=0, . . . ,L−1. (35)

REMARK1. Let{pα|α=0, . . . ,L−1}and{p_α^?|α=0, . . . ,L−1}be two bases ofL−1. Let us denote byφ_αandφ_α^?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−1X

β=0 M_αβφ_β 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=(c_αk)^k=−n_{α=0,... ,L}^{1+1,... ,−n}₋₁ ⁰⁻¹

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

c_αk= Xα

β=0 α

β

k^βMe_α−β,

whereMe_i = R

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

L−1X

β=0 L−1X

α=β α

β

v_αMe_α−βx^β

has n₁−n₀−1 distinct zeros; thus, recalling the relation (27), it is identically zero. This means Mv=0 where M =(M_{i j})is an upper triangular matrix with M_{i j} = ^j_iMe_j−i if j ≥i . M is non-singular, in fact det(M)=(R

e

ϕ(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=

( φ_k if k=0, . . . ,L−1, ϕ_0,k∗

0+k−L if k≥L. (37)

Observe that

V⁽⁺⁾=span{ϕ_0k: k≥L}. (38)

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

andVe₀ ⁺ . Setting k^∗=max

k^∗₀,k˜^∗₀ =max

−n₀+δ,− ˜n₀+ ˜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

eφ_β :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 V₀^BandeV₀^Bspanned by the so called boundary scaling functions as

V₀^B:=span

ϕ_0k: k=0, . . . ,eL−1 , Ve₀^B=span e

ϕ_0k: k=0, . . . ,eL−1 , (42)

and the spaces V₀^I andeV₀^Ispanned by the interior scaling functions as V₀^I :=span

ϕ_0k: k≥eL , eV₀^I :=span e

ϕ_0k: k≥eL . (43)

Thus we have

V₀ ⁺

=V₀^B⊕V₀^I, Ve₀ ⁺

=eV₀^B⊕Ve₀^I. (44)

Note that, if L< eL, the subspace V₀^I 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 V₀^B. As we already observed, the basis functions of V₀^IandVe₀^I are biorthogonal. Recalling (35) and (37), if 0≤k≤eL−1 there exist coefficientsα_kmsuch thatϕ_0k = P

m<k^∗α_kmϕ_0m; thus, if l≥eL, due to the position of the supports, we have

hϕ_0k,eϕ_0li = X m<k^∗

α_km Z

ϕ_0meϕ

0,k^∗+l−eLd x=0, i.e., V₀^B ⊂

e V₀^I⊥

. 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 V₀^B, say

η_0k: k=0, . . . ,eL−1 , and one ofVe₀^B, say

˜

η_0l: l=0, . . . ,eL−1 , such that

hη_0k,η˜_0li =δ_kl, k,l=0, . . . ,eL−1.

Settingη_0k=PeL−1

m=0d_kmϕ_0mandη˜_0k=PeL−1

m=0d˜_kmeϕ_0m, and calling X the Gramian matrix of components

X_kl= hϕ_0k,eϕ_0li, k,l=0, . . . ,eL−1, (45)

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

, satisfying

D XDe^T =I. (46)

A necessary and sufficient condition for (46) to have solutions is clearly the non-singularity of X , or equivalently V₀^B ∩

Ve₀^B ⊥

= {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 =

XDe^T −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{ϕ_0k}k≥0and{eϕ_0l}l≥0are dual biorthogonal bases.

Let us prove that the functionsϕ_0k, k≥0, form a p-stable basis of V₀ ⁺ . PROPOSITION3. We have

V₀ ⁺

=



v=X k≥0

α_kϕ_0k:{α_k}k∈ ∈`^p





with

kvkL^p( ⁺) k{α_k}k∈ k`^p, ∀v∈V₀ ⁺ . (47)

Proof. Letvbe any function in V₀ ⁺

; by (44) it can be written asv=v_B+v_I withv_B = PeL−1

k=0α_kϕ_0k andv_I = P

k≥eLα_kϕ_0k. The sequence{α_k}_k≥eL is p-summable thanks to the

“inclusion” of V₀^I in V₀( )and by (26), so the first part of the Proposition is proved.

To show (47), let us set K =max

|suppϕ_0k|: k=0, . . . ,eL−1 and note that kv_Bk_L^pp(0,K)

eL−1X

k=0

|α_k|^p kvk_L^pp( ⁺). (48)

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

XN

n=0 xnϕ_0n

L^p(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

|α_k|^p = Z

suppϕ˜0k

v(x)ϕ˜_0k(x)d x p

≤ Z

suppϕ˜0k

|v(x)|^pd x· Z

suppϕ˜0k

| ˜ϕ_0k(x)|^p0d x

!p/p⁰

kvk_L^pp( ⁺). Thus, by (48) and the p-stability ofϕ_0kon the line (26), we have

kvk^p_Lp( ⁺)= kv_B+v_Ik^p_Lp( ⁺) kv_Bk_L^pp(0,K)+ kv_Ik_L^pp( ⁺)X k≥0

|α_k|^p. On the other hand, we have

kv_IkL^p(0,K)= kv−v_BkL^p(0,K)≤ kvkL^p(0,K)+ kv_BkL^p(0,K) kvkL^p( ⁺) and so

kv_Ik^p_Lp( ⁺) kv_Ik^p_Lp(0,K)+ kv_Ik^p_Lp([K,+∞))

= kv_Ik^p_Lp(0,K)+ kvk_L^pp([K,+∞)) kvk^p_Lp( ⁺). Then

X

k≥0

|α_k|^p= eL−1X

k=0

|α_k|^p+X k≥eL

|α_k|^p kv_Bk_L^pp(0,K)+ kv_Ik_L^pp( ⁺) kvk^p_Lp( ⁺),

and the result is completely proven.

Similarly, it is possible to show that the dual basis{eϕ_0k}k≥0ofeV₀ ⁺

is a p⁰-stable basis.

Let us introduce the isometries T_j : L^{p +}

→ L^{p +}

and eT_j : L^{p0 +}

→ L^{p0 +} defined as

(T_jf)(x)=2^j/pf(2^jx) , eT_jf

(x)=2^j/p0f(2^jx) . (49)

and setϕ_{j k}=T_jϕ_0k,eϕ_{j k}=eT_jeϕ_0k,∀j,k≥0. We define the j -th level scaling function spaces as

V_j ⁺

:=T_j V₀ ⁺

, Ve_j ⁺

:=eT_j Ve₀ ⁺ . (50)

Let us now show that these are families of refinable spaces.

PROPOSITION4. For any j ∈ , one has the inclusions V_j ⁺

⊂ Vj+1 + and Ve_j ⁺

⊂eV_j+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

ϕ_0k(x)=ϕ_0,k∗

0+k−L(x)=2 1

2−¹_p X

m

h_m−2(k^∗

0+k−L)

T₁ϕ_0m

(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)= x^α for anyα, so that, by (33), (34) and (35), one has, for x≥0,

2^1/p(2x)^k = 2^1/p



φ_k(2x)+ X m≥k^∗

0

c_kmϕ_0m(2x)





= ϕ_1k(x)+ X m≥k₀^∗

c_km

T₁ϕ_0m

(x) .

Again, using (35),

ϕ_0k = 2^−(k+1/p)



ϕ_1k+ X m≥k^∗

0

c_kmϕ_1m



− X m≥k^∗

0

c_kmϕ_0m

= 2^−(k+1/p)



ϕ_1k+ X m≥L

c_k,k^∗

0+m−Lϕ_1m



− X m≥L

c_k,k^∗

0+m−Lϕ_0m, which completes the proof.

REMARK3. Note that, choosing the basis of monomials forL−1, the refinement equation for the modified border functions in V₀^Btakes the form

φ_α=2^−(α+1/p)T₁φ_α+ X k≥k^∗

0

H_αkϕ_1k (51)

where

H_αk =c_αk2^−(α+1/p)−2 1

2−¹

p

X

l≥k₀^∗

c_αlhk−2l, (52)

involving only the respective modified border function in V₁^B.