L. Levaggi A. Tabacco WAVELETS ON THE INTERVAL AND RELATED TOPICS

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1 Rend. Sem. Mat. Univ. Pol. Torino Vol. 57, 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 homogeneous 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, 1, 13, 10, ] deal only with orthogonal multiresolution analyses; in [] 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 s0, 1) = Bs,0 0, 1), see ) 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 difference 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, 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 13

2 14 L. Levaggi A. Tabacco 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, ) defined on a set V, we shall use the notation N 1 v) N v), if there exists a constant C > 0 such that N 1 v) C N v), for all v V. We say that N 1 v) N v), if N 1 v) N v) and N v) N 1 v). Moreover, for any x we will indicate by x or x ) the smallest largest) integer greater less) than or equal to x. Let denote 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 [3] see also [1]) and the Sobolev spaces W s,p ) = B s pp ), unless p = and s ), and H s ) = B s ). For the sake of completeness, we recall the definition of Besov space B s pq ). For v L p ), 1 < p < +, let us denote by r h the difference of order r \ 0 and step h, defined as r ) r h vx) = 1) r+ j r vx + jh), x rh = x : x + rh. j j=0 For t > 0, let ω r) p be the modulus of smoothness of order r defined as ω r) p v, t) = sup r h v L h t p rh ). For s > 0 and 1 < p, q < +, we say v B s pq ) whenever the semi-norm + [ v B s pq ) = t s ω r) ] ) q dt 1/q p v, t) 0 t 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 v B s pq ) := v L p ) + v B s pq ). Moreover, we recall the following real interpolation result: L p s ), B 1 pq ) ) s s1,q = Bs 1) pq ) where 1 < p, q < + and 0 < s < s 1. Finally, we will be interested in spaces of functions satisfying homogeneous boundary value conditions. To this end, let C0 ) denote the space of C )-functions whose support is a compact subset of ; here is 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, Bs pq,0 ) = Bs pq ), while, if s > 1 p, Bs 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 j v dx j = 0 on if 0 j < s 1 p.

3 Wavelets on the interval 15 Moreover, for s 0, 1 < p, q < +, let 3) B s pq,00 ) = v B s pq ) : supp v ; then, if s 1 p /, 4) B s pq,00 ) = Bs pq,0 ). For these spaces, we have the real interpolation result: L p ), B s ) 1 pq,0 ) = s /s 1,p Bs 5) pq,00 ), where 1 < p, q < + and 0 < s < s 1. In particular, if 1 < p < +, 0 < s < s 1, we have L p ), W s ) W s,p 00 ) if s 1 p, 1,p 0 ) = B s 6) s /s 1,p pq,0 ) if s, W s,p 0 ) if s, s 1 p /, where W s,p 0 ), W s,p 00 ) are defined similarly to the Besov spaces ) 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 + p 1 = q 1 + q 1 = 1). Then, we set B s pq ) = B s p q,0 ) ).. Multilevel decompositions We will recall how to define an abstract multilevel decomposition of a separable Banach space V, with norm denoted by, 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]..1. Abstract setting Let V j j = or = j : j j 0 ) be a family of closed subspaces of V such that V j V j+1, j. For all j, let P j : V V j be a continuous linear operator satisfying the following properties: 7) 8) 9) P j V,V j ) C independent of j), P j v = v, v V j, P j P j+1 = P j. Observe that 7) and 8) imply v P j v C inf u V j v u, v V, where C is a constant independent of j. Through P j, we define another set of operators Q j : V V j+1 by Q j v = P j+1 v P j v, v V, j,

4 16 L. Levaggi A. Tabacco 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 = j 0 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 j in V j+1, i.e., 10) V j+1 = V j W j. By iterating the decomposition 10), we get for any two integers j 1, j such that j 1 < j j 1 11) V j = V j1 W j, j= j 1 so that every element in V j 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 any v V. In fact, if 1) P j v v as j +, and 13) Pj0 1 = 0 if = j : j j 0, P j v 0 as j if =, then 14) and 15) + V = W j, j=inf v = Q j v, v V. j inf The decomposition 15) is said to be q-stable, 1 < q <, if 16) v j inf Q j v q 1/q For each j, let us fix a basis for the subspaces V j 17) j = ϕ jk : k j,, v V. and for the subspaces W j 18) j = ψ jk : k ˆ j, with j and ˆ j suitable sets of indices.

5 Wavelets on the interval 17 We can represent the operators P j and Q j in the form 19) P j v = v jk ϕ jk, Q j v = ˆv jk ψ jk, v V. k j k ˆ j Thus, if 1) and 13) hold, 15) can be rewritten as v = 0) ˆv jk ψ jk, v V. j inf k ˆ j The bases chosen for the spaces V j and W j ) are called uniformly p-stable if, for a certain 1 < p <, V j = α k ϕ jk : α k k l p j and k j α k ϕ jk α k k, α j k l p, l k p j the constants involved in the definition of being independent of j. If the multilevel decomposition is q-stable, the bases 18) of each W j are uniformly p- stable, 1) and 13) hold, we can further transform 16) as q/p 1/q v ˆv jk p, v V. j inf k ˆ j.. Characterization of intermediate spaces Let us consider a Banach space Z V, whose norm will be denoted by Z. We assume that there exists a semi-norm Z in Z such that 1) v Z v + v Z, v Z. In addition, we assume that ) V j Z, j. 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 Zq α, with 0 < α < 1 and 1 < q <, such that Z Z α q Z α 1 q 1 V, 0 < α 1 < α < 1, 1 < q 1, q <, with continuous inclusion. The space Zq α is defined as Zq α = V, Z) α,q = v V : v q α,q := [t α Kv, t)] q dt 0 t <,

6 18 L. Levaggi A. Tabacco where Kv, t) = inf z Z v z + t z Z, v V, t > 0. Zq α is equipped with the norm v α,q = semi-norm v α,q by an equivalent, discrete version as follows: v α,q j v q + v q α,q) 1/q. We note that one can replace the b αq j Kv, b j ) q 1/q, v V, where b is any real number > 1. We can characterize the space Zq α in terms of the multilevel decomposition introduced in the previous Subsection see [5, 6] and also [0]). 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 3) v Z b j v, v V j, j and the Jackson inequality 4) v P j v b j v Z, v Z, j hold. The Bernstein inequality is also known as an inverse inequality, since it allows the stronger norm v Z to be bounded by the weaker norm v, provided v V j. The Jackson inequality is an approximation result, which yields the rate of decay of the approximation error by P j for an element belonging to Z. Note that, if we assume Z to be dense in V, then the Jackson inequality implies the consistency condition 1). The following characterization theorem holds. THEOREM 1. Let V j, P j j be a family as described in Section.1. Let Z be a subspace of V satisfying the hypotheses 1) and ). If the Bernstein and Jackson inequalities 3) and 4) hold, then for all 0 < α < 1, 1 < q < one has with Zq α = v V : v α,q j inf b αq j Q j v q < + j inf b αq j Q j v q 1/q, v Z α q. If in addition both 1) 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 α q holds: q/p 1/q v α,q 1 + b αq j ) ˆv jk p, v Zq α, if =, j k ˆ j

7 Wavelets on the interval 19 or q/p 1/q v α,q P j0 v + b αq j ˆv jk p, v Zq α, j k ˆ j if = j : j j 0. It is possible to prove a similar result for dual topological spaces see e.g. [5]). Indeed, let V be reflexive and f, v denote the dual pairing between the topological dual space V and V. For each j, let P j : V V be the adjoint operator of P j and Ṽ j = Im P j V. It is possible to show that the family Ṽ j, P j satisfies the same abstract assumptions of V j, P j. Moreover, let 1 < q < and let q be the conjugate index of q 0 < α < 1, then one has: 1q + 1 q = 1). Set Z α q = Z α q ), for THEOREM. Under the same assumptions of Theorem 1 and with the notations of this Section, we have Zq α = f Z : b αq j Q j f q V < +, j j 0 with f Z α q P j0 f V + 1/q b αq j Q j v q V j j 0, f Z α q..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 functions ϕ L p ) and ϕ L p ) 1p + p 1 = 1) satisfying the following conditions. There exists a couple of finite real filters h = h n n 1 n=n 0, h = h n ñ1 n=ñ 0 with n 0, ñ 0 0 and n 1, ñ 1 0, so that ϕ and ϕ satisfy the refinement equations: 5) ϕx) = n 1 ñ 1 h n ϕx n), ϕx) = h n ϕx n), n=n 0 n=ñ 0 and supp ϕ = [n 0, n 1 ], supp ϕ = [ ñ 0, ñ 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, ϕ jk x) = j/p ϕ j x k), we have the biorthogonality relations ϕ jk, ϕ jk = ϕ jk x) ϕ jk x) dx = δ kk, j, k, k.

8 130 L. Levaggi A. Tabacco Thus j = ϕ jk : k are uniformly p-stable bases for the spaces V j = V j ) = span L pϕ jk : k = α k ϕ jk : α k k k and, for any v V j, we can write l p, v = k α k ϕ jk = k v jk ϕ jk with v jk = v, ϕ jk and 6) v L p ) k 1/p v jk p. We have V j V j+1, 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 j 7) L n 1 n 0 see [6], equation 3.)). We set ψx) = 1 ñ 0 n=1 ñ 1 g n ϕx n), with g n = 1) n h 1 n, and ψ jk x) = j/p ψ j x k), j, k. Then ψ jk, ψ j k = δ j j δ kk, j, j, k, k. The wavelet spaces W j = α k ψ jk : α k k l p, j, k satisfy L p ) = j 8) in addition, if p =, 9) W j. For any v L p ), this implies the expansion v = j,k v L ) ˆv jk ψ jk, with ˆv jk = v, ψ jk ; j,k 1/ ˆv jk, v L ). Next, let us recall that if ϕ B s 0 pq ) s 0 > 0, 1 < p, q < + ) then the Bernstein and Jackson inequalities hold for every space Z = B s pq ) with 0 s < mins 0, L). Indeed, we have 30) v B s pq ) js v L p ), v V j, j

9 Wavelets on the interval 131 and 31) v P j v L p ) js v B s pq ), v B s pq ), j. Thus, taking into account 1), we can apply the characterization Theorems 1 and to the Besov space Z. 3. Scaling function spaces for the half-line Starting from a biorthogonal decomposition on as described in Subsection.3, we aim to the construction of dual scaling function spaces V + ) j and Ṽ + ) j which will form a multilevel decomposition of L p + ) and of L p + ), 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 L and ñ 0 n 0 0 n 1 ñ 1 so that supp ϕ supp ϕ; 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 n 0, ϕ 0k have support contained in [0, + ). More precisely supp ϕ 0k = [n 0 + k, n 1 + k]. Let us fix a nonnegative integer δ and set k 0 = n 0 + δ; observe that k0 k = min : supp ϕ 0k [δ,+ ). Let us define 3) V +) = span ϕ [0,+ ) 0k : k k0 ; this space will be identified in a natural way with a subspace of V 0 ) and will not be modified by the subsequent construction. To obtain the right scaling space V + ) 0 for the half-line, we will add to the basis ϕ [0,+ ) 0k : k k0 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 1 and every fixed x, px) = k p 0k ϕ 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) = c αk ϕ 0k x) 33) = k n 1 +1 k 0 1 k= n 1 +1 c αk ϕ 0k x) + c αk ϕ 0k x), k k 0 where 34) c αk := p α ) 0k = p α y) ϕy k) dy, α = 0,..., L 1.

10 13 L. Levaggi A. Tabacco 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 35) φ α x) = k 0 1 k= n 1 +1 c αk ϕ 0k x), α = 0,..., L 1. REMARK 1. Let p α α = 0,..., L 1 and pα α = 0,..., L 1 be two bases of L 1. Let us denote by φ α and φα 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 for every α. L 1 φα = β=0 M αβ φ β PROPOSITION 1. The functions φ α, α = 0,..., L 1 are linearly independent. Proof. If δ is strictly positive, the linear independence of the boundary functions is obvious. Indeed, on [0,δ], they coincide with linearly independent polynomials. If δ = 0 let us observe that the functions ϕ 0k [0,+ ) involved in 35) have staggered support, i.e., supp ϕ 0k [0,+ ) = [0, n 1 +k], thus they are linearly independent. To obtain the linear independence of the functions φ α it is sufficient to prove that the matrix C = c αk ) k= n 1+1,..., n 0 1 α=0,...,l 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 α for any α, one has c αk = α β=0 ) α k β M α β, β where M i = x i ϕx)dx is the i-th moment of ϕ on. Let v be a vector in L such that C T v = 0, then the polynomial of degree L 1 L 1 L 1 β=0 α=β ) α v α M α β x β β has n 1 n 0 1 distinct zeros; thus, recalling the relation 7), it is identically zero. This means Mv = 0 where M = M ij ) is an upper triangular matrix with M ij = j i) M j i if j i. M is non-singular, in fact detm) = ϕx)dx) L = 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 functions ϕ 0k, one can easily check that PROPOSITION. The functions φ α, α = 0,..., L 1 and ϕ 0k [0,+ ), k k 0, are linearly independent.

11 Wavelets on the interval 133 Thus, it is natural to define 36) V 0 + ) = span φ α : α = 0,..., L 1 V +). We rename the functions in the following way 37) φk if k = 0,..., L 1, ϕ 0k = ϕ 0,k if k L. 0 +k L Observe that 38) V +) = span ϕ 0k : k L. We study now the biorthogonality of the dual generators of V 0 + ) and Ṽ 0 + ). Setting 39) k = max k 0, k 0 = max n0 + δ, ñ 0 + δ, let us observe that ϕ 0k : k k and ϕ 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 φ α : α = 0,..., L 1 ϕ 0k : k = k0,..., k 1 and by φ β : β = 0,..., L 1 ϕ 0k : k = k 0,..., k 1. This requirement can be translated into an explicit relation between δ and δ; indeed, we must have L k 0 = L k 0, i.e., 40) δ δ = L L + ñ 0 n 0. Since L L, we get k = ñ 0 + δ. REMARK. The two parameters δ and δ 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, between δ and δ, equal to zero and determine the other one from the relation 40). In particular, if L L + ñ 0 n 0 0, we set δ = 0 and δ = L L + ñ 0 n 0 ; in this case δ < L; whereas if L L + ñ 0 n 0 < 0, we choose δ = 0 and δ = L L + n 0 ñ 0. In analogy with the previous case, we will suppose 41) as 0 δ < L. Let us define the spaces V 0 B and Ṽ 0 B spanned by the so called boundary scaling functions 4) V B 0 := span ϕ 0k : k = 0,..., L 1, Ṽ B 0 = span ϕ 0k : k = 0,..., L 1, and the spaces V 0 I and Ṽ 0 I spanned by the interior scaling functions as 43) V I 0 := span ϕ 0k : k L, Ṽ I 0 := span ϕ 0k : k L.

12 134 L. Levaggi A. Tabacco Thus we have 44) V 0 + ) = V B 0 V I 0, Ṽ 0 + ) = Ṽ B 0 Ṽ I 0. Note that, if L < L, 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 I and Ṽ 0 I are biorthogonal. Recalling 35) and 37), if 0 k L 1 there exist coefficients α km such that ϕ 0k = m<k α kmϕ 0m ; thus, if l L, due to the position of the supports, we have ϕ 0k, ϕ 0l = α km ϕ 0m ϕ 0,k +l L dx = 0, m<k i.e., V B 0 Ṽ I 0 ). 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 V B 0, say η 0k : k = 0,..., L 1, and one of Ṽ B 0, say η 0l : l = 0,..., L 1, such that η 0k, η 0l = δ kl, k, l = 0,..., L 1. Setting η 0k = L 1 m=0 d kmϕ 0m and η 0k = L 1 m=0 d km ϕ 0m, and calling X the Gramian matrix of components 45) X kl = ϕ 0k, ϕ 0l, k, l = 0,..., L 1, this is equivalent to the problem of finding two L L real matrices, say D = d km ) and D = d km ), satisfying 46) DX D T = I. A necessary and sufficient condition for 46) to have solutions is clearly the non-singularity ) of X, or equivalently V 0 Ṽ B B 0 = 0. If this is the case, there exist infinitely many couples which satisfy equation 46); indeed if we choose D non-singular then it is sufficient to set D = X D ) 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 0 and ϕ 0l l 0 are dual biorthogonal bases. Let us prove that the functions ϕ 0k, k 0, form a p-stable basis of V + ) 0. PROPOSITION 3. We have V + ) 0 = v = α k ϕ 0k : α k k k 0 l p with 47) v L p + ) α k k l p, v V 0 + ).

13 Wavelets on the interval 135 Proof. Let v be any function in V + ) 0 ; by 44) it can be written as v = v B + v I with v B = L 1 k=0 α kϕ 0k and v I = k L α kϕ 0k. The sequence α k k L is p-summable thanks to the inclusion of V0 I in V 0 ) and by 6), so the first part of the Proposition is proved. To show 47), let us set K = max supp ϕ 0k : k = 0,..., L 1 and note that 48) L 1 v B p L p 0,K) k=0 α k p v p L p + ). Indeed, since for any N \ 0, the application x = x 0,..., x N ) N+1 N x n ϕ 0n, n=0 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 p α k p = vx) ϕ 0k x) dx supp ϕ 0k ) p/p vx) p dx ϕ 0k x) p dx v p L supp ϕ 0k supp ϕ p + ). 0k Thus, by 48) and the p-stability of ϕ 0k on the line 6), we have v p L p + ) = v B + v I p L p + ) v B p L p 0,K) + v I p L p + ) k 0 α k p. On the other hand, we have and so v I L p 0,K) = v v B L p 0,K) v L p 0,K) + v B L p 0,K) v L p + ) v I p L p + ) v I p L p 0,K) + v I p L p [K,+ )) = v I p L p 0,K) + v p L p [K,+ )) v p L p + ). Then L 1 α k p = α k p + α k p v B p L p 0,K) + v I p L p + ) k 0 k=0 k L v p L p + ), and the result is completely proven. Similarly, it is possible to show that the dual basis ϕ 0k k 0 of Ṽ + ) 0 is a p -stable basis. Let us introduce the isometries T j : L p + ) L p + ) and T j : L p + ) L p + ) defined as 49) T j f )x) = j/p f j x), T j f ) x) = j/p f j x).

14 136 L. Levaggi A. Tabacco and set ϕ jk = T j ϕ 0k, ϕ jk = T j ϕ 0k, j, k 0. We define the j-th level scaling function spaces as 50) V j + ) := T j V0 + )), Ṽ j + ) := T j Ṽ0 + )). Let us now show that these are families of refinable spaces. PROPOSITION 4. For any j, one has the inclusions V j + ) V j+1 + ) and Ṽ j + ) Ṽ 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 V + ) 0 V + ) 1. For k L, rewriting 5), one has ) 1 ϕ 0k x) = ϕ 0,k 0 +k L x) = 1 ) p h m k 0 +k L) T 1 ϕ 0m x). m As the filter h n is finite, we see that the first non vanishing term in the sum corresponds to ϕ 0,k 0 +δ+k L), which belongs to V +) for any k L, so that the function on the left hand side belongs to V + ) 1. Suppose now k < L; without loss of generality we can choose p α x) = x α for any α, so that, by 33), 34) and 35), one has, for x 0, 1/p x) k = 1/p φ k x) + c km ϕ 0m x) = ϕ 1k x) + m k0 c km T 1 ϕ 0m )x). Again, using 35), ϕ 0k = k+1/p) ϕ 1k + m k0 c km ϕ 1m c km ϕ 0m = k+1/p) m k0 m k0 ϕ 1k + c k,k 0 +m L ϕ 1m c k,k 0 +m L ϕ 0m, m L m L which completes the proof. REMARK 3. Note that, choosing the basis of monomials for L 1, the refinement equation for the modified border functions in V 0 B takes the form 51) φ α = α+1/p) T 1 φ α + H αk ϕ 1k where 5) k k0 ) H αk = c αk α+1/p) 1 1 p c αl h k l, l k 0 involving only the respective modified border function in V B 1.

15 Wavelets on the interval Projection operators Following the guiding lines of the abstract setting, we will define a sequence of continuous linear operators P j : L p + ) V j for j, satisfying 7), 8) and 9). By 50), it is obvious that the definition of P 0 gives naturally the complete sequence, by posing P j = T j P 0 Tj 1, where T j is the isometry defined in 49). For v L p + ), let us set P 0 v = v 0k ϕ 0k, with v 0k = vx) ϕ 0k x) dx. k 0 We will first prove that P 0 is a well-defined and continuous operator. PROPOSITION 5. We have P 0 v p L p + ) v 0k p v p L p + ), v L p + ). k 0 Proof. Let us write P 0 v = L 1 k=0 v 0k ϕ 0k + k L v 0k ϕ 0k. Observe that, by the Hölder inequality, L 1 L 1 v 0k p v p L p + ) ϕ 0k p = C v p L p + ) L p + ). k=0 k=0 Thus, by the p-stability property on the line 6), v 0k p v p L p + ). k 0 This implies P 0 v V 0 + ) and the result follows by 47). Since T j 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 V + ) j V + ) j+1, proven in Proposition 4. Similarly, one can define a sequence of dual operators, P j : L p + ) Ṽ + ) j, satisfying the same properties of the primal sequence 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 ). As in Section., we consider a Banach subspace Z of L p + ) and suppose that the scaling function ϕ belongs to Z. In the following, Z will be the Besov space B s 0 + ) pq, with s 0 > 0 and 1 < p, q < +.

16 138 L. Levaggi A. Tabacco PROPOSITION 6. For any 0 s s 0, the Bernstein-type inequality 53) holds. v B s pq + ) js v L p + ), v V + ) j, j, Proof. Applying the definition of the operator T j, it is easy to see that 54) T j v B s pq + ) = js v B s pq + ), v Bs + ) pq, j ; so, by 50), it is enough to prove the inequality for j = 0. Proceeding as in Proposition 3, we choose v V + ) 0 and write v = v B + v I. Let us estimate separately the semi-norms of the two terms. We have v B p B s pq + ) = L 1 p p L 1 v 0k ϕ 0k v 0k ϕ 0k B s k=0 pq + ) B s pq + ) k=0 L 1 k=0 v 0k p, the constants depending on the semi-norms of the basis functions and the equivalence of norms in L. Observe that v I is an element of V 0 = V 0 ); using the Bernstein inequality 30) and the p-stability on the line 6), we get v I B s pq + ) v I L p ) k L 1/p v 0k p. Thus, by 47), v p B s pq + ) ) v B p B s pq + ) + v I p B s pq + ) v 0k k p l p v L p + ). Next, we prove the generalized Jackson inequality, following the same ideas used in showing the analogous property 31) on the real line see [5]). PROPOSITION 7. For each 0 s < mins 0, L), we have 55) v P j v L p + ) js v B s pq + ), v Bs + ) pq, j. Proof. As before see 54)), it is enough to prove that v P 0 v L p + ) v B s pq + ), v Bs + ) pq. Let us divide the half-line into unitary intervals, + = l 0 I l where I l = [l, l+1], and estimate v P 0 v L p I l ). Recalling that polynomials up to degree L 1 are locally reconstructed through the basis of V + ) 0, it is easy to see that for any q L 1 there exists v q in V + ) 0 such that v q = q on I l. Then, using 8), we have v P 0 v L p I l ) = v q + P 0 q P 0 v L p I l ) v q L p I l ) + P 0 v q) L p I l ).

17 Wavelets on the interval 139 Moreover, from the compactness of the supports of the basis functions, setting R l = k : supp ϕ 0k I l = and J l = k Rl supp ϕ 0k, for any f L p + ), one gets P 0 f p p L p I l ) = f 0k ϕ 0k x) dx I l k R l ) p ) p max f L k R p supp ϕ 0k ) ϕ 0k L p l + ) ϕ 0k x) dx I l k R l Thus f p L p J l ). v P 0 v L p I l ) v q L p J l ), q L 1. Taking the infimum over all q L 1, we end up with 56) v P 0 v L p + ) inf q l 0 L 1 v q L p J l ). A local version of Whitney s Theorem see [1]) for a given interval of the real line J, states that, for any v L p J), 57) where inf v q L q p J) w L) v, J), L 1 [ w L) 1 J ) ] 1/p v, J) = dh L J h v p dx J JLh) and Js) = x J : x + s J. Observing that h = J l independent of l) and using 57), we have v P 0 v p L p + ) w L) ) p v, J l ) l 0 1 h = h dh h h L v p dx l 0 J l Lh) sup L h h + h v p dx = ω L) p v, h )) p, where ω L) p is the modulus of smoothness see [1]). To conclude the proof, it is enough to observe that, for any 1 < q <, ω L) p v, h ) jsq 1/q ω L) p v, j ) q ) v B s pq + ). j Since B s + ) pq is dense in L p + ), the following property immediately follows. COROLLARY 1. The union j V j + ) is dense in L p + ).

18 140 L. Levaggi A. Tabacco 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 W 0 + ) such that V 1 + ) = V 0 + ) W 0 + ) note that the sum is not, in general, orthogonal) and W 0 + ) Ṽ 0. To this end, let us consider the basis functions of V 1 + ) and let us write them as a sum of a function of V 0 + ) and a function which will be an element of W 0 + ). 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]): 58) 59) ϕ 1k = 1 p 1 h k m ϕ 0m + g k m ψ 0m, ñ 0 k m ñ 1 1 n 1 k m 1 n 0 ψ 0m = 1 1 p 1+m ñ 0 l=1+m ñ 1 g l m ϕ 1l. Interior wavelets. Since V + ) 0 contains the subspace V +) defined in 3), V + ) 1 contains the subspace T 1 V +) = ϕ [0,+ ) 1k : 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 k 0. This is equivalent to m k0 + ñ 1 1, so we set 60) k m 0 + ñ 1 1 =: m 0. Since, see.9) in [11]), 61) h n h n k = δ 0k, k, n it is easy to see that ñ 1 n 0 is always odd; thus, by 39), 6) m 0 = ñ1 n δ. Let us set 63) W I 0 := span ψ 0m [0,+ ) : m m 0 ; we observe that W0 I can be identified with a subspace of W 0 ), thus it is orthogonal to Ṽ 0 and W 0 I W + ) 0. The functions ψ 0m := ψ 0m [0,+ ) are called interior wavelets. Border wavelets. Let us now call W 0 B a generic supplementary space of W 0 I in W + ) 0 and set V1 B = T 1V0 B. PROPOSITION 8. The dimension of the space W B 0 is m 0.

19 Wavelets on the interval 141 Proof. Let K > 0 be an integer such that on [K,+ ) all non-vanishing wavelets and scaling functions are interior ones. Then V1 B span ϕ 1k : k0 k < n 0 + K = [ V0 B span ϕ 0k : k0 k < n 0 + K ] [ W0 B span ψ 0m : m 0 m K 1], since, by 58), the first interior wavelet used to generate ϕ 1, n0 +K is ψ 0K. Then the result easily follows. To build W0 B we need some functions that, added to W 0 I, will generate both V B 1 and the interior scaling functions that cannot be obtained in 58) using V +) ) and W0 I. Thanks to 51), we only have to consider the problem of generating interior scaling functions. Let us now look for the functions ϕ 1k generated by ϕ 0m, for m k 0, and by ψ 0m, with m m 0. Let us work separately on the two sums of 58): a) we must have m k ñ 1 )/. Imposing m k0 and seeking for integer solutions, we get k ñ1 64) k 0 = n 0 + δ ; b) similarly, we obtain m n k)/. Again, we want m m 0, so we must have n0 1 + k m 0. Using 6), this means n0 1 + k 65) n 0 + ñ δ. Since 64) and 65) have to be both satisfied, we obtain the following condition 66) k n 0 + ñ 1 + δ 1 = k 0 + ñ 1 1. Indeed, this can be seen considering all possible situations. For instance, if n 0 and δ are even, then ñ 1 is odd and δ = δ. 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 k ñ1 = k ñ and n0 1+k = n 0 1+k + 1, and 66) easily follows. The other cases are dealt with similarly. Let us set 67) so that k = k 0 + ñ 1 1, span ϕ [0,+ ) 1k : k k V +) W0 I. We are left with the problem of generating some functions of V + ) 1, precisely ϕ 1k with k0 k < k. Observe that we have to generate k k 0 functions using a space of dimension m 0 = k k 0. In fact one can show that one out of two ϕ 1k, for k = k0,..., k 1, depends on the previous ones through elements of level zero.

20 14 L. Levaggi A. Tabacco PROPOSITION 9. Let e = 0 if δ is even, e = 1 if δ is odd. For all 1 m m 0 e, one has ϕ 1,k m [0,+ ) S m V I 0 W I 0, with S m := span ϕ [0,+ ) : 1 l m. 1,k l+1 Proof. Let us set ñ 1 n 0 = r + 1 with r > 0 recall that ñ 1 n 0 is 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 ) 1p ϕ = 1 1,k l h k l n ϕ 0n + g n k0 l k l n ψ 0n n m 0 + δ l 68) 1p = ) 1 hñ1 1ϕ 0,k 0 l + hñ1 3ϕ 0,k 0 l ) + g n0 ψ 0,m g δ n l 0 ψ ,m 0 + δ l+1 and 69) ) 1p ϕ = 1 1,k l+1 h k l+1 n ϕ 0n + g n k0 l k l+1 n ψ 0n n m 0 + δ l 1p = ) 1 hñ1 ϕ 0,k0 l + hñ1 ϕ 0,k 0 l ) + g n0 +1ψ 0,m g δ n l 0 1ψ ,m 0 + δ. l+1 Let us prove the stated result by induction on m. For m = 1 we consider the linear combination h n0 ϕ 1,k + h n0 +1ϕ 1,k 1 = 1p 1 ) h n0 hñ1 1 + h n 0 +1 hñ1 ) ϕ 0,k ) h n0 g n0 + h n0 +1 g n0 +1 ψ0,m δ 1 n k 0 c 1n ϕ 0n d 1nψ 0n, n m 0 + δ for some coefficients c 1n and d 1n. Writing 61) with k = n 0 ñ 1 +1 = 0 and g n h n k = 0 n see 3.9) in [11]) with k = n 0, we have h n0 hñ1 1 + h n0 +1 hñ1 = 0, h n0 g n0 + h n0 +1 g n0 +1 = 0.

21 Wavelets on the interval 143 Thus we have h n0 ϕ 1,k [0,+ ) + h n0 +1ϕ 1,k 1 [0,+ ) V I 0 W I 0, and the result follows because h n0 = 0. Set now 1 < m m 0 e. As before, we choose a certain linear combination of the scaling functions ϕ and ϕ with 1 l m. Then we use 68), 69) to represent them 1,k l 1,k l+1 through functions of level 0. More precisely h n0 ϕ + h 1,k m n0 +1ϕ h 1,k m+1 n0 +m ϕ + h 1,k n0 +m 1ϕ 1,k 1 ) = h n0 hñ1 1 + h n 0 +1 hñ1 ϕ 0,k 0 m ) + h n0 hñ1 3 + h n0 +1 hñ1 + h n0 + hñ1 1 + h n0 +3 hñ1 ϕ 0,k 0 m+1 70) +... ) + h n0 hñ1 m+1 + h n0 +1 hñ1 m h n0 +m 1 hñ1 ϕ 0,k c mn ϕ 0n n k 0 + h n0 g n0 + h n0 +1 g n0 +1) ψ0,m 0 + δ m + h n0 g n0 + h n0 +1 g n0 1 + h n0 + g n0 + h n0 +3 g n0 +1) ψ0,m 0 + δ m h n0 g n0 m+ + h n0 +1 g n0 m h n0 +m g n0 +1) ψ0,m 0 + δ 1 + n m 0 + d mnψ 0n, δ for some c mn and d mn. The coefficients of the functions ϕ 0,k 0 m,...,ϕ 0,k0 can be written 1 as 71) h n h n k = δ 0k, n with k = n 0 ñ 1 +1,..., n 0 ñ m 1 = r,, r + m 1, respectively. Similarly, the coefficients of the functions ψ 0,m δ,..., ψ 0 + m 0,m δ can be written as h n k g n = 0, n with k = n 0,, n 0 m + 1, respectively. Observe now that m m 0 e = r + δ. If m r, all indices k in 71) are negative, so ) [0,+ ) h n0 ϕ + h 1,k m n0 +1ϕ 1,k m+1 h n0 +ϕ + h 1,k m+ n0 +3ϕ ,k m+3 + h n0 +m ϕ 1,k + h n0 +m 1ϕ 1,k 1 ) [0,+ ) + V I 0 W I 0

22 144 L. Levaggi A. Tabacco and the result is proven by induction since h n0 = 0. If m > r i.e., δ ), we get ) [0,+ ) h n0 ϕ + h 1,k m n0 +1ϕ 1,k m+1 h n0 +ϕ + h 1,k m+ n0 +3ϕ ,k m+3 + h n0 +m ϕ 1,k + h n0 +m 1ϕ 1,k 1 ) [0,+ ) + 1p 1 ) ϕ 0,k 0 m+r [0,+ ) + V0 I W 0 I. Therefore, by the induction hypothesis, we only have to prove that ϕ 0,k 0 m+r [0,+ ) S m V0 I W 0 I. We immediately get m r m 0 e r = δ, so we show that 7) ϕ 0,k 0 l [0,+ ) S l, by induction on l. If l = 1, from 69) we have 1 l δ, ) 1 ϕ 0,k 0 1 [0,+ ) = 1 p hñ1 ϕ [0,+ ) + c 1,k 1 1n ϕ [0,+ ) 0n n k0 + d 1n ψ n m 0 + 0n [0,+ ) S 1 V0 I W 0 I δ 1 for some c 1n and d 1n ). If l > 1, using induction, we similarly get ) 1 ϕ 0,k 0 l [0,+ ) = 1 p hñ1 ϕ [0,+ ) + 1,k l+1 c l,n ϕ 0n n k0 l+1 + d l,n ψ n m 0 + 0n [0,+ ) S l V0 I W 0 I δ l [0,+ ) again c ln and d ln are fixed coefficients). Thus we have proven 7), and this completes the proof. 73) and Using this result, setting ψ 0,m 0 l := ϕ ) 1,k l+1 [0,+ ) P 0 ϕ [0,+ ), l = 1,..., m 1,k l+1 0, 74) W B 0 = ψ 0m m = 0,..., m 0 1, we get 75) W 0 + ) = W B 0 W I 0,

23 Wavelets on the interval 145 with W I 0 as defined in 63). With the same process we can build W 0 + ) = W B 0 W I 0. As for the scaling functions, we must find a couple of biorthogonal bases for the spaces W + ) 0 and W + ) 0. To fix notations, we suppose that m 0 m 0 otherwise we only have to exchange m 0 and m 0 in what follows). From the biorthogonality properties on the real line we have ψ 0m, ψ 0n = δ mn, m, n m 0. Note, however, that the modified wavelets we have defined are no longer orthogonal to the interior ones. In fact, from definition 73), it follows that ψ 0m, ψ 0n = ϕ 1,k m 0 m)+1, ψ 0n, m = 0,..., m 0 1, n m 0. Using the refinement equation for wavelets on the real line, we easily show that k ψ 0m, ψ 0n = 0, n = 0,..., m 0 1 if m + ñ 1 1 =: m 76) ψ 0m, ψ 0n = 0, m = 0,..., m 0 1 ifn k + n 1 1 =: m. Observing that m m, it is sufficient to find two m m matrices E = e mr ) and Ẽ = ẽ ns ) such that m 1 m 1 e mr ψ 0r, ẽ ns ψ 0s = δ mn, m, n = 0,..., m 1. r=0 s=0 Calling Y the m m matrix of components Y mn = ψ 0m, ψ 0n, this condition is equivalent to EY Ẽ T = 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 we immediately get the result observing that det Y = 0 iff W 0 + ) W 0 + ) = 0, W 0 + ) W 0 + ) V1 + ) Ṽ 1 = 0. Moreover indeed, for any v L p + ), we have W 0 + ) Ṽ 0 ; v P 0 v, ϕ 0k = v 0k l 0 v 0l ϕ 0l, ϕ 0k = 0. Finally, for any j, we set 77) W j + ) = T j W 0 + ) and W j + ) = T j W 0 + ) ;

24 146 L. Levaggi A. Tabacco setting ψ jm = T j ψ 0m, ψ jm = T j ψ 0m for every j, m, it is easy to check that the biorthogonality relations ψ jm, ψ j n = δ j j δ mn, j, j, m, n 0, hold. Moreover, with a proof similar to the one of Proposition 3, one has PROPOSITION 10. The bases j = ψ jm : m of W + ) j, are uniformly p-stable bases and the bases j = ψ jm : m of W + ) j, 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 V + ) ) j, Ṽ j j 0 as described in Section 3. With the notation of the Introduction, for 1 < p, q < +, let us set B s + ) X s pq if s 0, pq := B s + )) p q if s < 0, and denote by + ) the space of distributions. THEOREM 3. Let ϕ B s 0 + ) pq for some s 0 > 0, 1 < p, q < +. For all s S := mins 0, L), mins 0, L)) \ 0, the following characterization holds: v L p + ) : j 0 sq j k 0 v jk p) q/p < + if s > 0, where X s pq = v + ) : v X s pq, for some s S and j 0 sq j k 0 v jk p) q/p < + ˆv v jk = jk = v, ψ jk if s > 0, ˆṽ jk = v, ψ jk if s < 0. In addition, for all s S and v X s pq, we have 78) v X s pq sq j q/p 1/q v jk p. j 0 k 0 if s < 0, Finally, if p = q =, the characterization and the norm equivalence hold for all index s mins 0, L), mins 0, L)). Proof. It is sufficient to apply Theorems 1 and since the Bernstein- and Jackson- type inequalities have been proven in Proposition 6 and 7, respectively) and remember the interpolation result 1). REMARK 4. It is possible to obtain a characterization result using the same representation of a function v X s pq for both positive and negative s. Indeed, if ϕ Bs 0 pq ), ϕ B s 0 pq ), given any s min s 0, L), mins 0, L)) \ 0 and v = j,k 0 ˆv jkψ jk X s pq, we have the same norm equivalence as in 78) with ˆv jk instead of v jk see, e.g., [14]). We also observe that, in general, s 0 < s 0 and thus Theorem 3 gives characterization for a larger interval.

25 Wavelets on the interval Boundary values of scaling functions and wavelets The aim of this section is to construct scaling functions and wavelets satisfying certain boundary value conditions, in view of the characterization of spaces arising from homogeneous boundary value problems. More precisely, we will see that it is possible to construct a basis of scaling functions in such a way that only one scaling function is non-zero at zero both for the primal and the dual systems. A similar property will be shown for the wavelet basis. Let us start considering the scaling function case. Observe that all the interior scaling functions are zero at zero, while the value at zero of the boundary scaling functions depends upon the choice of the polynomial basis p α of L 1 and the biorthogonalization. If we start with a properly chosen polynomial basis e.g., p α x) = x α ), only ϕ 00 and ϕ 00 see 35)) do not vanish at zero. Thus, the idea is to biorthogonalize first the functions in the sets = ϕ 0k : k = 1,..., L 1 and = ϕ 0k : k = 1,..., L 1, so that the resulting systems contain functions which all vanish at zero. To do this, we need to check the non-singularity of the Gramiam matrix ϕ 0k, ϕ 0k k,k =1,..., L 1 obtained from the matrix X see 45)) by deleting the first row and column. As for the non-singularity of the whole matrix we have to check this case by case. For example, for the B-splines case, this property is satisfied due to the total positivity of the associated matrix X see Proposition 1 and also [16]). From now on we suppose this property is verified and biorthogonalize and. For simplicity, we will maintain the same notations for the new basis functions. The second step consists in the biorthogonalization of the complete systems, keeping invariant the functions in and. Precisely, we have the following general result. PROPERTY 1. Let and be the two biorthogonal systems described above. Consider = ϕ 00, = ϕ 00 and suppose that the matrix ϕ 0k, ϕ 0k ), k, k = 0,..., L 1, is non-singular. Then, it is possible to construct new biorthogonal systems spanning the same sets as and, respectively, in which only the two functions ϕ 00, ϕ 00 have been modified. Proof. Let us set L 1 ϕ 00 # = k=1 α k ϕ 0k + α 0 ϕ 00, L 1 ϕ 00 # = l=1 β l ϕ 0l + β 0 ϕ 00. We want to prove that we can find α k, β l, k,l = 0,..., L 1 so that α 0 β 0 = 0, 79) and 80) ϕ # 00, ϕ 0l = ϕ 0k, ϕ # 00 = 0, k, l = 1,..., L 1, ϕ # 00, ϕ# 00 = 1. Imposing the conditions 79) and using the biorthogonality of the systems,, we get α k = α 0 ϕ 00, ϕ 0k and β l = β 0 ϕ 0l, ϕ 00, k, l = 1,..., L 1. Substituting these relations in 80), we end up with the identity L 1 Kα 0 β 0 = 1 where K = ϕ 00, ϕ 00 ϕ 00,ϕ 0k ϕ 0k. k=1

26 148 L. Levaggi A. Tabacco To conclude the proof it is enough to show that K = 0. Indeed, if K = 0, the function η = ϕ 00 L 1 k=1 ϕ 00,ϕ 0k ϕ 0k span ), would also be orthogonal to ϕ 00 ; moreover η span and the systems and are biorthogonalizable, i.e., span ) span ) = 0; this would mean η = 0, contradicting the linear independence of the functions in. Next we consider the wavelet case. Suppose we have constructed biorthogonal wavelets ψ jk k 0, ψ jk k 0 starting from biorthogonal scaling systems ϕ jk k 0, ϕ jk k 0 such that 81) ϕ j0 0) ϕ j0 0) = 0 and ϕ jk 0) = ϕ jk 0) = 0, k 1, j 0. Recalling that ϕ j+1,k = T 1 ϕ jk, one has 8) ϕ j+1,0 0) = 1/p ϕ j0 0), ϕ j+1,0 0) = 1/p ϕ j0 0). We want to prove that we can modify the wavelet systems and obtain a property similar to 81). To this end, we report general observations about biorthogonal bases that can be found in the Appendix of [7]. Let S, S be two spaces of functions defined on some set with biorthogonal bases E = η l l and Ẽ = η l l here is some set of indices) with respect to some bilinear form S, S on S S. Let F = ν l l and F = ν l l be two other bases such that ν l = K lm η m, ν l = K lm η m, where K and K are suitable generalized matrices. It is easy to check that, to preserve the biorthogonality, we must have K = K T. LEMMA 1. With the previous notation, suppose the elements of E and Ẽ are continuous functions, then the quantity η l x) η l x), x l is invariant under any change of biorthogonal basis. Proof. Let us denote by ex) the vector η l x)) l, and similarly for ẽx), f x), f x). Note that f x) = K ex) and f x) = K T ẽx); thus ν l x) ν l x) = f x) T f x) = ex) T K T K T ẽx) = ex) T ẽx) = η l x) η l x). l l We will apply this result to our biorthogonal wavelets. COROLLARY. Suppose the scaling functions ϕ, ϕ are continuous on 83) ψ 0k 0) ψ 0k 0) = 0. k 0, then Proof. Let S = V + ) 1, S = Ṽ + ) 1. Using the relations V + ) 1 = V + ) 0 W + ) 0 and Ṽ + ) 1 = Ṽ + ) 0 W + ) 0, we have two couples of biorthogonal bases on S and S: E = ϕ 1k k 0, Ẽ = ϕ 1k k 0 and F = ϕ 0k k 0 ψ 0k k 0, F = ϕ 0k k 0 ψ 0k k 0. Using the previous Lemma, 81) and 8), we have ϕ 10 0) ϕ 10 0) = ϕ 00 0) ϕ 00 0) = ϕ 00 0) ϕ 00 0) + k 0 ψ 0k 0) ψ 0k 0).

27 Wavelets on the interval 149 Thus ψ 0k 0) ψ 0k 0) = ϕ 00 0) ϕ 00 0) = 0. k 0 This implies that we can always find k 0 such that ψ 0k 0) ψ 0k 0) = 0. Without loss of generality, we can suppose k = 0. Let us define, for k 1, 84) and 85) ψ 0k x) := ψ 0kx) ψ 0k0) ψ 00 0) ψ 00x) =: ψ 0k x) c k ψ 00 x) ψ 0k x) := ψ 0k x) ψ 0k 0) ψ 00 0) ψ 00 x) =: ψ 0k x) c k ψ 00 x). Observe that ψ0k 0) = ψ 0k 0) = 0, k 1; moreover only a finite number of wavelets are modified, since all the interior ones vanish at the origin. Thus, to end up our construction, it is enough to show that it is possible to biorthogonalize the systems ψ 0k Indeed LEMMA. The matrix Y = ψ0k, ψ 0l m 1 is non-singular. k,l=1 Proof. Let us set M = m 1; for k, l = 1,..., M we have ψ 0k, ψ 0l = 1 + ck c k if k = l c k c l if k = l. k 1 and ψ 0k k 1. It is easy to see that Y has only two different eigenvalues: λ 1 = 1, with multiplicity M 1 and λ = 1 + M k=1 c k c k, with multiplicity 1. Thus, by Corollary, det Y = 1 + M c k c k = k=1 k 0 ψ 0k0) ψ 0k 0) = 0. ψ 00 0) ψ 00 0) Finally we construct our wavelet systems as described in Property Characterization of Besov spaces B s + ) pq,00 We now prove that we can build multiresolution analyses on L p + ) of functions satisfying homogeneous boundary value conditions in zero. This can be done by choosing sufficiently regular scaling functions ϕ and ϕ and by building boundary functions starting from particular bases of polynomials. Suppose ϕ B s 0 pq ) or W s0,p )) and S = [s 0 ] < L. Let us consider the basis px) = x α of the monomials, and let us build the functions on the boundary as in 35). We remove the first S boundary functions and define see 4)) 86) 0V B 0 + ) = span ϕ 0k : S k L 1 := span 0 0

28 150 L. Levaggi A. Tabacco and 87) 0Ṽ 0 B + ) = span ϕ 0k : S k L 1 := span 0 0. The systems 0 0 and 0 0 can be biorthogonalized if the matrix 0 X = ϕ 0k, ϕ 0l ), with k, l = S,..., L 1 is non-singular. For example, this condition is verified in the B-spline case see Proposition 1 and also [16]). As before, we set + 0V ) j := T j 0V + )) 0 and + 0Ṽ ) j := T + )) j 0Ṽ0. Let us define 0 P 0 : L p + ) 0 V + ) 0 as 88) 0 P 0 v = v 0k ϕ 0k, v L p + ). k S For any level j > 0, let us set 89) 0 P j = T j 0 P 0 T 1 j. Similar definitions hold for the dual operators 0 P j. These sequences of operators satisfy the requirements 7), 8) and 9). Following the same construction as in Sections 3 and 5, we build a multiresolution analysis that will be used to characterize the Besov spaces B s pq,00 see 3)). For the scaling spaces the situation is basically the same, in fact we have only dropped some functions on the boundary. Many results hold in this context; for example, since 0 V + ) 0 is a subspace of V + ) 0, one has 1/p v L p + ) v 0k p k S, for any v 0 V 0 + ) see Proposition 5). We note that while the number of boundary wavelets does not change, they are defined in a different way since their definition depends on the projector 0 P 0. In fact, one has for k = 1,..., m 0. 0ψ 0,m 0 k := ϕ 1,k k+1 [0,+ ) 0 P 0 ϕ 1,k k+1 [0,+ ) S 1 = ψ 0,m 0 k + ϕ, ϕ 1,k k+1 0l ϕ 0l l= Bernstein and Jackson inequalities In order to use the characterization Theorems 1 and, we will prove Jackson- and Bernstein-type inequalities for Besov spaces B s + ) pq,0 or Sobolev spaces W s,p + ) 0. Let us observe that the Bernstein inequality follows from 53), because 0 V + ) 0 V + ) 0 and B s + ) pq,0 B s + ) pq with the same semi-norms. It only remains to prove a Jackson-type inequality, that cannot be deduced directly from 55) because it depends on the projectors 88).

29 Wavelets on the interval 151 PROPOSITION 11. For each s such that 0 < S s s 0 < L, we have v 0 P j v L p + ) js v B s pq + ), v Bs + ) pq,0, j. Proof. We can proceed as in Proposition 7, the only difference being on the first interval I 0 = [0, 1] and on the space of polynomials to be considered. Indeed, we consider the subspace 0 L 1 of L 1, i.e., the space of polynomials which are zero at zero with all their derivatives of order less than S. Then, inequality 57) holds for every v B s pq,0 + ) straightforward modifications of the proof of Theorem 4., p. 183, in [19]). Finally, we state a characterization theorem for Besov spaces B s pq,00 + ) based on the biorthogonal multilevel decomposition 0V j + ), 0 Ṽ j + )) as described before. This result immediately follows from Theorem 1. we have 90) THEOREM 4. Let ϕ B s 0 pq + ) for some 0 < s 0 < L, 1 < p, q <. For all 0 < s < s 0, B s + ) pq,00 = v L p + ) : sq j j 0 ˆv jk p k S q/p < and 91) v B s pq + ) sq j q/p 1/p ˆv jk p, v B s + ) pq,00. j 0 k S REMARK 5. Since, for any s, 1 < p, q <, 1 p + 1 p = 1 q + 1 q = 1, B s + )) pq,00 = B s + ) p q, see [3] p. 35) the extension of the previous theorem to the dual spaces negative s) gives the same result of Theorem 3. REMARK 6. As usual, if p = q =, s can assume the value 0. In this particular case, if 0 s 0 < L, we get H00 s + ) = v L + ) : sj ˆv jk < + j 0 k S H0 s + ) if s 1 /, = H00 s + ) if s 1, and 9) v H s + ) 1/ 1 + sj ) ˆv jk j 0 k S, v H00 s + ).