Wavelets and numerical methods. Politecnico di Torino Corso Duca degli Abruzzi, 24 Torino, 10129, Italia

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1 Wavelets and numerical methods Claudio Canuto Anita Tabacco Politecnico di Torino Corso Duca degli Abruzzi, 4 Torino, 9, Italia 7

2 Wavelets and Numerical methods Claudio Canuto Anita Tabacco Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi 4 9, Torino, Italy Workshop Wavelets and Applications Barcelona, July -6,

3 Chapter Abstract multilevel decompositions. Introduction The multilevel decomposition of functional spaces has become a fast-growing area of pure and applied research in Mathematics, during the last few decades. Atomic decompositions, splines, wavelets, stationary subdivision techniques, hierarchical finite elements, multigrid algorithms are examples that fit into the general framework of multilevel decompositions. The basic idea is to consider a sequence of nested subspaces of a functional space V, providing a better and better approximation of V. Each subspace is associated to a level of approximation; changing the level is efficiently accomplished by adding or removing appropriate details. An important feature of multilevel decompositions is the possibility of characterizing the regularity of a function in V by the weighted summability of certain norms of its details. Such a feature turns out to be very useful in the investigation of adaptive approximations of functions, as well as in other areas of analysis and numerical analysis. The characterization of families of functional spaces is based on generalized inequalities of the Bernstein and Jackson type. These families are not restricted to Hilbert or Banach spaces (typically, spaces built on L p -summability with p ), since they include quasi-banach spaces (i.e., spaces built on L p -summability with < p < ). However, different approaches may be required to deal with the Banach case or the quasi-banach case. Although originated in different areas of Mathematics, multilevel decompositions share many aspects and properties. Several abstract frameworks have been proposed in the literature to point out what is common to apparently different constructions. Inspired by the work of Dahmen and DeVore, we present an abstract multilevel setting, which aims at giving a unified approach to both the Banach case (p ) and to the quasi-banach one (p < ). We begin with a minimal set of assumptions, and we deduce from them a multilevel decomposition of V and a fairly general form of a characterization theorem of the DeVore-Popov type for a scale of spaces imbedded in V (Theorem.). Next, we show how extra assumptions lead to more precise results. Finally, as an application, we provide an accurate construction of biorthogonal decompositions of L p (IR) for < p < +, based on 9

4 compactly supported wavelets ([33]); we check that the abstract assumptions are fulfilled and we deduce from the general result a characterization of the non-homogeneous Besov spaces B s pq(ir) ( < p, q, + ), as well as B s σσ(ir) (τ = s + p, p ). The presentation we follow in this chapter is contained in [6, 7]... Notation and definitions Throughout these notes, C will denote a strictly positive constant, which may take different values in different places. Given two functions N i : V R + (i =, ) defined on a set V, we shall use the notation if there exist two constants c, c > such that N (v) N (v) v V (..) c N (v) N (v) c N (v) v V. (..) We recall the definition and the main properties of non-homogeneous Besov spaces. For f L p (IR), < p +, let us denote by r h the difference of order r IN and step h IR, defined as r ( ) r r h f(x) = ( ) r+ f(x + h) (..3) and by ω p (r) (f, t), t >, the modulus of smoothness of order r defined as = ω p (r) (f, t) = sup r h f( ) p, (..4) h t where p denotes the quasi-norm in L p (IR). If s, p, q >, we say f B s pq(ir) L p (IR) whenever the semi-quasi-norm f B s pq (IR) = ( + [t s ω p (r) (f, t)] q dt ) /q (..5) t is finite, provided r is any integer > s (different values of r give equivalent semi-quasinorms). Actually, (..5) is a quasi-norm in B s pq(ir). However, B s pq(ir) will be endowed with the quasi-norm f B s pq (IR) = f p + f B s pq (IR), (..6) for which it is complete. In the sequel we shall repeatedly use an equivalent discrete version of (..6), namely f B s pq (IR) f p + + = [ s ω (r) p We shall also use the following real interpolation result: (f, )] q /q. (..7) (L p (IR), B s pq(ir)) s /s,q = B s pq(ir), (..8)

5 where < p, q < + and < s < s. Finally, we recall that B s (IR) = Hs (IR), the Sobolev space of order s built on L (IR), whereas for all non-integers s and p >, B s pp(ir) = W s p (IR), the Sobolev space of order s built on L p (IR).. Abstract multilevel decompositions Let V be a separable, quasi-banach space with quasi-norm denoted by. Assume that a family of closed subspaces {V } J is given, where J is either Z or N, such that V V +, J. (..9) For all J, an operator P : V V is given, which satisfies the following condition v P (v) C inf u V v u, v V, (..) where C is a constant independent of. Using the quasi-triangle inequality, this condition implies P (v) C v, v V ; (..) besides, (..) also implies P (v) = v, v V, i.e., P = P. (..) Thus, P is a proection operator onto V. Note that neither we require P to be continuous, nor to be linear. Given v V, we shall call P (v) the approximation of v of level. For each J, let us introduce the operator Q : V V + defined by Observe that, again, we have We set W := Im Q, so that Q (v) = P + (v) P (v), v V. (..3) Q (v) C v, v V. (..4) V + = V + W. (..5) This decomposition easily follows from (..3), now written in the form P + (v) = P (v) + Q (v), v V, (..6) taking v V + and using (..). The relation (..6) means that the element Q (v) W is the detail to be added to the approximation of level of v in order to get the approximation of level +. By iterating the decomposition (..5), we get for any two integers, J such that < J V J = V + (W + W W J ), (..7)

6 or, equivalently, P J (v) = P (v) + J = Q (v), v V. (..8) In order to obtain the decomposition of any element v V along all the levels, let us make the following assumptions: for all v V P (v) v as + ; (..9) { P (v) as, if J = Z P (v) = if J = N. (..) It is easy to see that, due to (..), the assumption (..9) is equivalent to the fact that J V is dense in V, whereas the assumption (..) implies that J V = {} (it is indeed equivalent to V = {} if J = N). Taking the limit in (..8), one gets the multilevel decomposition v = J Q (v), v V. (..) Thus, any element in V can be split as the sum of all its details, corresponding to all levels in J. The decomposition (..) is said to be q-stable (for a certain < q ) if v Q (v) q J /q, v V. (..) In some applications, the assumption (..) might not be true. In this case, instead of (..), we merely have the multilevel decomposition v = P (v) + Q (v), v V, (..3) where is any index in J. A more detailed representation of the elements of V is obtained by introducing a basis in each subspace V ( J ) and by expanding Q (v) along this basis. To be precise, let us choose a basis in V {ϕ k k K }, (..4) that we consider as the canonical basis for this subspace; here, K is a certain subset of Z d, for a suitable d. We say that the basis (..4) is a uniformly p-stable basis for V (for a certain < p ) if V = { k K α k ϕ k {α k } k K l p } (..5) and k K α k ϕ k {α k } k K l p, {α k } l p, (..6) the constants involved in the definition of being independent of.

7 Given v V, let us represent P (v) V and Q (v) V + as P (v) = k K v k ϕ k, Q (v) = k K + v k ϕ +,k. (..7) Due to the assumption (..9), there exist coefficients h () km (k K +, m K ) such that ϕ m = h () km ϕ +,k, m K. (..8) k K + Thus, the relation (..6) is equivalent to v +,k = v k + m K h () km v m, k K + (..9) (the series is formal; in the applications, a proper decay of the coefficients h () km as k m + will guarantee its convergence). Using the representation (..7) for Q (v), (..) becomes v = J v k ϕ +,k, (..3) whereas (..3) becomes k K + v = v,kϕ,k + k K k K + v k ϕ +,k. (..3) If the multilevel decomposition is q-stable and the canonical basis of each V is uniformly p-stable, and if both assumptions (..9) and (..) are satisfied, one has the equivalent representation of the norm v J k K +.3 Characterization of intermediate spaces q/p /q v k p, v V. (..3) Let us consider a quasi-banach space Z V, whose quasi-norm will be denoted by Z. We assume that there exists a semi-quasi-norm Z in Z such that v Z v + v Z, v Z. (.3.33) In addition, we assume that V Z, J. (.3.34) Thus, Z is included in V with continuous imbedding and dense image. 3

8 We recall that the real interpolation method ([7]) allows us to define a family of intermediate spaces Z α q, with < α < and < q, such that Z Z α q Z α q V, α < α, q, q arbitrary, (.3.35) with continuous inclusion. The space Z α q is defined as where Z α q = (V, Z) α,q = {v V v q α,q := [t α K(v, t)] q dt t < }, (.3.36) K(v, t) = inf z Z { v z + t z Z}, v V, t >. (.3.37) Z α q is equipped with the quasi-norm v α,q = ( v q + v q α,q) /q. We note that one can replace the semi-quasi-norm v α,q by an equivalent, discrete version, as follows: v q α,q b αq K(v, b ) q J /q, v V, (.3.38) where J is either N or Z and b is any real number >. We want to characterize the space Zq α in terms of the multilevel decomposition introduced in the previous Section. This will be accomplished using a general result, due to DeVore and Popov ([57]), which is based on two inequalities classically known as Bernstein and Jackson inequalities. In our framework, these inequalities read as follows: there exist constants C > and b > such that the Bernstein inequality and the Jackson inequality v Z C b v, v V, J (.3.39) v P (v) C b v Z, v Z, J (.3.4) hold. The Bernstein inequality is also known as an inverse inequality, since it allows the stronger quasi-norm v Z to be bounded by the weaker quasi-norm v, provided v V. The Jackson inequality is an approximation result, which yields the rate of decay of the approximation error by P 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 (..9). The following theorem is a reformulation of the DeVore and Popov result, within our functional setting. Theorem. Assume that the family {V, P } J satisfies the assumptions (..9)-(..) and the assumption (..) if J = N. In addition, let the space Z satisfy the hypotheses (.3.33) and (.3.34). If the Bernstein and Jackson inequalities (.3.39) and (.3.4) hold, then for all < α <, < q one has Z α q = {v V J b αq Q (v) q < + } (.3.4) 4

9 with v α,q ( J b αq Q (v) q ) /q, v Z α q. (.3.4) Corollary. Under the assumptions of Theorem., let the canonical basis of each V ( J ) be uniformly p-stable (for suitable < p < ). Then, the following representation of the semi-quasi-norm of Zq α holds: v α,q J b αq k K + q/p /q v k p, v Zq α. (.3.43) If in addition both (..9) and (..) are satisfied and the multilevel decomposition (..) is q-stable, the following representation of the quasi-norm of Z α q holds: v α,q ( + b αq ) J k K + q/p /q v k p, v Zq α. (.3.44).4 Multilevel decompositions based on linear proections In this section, we specialize the abstract framework previously described to the case in which the operators P : V V ( J ) are linear continuous, and satisfy the assumptions P L(V,V ) C (independent of ), (.4.45) P v = v, v V, (.4.46) P P + = P. (.4.47) Note that (.4.46) is nothing but (..), (.4.45) is equivalent to (..) thanks to (.4.46), whereas the commutativity condition (.4.47) is an additional assumption, which is equivalent to P P l = P for all l. Also note that in the Hilbertian case, (.4.45)- (.4.47) are certainly satisfied if P is the orthogonal proection upon V. Under the previous assumptions, the closed subspace W = Im Q is a supplementary space of V in V +, i.e., V + = V W. (.4.48) Indeed, if w V W, then w = P + v P v for a suitable v V, hence, w = P w = P P + v P v =. In addition, Q = Q, (.4.49) i.e., Q is a proection on W. 5

10 By iterating (.4.48), (..7) becomes V J = V ( J = W ); (.4.5) The assumptions (..9)-(..) yield the multilevel decomposition of V if only (..9) holds, then for any J V = J W ; (.4.5) V = V ( W ). (.4.5) Let us fix a basis in each W {ψ k k ˆK }, (.4.53) where ˆK is a certain subset of Z d. We call it a uniformly p-stable basis for W (for a certain < p ) if W = { k ˆK α k ψ k {α k } k ˆK l p } (.4.54) and k ˆK α k ψ k {α k } k ˆK l p {α k } l p, (.4.55) the constants involved in the definition of being independent of. Due to (.4.48), we have in V + two bases: the canonical basis (..4) and the hierarchical basis {ϕ k k K } {ψ k k ˆK }. (.4.56) If v = k K + v +,k ϕ +,k = k K v k ϕ k + k ˆK ˆv k ψ k. (.4.57) is the decomposition of any v V + along the two bases, we call the mapping v {{ v k } k K, {ˆv k } k ˆK } (.4.58) the two-level analysis/synthesis associated to the decomposition (.4.48). The mapping { v +,k } k K+ {{ v k } k K, {ˆv k } k ˆK } (.4.59) is the corresponding two-level hierarchical transform. Due to (.4.48), there exist coefficients g () km (k K +, m ˆK ) such that ψ m = g () km ϕ +,k, m ˆK ; (.4.6) k K + conversely, there exist coefficients χ () mk (m K, k K + ) and γ () mk (m ˆK, k K + ) such that ϕ +,k = mk ϕ m + mk ψ m, k K +. (.4.6) m K χ () m ˆK γ () 6

11 Then, the hierarchical transform (.4.59) can be expressed as follows: v m = mk v +,k, m K ; (.4.6) ˆv m = v +,k = m K h () k K + χ () k K + γ () km v m + mk v +,k, m ˆK ; (.4.63) m ˆK g () kmˆv m, k K +, (.4.64) (the previous remark about the series applies here as well). In applications, often the hierarchical transform can be accomplished in a fast way, by exploiting particular properties of the coefficients h, g, χ and γ. Given any v V, in lieu of (..7) let us now represent Q v as Q v = k ˆK ˆv k ψ k. (.4.65) For any two indices < J, the approximation P J v can be represented according to the multilevel decomposition (.4.5) as P J v = k K J v Jk ϕ Jk = k K v,kϕ,k + J = k ˆK ˆv k ψ k. (.4.66) A recursive application of the two-level hierarchical transform (.4.6)-(.4.64) yields the multilevel hierarchical transform associated to (.4.5). At last, the multilevel decomposition (..) is expressed in the hierarchical basis as v = ˆv k ψ k, v V. (.4.67) J k ˆK We recall that this representation holds provided both assumptions (..9) and (..) are satisfied. If the multilevel decomposition is q-stable and the bases (.4.5) are uniformly p-stable, then we have by (..) v q/p /q ˆv k p, v V. (.4.68) J k ˆK Finally, if the bases (.4.5) are uniformly p-stable we have by (.3.4) the following equivalent representation for the semi-quasi-norm of the intermediate space Z α q v α,q J b αq q/p /q ˆv k p, v Zq α. (.4.69) k ˆK 7

12 .5 Dual biorthogonal decompositions In this section, we adapt the dual multilevel construction of Dahmen (see, e.g., [37], [38]) to our abstract setting. We assume that V is a reflexive Banach space, and the linear proectors P : V V ( J ) satisfy the assumptions (.4.45)-(.4.47) and (..9)- (..). Let us denote by f, v the duality pairing between the topological dual space V and V. For each J, let P : V V be the adoint operator of P, and let Ṽ := Im P V. Condition (.4.47) easily implies that Ṽ Ṽ+. Proposition.3 The sequence (Ṽ, P ) ( J ) satisfies the abstract assumptions (.4.45)- (.4.47) and (..9)-(..); thus it generates a multilevel decomposition of V. Let us set Q = P + P and W := Im Q V. We thus obtain the multilevel decomposition of V V = J W, i.e., f = J Q f, f V, (.5.7) where the series converges in the strong topology of V. It is easy to check the following biorthogonality condition g, w =, g W l, w W with l. (.5.7) It follows that we have the representation of the duality pairing f, v = l Q l f, Q v = Q f, Q v f V, v V. (.5.7) Coming now to the bases, suppose that V possesses a uniformly p-stable basis (..4) for some < p < +. Then, if f = P f Ṽ, we have f, v = f, P v = k K v k f, ϕ k, v V. Thus, if we define the forms Φ k Ṽ by the condition Φ k, v = v k, v V, we have f = k K fk Φ k with f k = f, ϕ k, for all f Ṽ, and {Φ k k K } is a uniformly p -stable basis of Ṽ (where p is the conugate exponent of p). Similarly, if (.4.53) is a uniformly p-stable basis of W, we define Ψ k W by setting Ψ k, v = ˆv k, v V. Then, f = k ˆK ˆfk Ψ k with ˆf k = f, ψ k, for all f W, and {Ψ k k ˆK } is a uniformly p -stable basis of W. Moreover, we have the biorthogonality relations Ψ k, ψ k = δ δ kk,, k, k ˆK. (.5.73) Finally, we consider the characterization of the dual spaces of the interpolation spaces defined in (.3.36). From now on, we suppose that all the hypotheses of Theorem. are 8

13 satisfied. We assume that < q < +, and set Z α q = (Z α q ), where q + q =. Note that the continuous inclusions (with dense images) Z Zq α V imply the continuous inclusions (with dense images) V Z α q Z. Let us denote by Z f, v Z (resp., Z α f, v Z α q q ) the duality pairing between Z and Z (resp., between Z α q and Z α q ). We observe that for any f Z, we can define P f V by P f, v = Z f, P v Z ; indeed, P L(V, Z) by the Bernstein inequality. Thus, Q f is defined as an element of V, for any J and any f Z. Theorem.4 Under the assumptions of this section, for all < α < and < q < +, one has with Z α q = {f Z b αq Q f q V < + } (.5.74) f Z α q [ P f V + ( b αq Q f q V ) /q ], f Z α q. (.5.75) Moreover, Z α f, v Z α q q = Q f, Q v, f Z α q, v Z q α. (.5.76) J Remark. If J = IN, then P f V = in (.5.75). If J = Z, then P f V can be replaced by f Z in the same formula, as a consequence of the Bernstein inequality on V..6 Multilevel decompositions on the real line A biorthogonal system of compactly supported wavelets ([33], see also [5], [6]) can be introduced by assigning two trigonometric polynomials m (ξ) = n ñ h n e inξ, m (ξ) = hn e inξ n=n n=ñ (.6.77) with real coefficients h n, h n such that h n h n and hñ hñ ; the two polynomials are related by the identity m (ξ) m (ξ) + m (ξ + π) m (ξ + π) =, ξ IR, (.6.78) and satisfy m () = m () =, m (π) = m (π) =. We assume that m and m vanish at ξ = π with a zero of order L and L, respectively. Thus, ( ) + e iξ L ( + e iξ m (ξ) = F(ξ), m (ξ) = ) L F(ξ) (.6.79) 9

14 with F(π), F(π). We also assume that there exists integers l, l such that, if we set max F(ξ)... F( l ξ) ξ = lτ max F(ξ)... F( l ξ) = l τ ξ (.6.8) we have τ < L, τ < L. We define σ = L τ > e σ = L τ >. Starting from the polynomial m, we shall define a scaling function ϕ, a mother wavelet ψ and the corresponding spaces of dilates and translated. A parallel construction is done starting from m ; the functions and spaces being generated in this way will carry a superscript (i.e., ϕ, ψ and so on). For the sake of conciseness, we systematically omit to indicate the parallel construction, unless differences appear. The Fourier transform of the real-valued scaling function ϕ is defined as ϕ(ξ) = m ( ξ); (.6.8) = consequently, ϕ satisfies the refinement equation ϕ(x) = n n=n h n ϕ(x n) (.6.8) as well as the normalization condition IR ϕ(x)dx = ; furthermore, ϕ is compactly supported with support equal to [n, n ] (see [7]). The Fourier transform ϕ decays at infinity according to the estimate ϕ(ξ) C( + ξ ) / σ, (.6.83) which implies ϕ L (IR), indeed ϕ H s (IR) s < σ. Since we will be interested in working in L p -spaces, with p, we now make the additional assumption σ /; the Sobolev imbedding theorem implies that ϕ L p (IR) for all p (, + ). The same assumption is made on σ. Defining ϕ k (x) = ϕ(x k) for k Z, one can prove the biorthogonality relation ϕ k (x) ϕ k (x)dx = δ kk, k, k Z. (.6.84) IR Finally, as a consequence of (.6.79), the integer translates of ϕ generate locally the polynomials of degree L..6. The spaces V For < p < +, let us set V = L p (IR) with norm p, and V = span L p (IR){ϕ k k Z}. (.6.85) 3

15 In order to define the spaces V for, let us introduce the isometries in L p (IR) It will be also useful to define the mappings T such that (T v)(x) = /p v( x). (.6.86) ( T v)(x) = /p v( x), where p + p =. (.6.87) If p, these are isometries in L p (IR). Then define V = V (p) = {T v v V } = span L p (IR){ϕ k k Z}, (.6.88) where ϕ k (x) = T ϕ k (x) = /p ϕ( x k). Due to the refinement equation (.6.8) the relation (..8) becomes ϕ m = / /p k Z h k m ϕ +,k, (.6.89) which implies the inclusion (..9). Furthermore, (.6.8) yields ϕ k (x) ϕ k (x)dx = δ kk, k, k Z, Z. (.6.9) IR We shall now check the uniform p-stability of the basis {ϕ k } k Z of V. The following result will be useful. Lemma.5 Set K = {k Z supp ϕ k (, ) }. (.6.9) Then, the restrictions of the functions ϕ k, k K, to the interval [, ] are linearly independent. Proof. It is proven in [33] that Z V () is dense in L (IR), Z V () = {} and {ϕ k k Z} is a Riesz basis of V (). Then, the result follows from [7], Corollaire 3. Proposition.6 The basis {ϕ k k K = Z} of V is uniformly p-stable. Proof. Since T is an isometry in L p (IR), it is enough to prove that V = { k Z α k ϕ k {α k } l p }, (.6.9) with k Z α k ϕ k p {α k } l p, {α k } l p. (.6.93) 3

16 To this end, let v = k Z α kϕ k have only a finite number of α k different from. For any l Z, denote by I l the interval [l, l + ]. Then v(x) p dx = v(x) p dx = α k ϕ k (x) p dx IR l Z I l l Z I l k l+k = α l+m ϕ m (x) p dx. l Z m K If β = {β m } m K, the mapping ( β N(β) = β m ϕ m (x) p dx m K is a quasi-norm in IR M, with M = card K. Indeed, N(β) = implies β = thanks to Lemma.5; the other conditions of quasi-norm are trivially satisfied. Since all the quasinorms are equivalent in a finite dimensional space, we have N(β) β l p, β IR M ; the constants in only depend on M, i.e., on the size of the support of ϕ. We conclude that l Z ) /p ( ) /p ( ) /p v p α l+m p α k p. m K Using this equivalence and the completeness of L p (IR) and l p, it is a straightforward matter to obtain the characterization (.6.9), whence the result. k Z.6. The proectors P The proectors P : L p (IR) V are defined by the commutativity relation P (v) = T P (T v), v L p (IR), (.6.94) once we have given the definition of P. In turns, this definition depends on whether p or p <. If p, we set P v = v k ϕ k with v k = v(x) ϕ k (x)dx. (.6.95) k Z IR Note that P v V ; indeed, v k = v(x) ϕ k (x)dx, supp ϕ k whence ( p/q v k p v(x) p dx ϕ(x) dx) q ; supp ϕ k supp ϕ 3

17 since any point on the real line belongs to at most a fixed number of supports of translates of ϕ, there exists a constant C > such that v k p v(x) p dx, k Z IR i.e., recalling (.6.93) P v p C v p, v V. (.6.96) The explicit expression for P v is P v = v k ϕ k with v k = k Z IR v(x) ϕ k (x)dx. (.6.97) The inequality (.6.96) implies that the condition (.4.45) is satisfied. The condition (.4.46) follows from the biorthogonality relation (.6.84), whereas (.4.47) is a consequence of the refinement equation (.6.8). If v, w = IR v(x)w(x)dx denotes the duality pairing between Lp (IR) and L p (IR), it is easily seen that the following relation holds P v, w = v, P w = k Z v k wk, v L p (IR), w L q (IR), Z. (.6.98) Thus, the operator P : L p (IR) Ṽ defined by the parallel formula of (.6.97) coincides with the adoint operator of P, as defined in section 5, once we identify the dual space of L p (IR) with the space L p (IR). (Note that the arguments in the duality pairing are written here in the reverse order with respect to that section.) Obviously, the function ϕ k coincides with the form Φ k defined therein. Let us now consider the case p <. For each l Z, let us set V,l = {v Il v V }, (.6.99) where again I l = [l, l + ]. Note that V,l has finite dimension, independent of l. Next, let us set V = {η L p (IR) l Z, η Il V,l }. (.6.) Obviously, V V ; furthermore, V contains the space X = {η L p (IR) l Z, η Il P L } of the piecewise polynomial functions of degree L. Let us define the operator R : L p (IR) V by selecting for each v L p (IR) and l Z a function r V,l satisfying the condition v r L p (I l ) C inf q V,l v q L p (I l ), (.6.) (where C is a fixed constant), and setting R (v) Il = r. One has v R (v) p C inf η V v η p, (.6.) 33

18 which implies R (v) p C v p. (.6.3) Next, we define the operator P : V V by setting P η = α k ϕ k, with α k = k Z IR η(x) ϕ k (x)dx. (.6.4) Note that P is nothing but (.6.95). Indeed, since ϕ, ϕ L (IR), we have P is well-defined and bounded in the L p (IR)-norm. α k C η L (supp ϕ k ); (.6.5) since supp ϕ k is bounded, the space of the restrictions to supp ϕ k of the functions in V has finite dimension (independent of k); hence, for the equivalence of quasi-norms α k C η L p (supp ϕ k ). (.6.6) Summing-up over k and recalling Proposition.6 we conclude that P η p C η p, η V. (.6.7) Finally, we define the operator P : L p (IR) V by setting P (v) = P R (v), v L p (IR). (.6.8) Let us check that the assumption (..) is satisfied, i.e., v P (v) p C inf v V v v p, v L p (IR). (.6.9) To this end, let us consider any v V ; then v P (v) p C[ v v p + v P R (v) p ] = C[ v v p + P ( v R (v)) p ] (by (.6.96) for P ) C[ v v p + v R (v) p ] (by (.6.7)) C[ v v p + v R (v) p ] C[ v v p + inf η V v η p ] (by (.6.)) C[ v v p + inf η V v η p ] (since V V ). Taking the infimum over all v V, we get (.6.9)..6.3 The spaces W Let us now go back to the case p, in order to give a more precise description of the supplementary spaces W. Firstly, let us give the following characterization. Lemma.7 W = {v V + v, ṽ =, ṽ Ṽ} (.6.) 34

19 Proof. If w = P + v P v for a certain v V, then by (.6.98) w, ṽ = v, P + ṽ P ṽ = v, ṽ ṽ =. Conversely, if v V + satisfies v, ϕ k = k Z, then P v =, hence, v = Q v. Next, we want to provide a basis for W. To this end, let us define the wavelet ψ(x) = ñ n= ñ g n ϕ(x n), with g n = ( ) n h n (.6.) (remember that we have a parallel definition of ψ with the obvious changes in notation). Setting ψ k (x) = ψ (p) k (x) = /p ψ( x k), one can prove the following biorthogonality relations (see [33]) Furthermore, (.6.) yields ψ k, ϕ k = ψ k, ϕ k =, Z, k, k Z, (.6.) ψ k, ψ k = δ δ kk, Z, k, k Z. (.6.3) ψ m = / /p k Z g k m ϕ +,k, (.6.4) which is precisely (.4.6). Let us set W = span L p (IR){ψ k k Z}. (.6.5) The following result is the analog of Lemma.5 for the functions ψ k. Proposition.8 The spaces W and W coincide. Moreover, {ψ k k ˆK = Z} is a uniformly p-stable basis of W and (.4.65) holds with ˆv k = v, ψ k. Note again that the function ψ k coincides with the form Ψ k defined in Section The Bernstein and Jackson inequalities From now on, let us consider again the general case < p < +. In view of the characterization of the Besov spaces, let us assume that ϕ B s pq(ir) for some s > and < p < +, < q +. The following Proposition states that each V is contained in B s pq(ir) and the Bernstein inequality is satisfied. Proposition.9 For each s s, there exists a constant C > such that v B s pq (IR) C s v p, v V, Z. (.6.6) 35

20 Proof. It is enough to prove the inequality for =. Indeed, recalling the definition (.6.88) of V, if T v V for a certain v V, one has h (T v) = T ( hv), which implies (T v, t) = ω (r) (v, t), hence, ω (r) p p T v B s pq (IR) = s v B s pq (IR), v V, Z. (.6.7) Recalling that T is an isometry in L p (IR), (.6.6) for follows from the inequality In order to prove this inequality, let us observe that ω p (r) (v, t) = ( sup r h v p dx) /p ( h t I l l Z l Z v B s pq (IR) C v p, v V. (.6.8) sup r h v p dx) /p. (.6.9) h t I l Now, for each l Z, if v = k α kϕ k with {α k } l p, we have by the Minkowski inequality, ( sup r h v p dx) /p C sup α m+l ( h t I l h t m K I l r h ϕ,m+l p dx) /p (.6.) C m K α m+l ( sup h t r h ϕ m p dx) /p ; in these inequalities, the constant C is if p, whereas it depends only on the cardinality of K if p <. Let us set β m (t) = ( sup h t and note that, if p ( m K α m+l β m (t)) p r h ϕ m p dx) /p, m K, α m+l p ( β q k (t))p/q C m K m K m K α m+l p max m K β p k (t) due to the equivalence of the l q and l norms in finite dimension; if p < ( α m+l β m (t)) p α m+l p βm(t) p α m+l p max β p m K k (t). m K m K m K Using the last inequalities in (.6.) and then recalling (.6.9), we easily conclude that v B s pq (IR) C( α m+l p ) /p ( l Z m K t qs max m K β q k (t)dt t )/q C v p. Here, we have used the p-stability of V and the assumption ϕ B s pq(ir) Bpq(IR). s The proof of (.6.8) is complete. The next Proposition states that the Jackson inequality (.3.4) is satisfied. 36

21 Proposition. Assume that s < L, where L is defined in (.6.79). Then there exists a constant C > such that v P (v) p C s v B s pq (IR), v B s pq(ir), Z. (.6.) Proof. Again, it is enough to consider the case = : the estimate for, follows from (.6.94), (.6.7) and the isometry property of T in L p (IR). We shall detail the proof for p < only. For p, the proof is simpler and it can be easily obtained from the following arguments. Let us fix l Z, and let us estimate v P (v) L p (I l ) = v P R (v) L p (I l ). Recalling the definition (.6.4) and the bound (.6.6) there exists a finite set of indices K # (independent of l) such that, if J l = k K # I l we have P η L p (I l ) C η L p (J l ), η V, l Z. Let q P L be a fixed polynomial: there exists v q V such that v q = q on J l ; furthermore, q = P q on I l. Thus, v P (v) L p (I l ) = (v q) + P (q R (v)) L p (I l ) C[ v q L p (I l ) + q R (v) L p (J l )] C[ v q L p (J l ) + v R (v) L p (J l )]. Now, v R (v) p L p (J l ) = v R (v) p L p (I l+k ) k K # C inf v η p η V L p (I l+k ),l+k k K # C v q p L p (I l+k ) k K # = C v q p L p (J l ). Therefore, we obtain v P (v) L p (I l ) C inf v q L q P p (J l ). (.6.) L Now we recall the following local version of Whitney s Theorem, which can be found in [58]: given an interval J of the real line, define ( w L (v, J) = J J J /p dh L h dx) v p, v L p (J), (.6.3) J(Lh) where J(s) = {x J x + s J}; then, there exists a constant C > such that inf q P L v q L p (J) Cw L (v, J), v L p (J). (.6.4) 37

22 Using (.6.), and setting J = J l (independent of l), we get v P (v) p L p (IR) C l Z w p L (v, J l) C J dh J J l Z C sup L h v p dx h J IR = Cω p L (v, J ). J l (Lh) L h v p dx Observing that ω L (v, J ) C( Z sq ω L (v, ) q ) /q, we conclude the proof of (.6.) for =. The first consequence of Proposition. is that the assumption (..9) is satisfied. Proposition. Z V is dense in L p (IR). Proof. Given v L p (IR) and ɛ >, there exists v ɛ C (IR) Bs pq(ir) such that v v ɛ p < ɛ/. According to (.6.), there exists Z such that v ɛ P (v ɛ ) p < ɛ/, whence the result. We consider now the behaviour for of the multilevel decomposition (V, P ) Z. Proposition. For < p < +, if p >, then Z V = {}; (.6.5) P (v) as, v L p (IR). (.6.6) Proof. Assume that v V for all Z. Letting in (.6.6), we obtain v s B pq (IR) = ; this implies v =. Assume now that p >, and let v Lp (IR). Given ɛ >, there exists v ɛ C (IR) such that v v ɛ p < ɛ. Then, using (.4.45), P v p P v ɛ p + P (v v ɛ ) p P v ɛ p + Cɛ. (.6.7) In order to estimate P v ɛ p, we consider the coefficient (v ɛ ) k = v ɛ (x) ϕ k (x)dx = supp /p v ɛ (x) ϕ( x k)dx. ϕ k IR Let us fix a real number p satisfying < p < p, and let p > p be the conugate exponent of p. Then (v ɛ ) k /p ( v (x) p ɛ dx) / p ( ϕ( x k) p dx) / p supp ϕ k supp ϕ k = (/p / p ) ( v (x) p ɛ dx) / p ( ϕ( x k) p dx) / p. supp ϕ k supp ϕ k 38

23 Thus P v ɛ p C (/p / p ) ( k Z C (/p / p ) v ɛ p ) /p ( v (x) p ɛ dx) p/ p supp ϕ k and we can choose ɛ such that P v ɛ p < ɛ for all < ɛ. This and (.6.7) imply (.6.6). The result (.6.6) is not necessarily true if p (see [6]..6.5 Characterization of the Besov spaces B s pq(ir) Let us come now to the representation of an element v L p (IR). If p, for any integer Z, we have from (..3) v = k Z v,kϕ,k + k Z v k ϕ +,k ; (.6.8) we recall that the coefficients v k are defined by the condition P (v) = k Z v kϕ k, whereas, from (..9) and (.6.89), v k = v +,k / /p m Z h k m v m. (.6.9) If p >, then from (.4.67) v = ˆv k ψ k, with ˆv k = Z k Z IR v(x) ψ k (x)dx. (.6.3) If p =, the multilevel decomposition is known to be -stable (see [33]), hence, one has v / ˆv k, v L (IR). (.6.3) Z k Z Finally, we come to the characterization of Besov spaces. Theorem.3 Assume that ϕ B s pq(ir). For all s such that < s < s = min(s, L), the following characterization holds: B s pq(ir) = {v L p (IR) Z sq ( k Z v k p ) q/p < + } (.6.3) and v B s pq (IR) ( ) q/p sq v k p Z k Z /q, v B s pq(ir). (.6.33) 39

24 If p, then each v k can be replaced by ˆv k in the previous formulae. Finally, if p =, then v H s (IR) ( + 4 s ) ˆv k Z k Z /, v H s (IR). (.6.34) Proof. For any s such that < s < s, let us choose s satisfying s < s < s (actually, s can be equal to s if s < L). Set Z = B s pq(ir). Then, assumption (.3.34) is satisfied and both the Bernstein and the Jackson inequalities hold. Moreover, B s pq(ir) = (L p (IR), B s pq(ir)) α,q with α = s/ s. Then we get the result applying Theorem., Corollary. and (.4.69). From Theorem.4 we obtain the following characterization of the dual Besov spaces. Corollary.4 Let < p, q < +, and set B s p q (IR) = (B s pq(ir)). For all s such that < s < s = min(s, L), if s < s < s, the following characterization holds: B s p q (IR) = { f B s p q (IR) sq ( ˆf k p ) q /p < + }, (.6.35) k Z where ˆf k = f, ψ k, and f B s p q (IR) f B s p q (IR) + ( ) q /p /q sq ˆf k p, f B s p q (IR). k Z (.6.36).6.6 Another characterization of the Besov spaces B s σσ(ir) In different applications (see, e.g., [54], [5], [8], [36]), the natural functional setting is the scale of non-homogeneous Besov spaces Bσσ(IR), s where s >, τ = τ(s) is defined by the relation τ = s + p and p is a fixed real number. Since Bσσ(IR) s L p (IR) with continuous inection (see [58]), it is natural to use the multilevel decompositon of L p (IR) (rather than the one of L τ (IR)) in order to provide another characterization of Bσσ(IR). s This result is indeed well-known in the literature, as it can be obtained from the Littlewood- Paley theory (see, e.g., [6]). We present hereafter a simple, self-contained proof, coherent with the framework we have introduced. For the sake of clarity, we shall append the suffix (p) (resp., (τ) ) to any obect related to the decomposition of L p (IR) (resp., L τ (IR)). Thus, if v Bσσ(IR), s we can define the proections P (p) p v, ψ k. v V (p), as well as the details Q (p) 4 v = k ˆv(p) k ψ(p) k W (p), with ˆv (p) k =

25 Theorem.5 Assume that ϕ B s τ τ (IR) for some s > and τ = τ(s ). For all s such that < s < s = min(s, L), the following characterization holds: and B s σσ(ir) = {v L p (IR) v B s σσ (IR) ˆv (p) k τ,k Z /τ,k Z ˆv (p) k τ < + } (.6.37), v B s σσ(ir). (.6.38) Proof. If τ, the result is ust a rephrasing of Theorem.3, since by (.6.33) ( v B s σσ (IR) ) /τ Z sτ k Z ˆv(τ) k τ (τ), and ˆv k = s ˆv (p) k,, k Z. If τ <, the coefficients (τ) v k are not directly related to ˆv (p) k, so we resort to a different argument, based on the characterization v B s σσ (IR) ( Z sτ Ē (v) τ ) /τ, where Ē (v) = inf v V (τ) P (p) v v τ. First of all, let us prove that P (p) v V (τ) v = k v(p) k ϕ(p) k = s k v(p) k ϕ(τ) k, with if v B s σσ(ir). We have v (p) k supp = v(x) ϕ (p ) ϕ (p ) k (x)dx C v C v L p (supp ϕ (p ) k ) Bττ s (supp ϕ (p ) ). k k Since each x R belongs at most to a fixed number (independent of ) of supports of ϕ (p ) k, we conclude that P (p) v τ C s ( k v (p) k τ ) /τ C s v B s σσ(ir). It follows that Ē(v) v P (p) v τ, which implies the inequality v B s σσ (IR) C ˆv (p) /τ.,k Z k τ In order to establish the reverse inequality, let us prove that Q (p) v τ C s v B s σσ (IR), v B s σσ(ir). (.6.39) By the usual scaling argument, it is enough to prove the inequality for =. Q (p) v = k ˆv k ψ k, where ˆv k = v(x) ψ k (x)dx [v(x) q(x)] ψ k (x)dx, q P L ; supp ψ k supp ψ k Now, the latter inequality is a consequence of (.6.) and the fact that V locally contains the polynomials of degree L. Thus, setting J k = supp ψ k, we have ˆv k C inf v q L q P p (J k ) C inf v q B s L q P ττ (J k ). L 4

26 Now, using Whitney s theorem (.6.4) with J = J k, we get inf v q B s q P ττ (J k ) inf v q L L q P p (J k ) + v B s ττ (J k ) C v B s ττ (J k ). L By Proposition.8 (which holds for an exponent of summability τ < as well), we conclude Q (p) v τ C( k ˆv k τ ) /τ C v B s σσ (IR). Thus, (.6.39) is proven. This inequality can actually be written as Q (p) v τ C s v v B s σσ (IR), v V (τ). Let v V (τ) be such that v v τ = Ē(v). Note that Ēl(v v ) = Ē(v) if l, whereas Ēl(v v ) = Ēl(v) if l >. Thus, v v τ B s σσ(ir) l sτl Ē l (v v ) τ C l sτl Ē l (v) τ, so that sτ Q (p) v τ τ C This concludes the proof of the theorem. sτ l sτl Ē l (v) τ C l sτl Ē l (v) τ..7 Examples We present now some significant examples of orthogonal and biorthogonal bases satisfying the assumptions required in Section.: the Haar basis, two different splines bases (the orthogonal wavelets of Battle-Lemarié and the biorthogonal wavelets introduced by Cohen, Daubechies and Feauveau), and the compactly supported wavelets introduced by Daubechies. In each case one can verify conditions (.6.77)-(.6.8) and thus one can get all the property obtained in Section...7. The Haar basis Let if x [, ), ϕ(x) = χ [,) (x) = otherwise, be the scaling function of the Haar basis. Then, V = {v L (IR) k Z, v Ik is constant} = { k Z α k ϕ k {α k } k Z l ( Z).} (.7.4) (.7.4) 4

27 The refinement equations read ϕ k (x) = [ϕ +,k (x) + ϕ +,k+ (x)],, k Z. (.7.4) The Haar wavelet is (see Figure.). if x < /, ψ(x) = if / x <, otherwise (.7.43).5 ϕ(x).5 ψ(x) Here the refinement equation are Figure.: The Haar scaling function ϕ and wavelet ψ. ψ k (x) = [ϕ +,k (x) ϕ +,k+ (x)],, k Z. (.7.44) See Figure.,.3,.4 and.5 for examples of approximation of functions with the Haar basis. We define so that m (ξ) = m (ξ) = + e iξ h n = { n =,, otherwise., (.7.45) (.7.46) Condition (.6.77) is immediat, L =, F, thus τ = and σ = /. The Haar basis is the simplest example of wavelet basis. Unfortunately, the functions are not regular and they have only one zero moment. Thus, it is natural to look for a generalization of the Haar system, to obtain more regular wavelets orthogonal to a greater number of polynomials. 43

28 .5.5 v(x) P v(x) P v(x) P v(x) P 3 v(x) P 4 v(x) P 5 v(x) P 6 v(x) P 7 v(x) Figure.: The top left panel shows a regular function v on the unit interval. The other panels show some piecewise constant approximations P v. 44

29 v(x) P v(x) Q v(x) Q v(x) Q v(x) Q 3 v(x) Q 4 v(x) Q 5 v(x) Q 6 v(x) Figure.3: Detail functions Q v = P + v P v for the example in the previous figure. The function v is reported in the top left panel, and the coarsest approximation P v is shown in the top middle panel. 45

30 v(x) P v(x) P v(x) P v(x) P 3 v(x) P 4 v(x) P 5 v(x) P 6 v(x) P 7 v(x) Figure.4: The top left panel shows a function v with a discontinuity. The other panels show some piecewise constant approximations P v. 46

31 v(x) P v(x) Q v(x) Q v(x) Q v(x) Q 3 v(x) Q 4 v(x) Q 5 v(x) Q 6 v(x) Figure.5: The top left panel shows the function v of the previous figure. The top middle panel shows its coarsest approximation P v, and the other panels show the details Q v. 47

32 .7. Bases arising from splines We want to construct a system of scaling functions and wavelets generated from B-splines functions. We recall that a spline function of degree l associated to the integer nodes of the real line is a function which coincides with a polynomial of degre l on each interval of the form [k, k + ) (k Z), is continuous on IR together with its derivatives up to the order l. Such function are generated from the integer translates of the function χ l [,) = χ [,) χ [,). }{{} l convolutions (.7.47) The function χ l [,) has support on the interval [, l + ) and it is symmetric with respect to the point x = l+. It will be more convenient to use its translate which has center of symmetry in if l is odd, or in / if l is even. Thus, we set ( Φ l (x) = χ l [,) x + [ l + Recalling the expression of the Fourier transform of χ [,), we have ]). (.7.48) Φ l (ξ) = = l+ e i[ ]ξ π ( e iξ iξ ) l+ ( ) sin ξ/ l+ e ikξ/, π ξ/ (.7.49) with k = if l is even, k = if l is odd. Notice that, if l >, the integer translates of Φ l do not constitute an orthogonal system (since they are positive functions with the supports not necessarily disoint). We can procede in two different directions: A. we can build an orthogonal system through the orthogonalization of the translates of Φ l ; B. we can loose the orthogonality property and gain a more flexible condition of biorthogonality and thus we can build a dual function Φ l. The orthogonal wavelets of Battle-Lemariè. We present the orthogonalization procedure of the translates of Φ l, which allows us to define the scaling function ϕ l and the corresponding function m,l (see [6] and [7] for more details). It is possible to show that there exists C l > such that C l m Φ l (ξ + πm), ξ IR; (.7.5) 48

33 thus, we can define ϕ l (ξ) = Φl (ξ) ( π ) /, (.7.5) m Φ l (ξ + πm) so that the integer translates of ϕ l are orthogonal. If ϕ l is a scaling function generated from a periodic function m,l, using (.6.8) it has to satisfy the relation from which we (formally) get ϕ l (ξ) = ϕ l (ξ)m,l (ξ) m,l (ξ) = ϕ l(ξ) ϕ(ξ). (.7.5) It is possible to show that this formal construction is well defined. Moreover m,l satisfies conditions (.6.78) and (.6.79). Finally, setting l = ν in (.6.8), we have τ(ν) as ν and, consequently, σ(ν) l +. This procedure generates an orthonormal system in L (IR). What we loose is the compact support property of the original B-splines basis. The biorthogonal splines wavelets We show now how to construct a biorthogonal system of splines functions in such a way that the scaling functions and wavelets have compact support (see [33] for more details). For each l, the scaling function ϕ l will coincide with Φ l. Consequently, ϕ l is given by equation (.7.49); thus l+ m,l (ξ) = e i[ ]ξ = ( + e iξ (cos ξ/) l+ e i ξ (cos ξ/) l+ ) l+ l odd, l even. Observe that m,l satisfies (.6.77)-(.6.8) with L = l + and F l (ξ) = e i[ L ]ξ, τ = for each integer ν, σ = L / = l + /. It is possible to prove that, for any choice of L such that L + L is even, m,l is of the form (cos ξ ) L p,l (cos ξ) L even, m,l (ξ) = e i ξ (cos ξ ) L p,l (cos ξ) L odd. where p,l is the polynomial satisfying P (x) = p,l ( x) with ( µ P (x) = n= µ + n n 49 ) x n, µ = L + L. (.7.53)

34 Moreover ( ) 3 τ = log P. (.7.54) 4 The numerical values of τ are reported in the following table. µ τ For any couple (L, L) with the same parity, we report the values of σ (N.V. means that the condition (.6.8) is not fulfilled). L/ L NV NV NV NV.3 The plots of some biorthogonal spline scaling functions and wavelets are reported in Figures.6,.7, The compactly supported Daubechies wavelets It is possible to construct orthogonal systems (and not only biorthogonal ones) of scaling functions and wavelets having compact support. This construction in due to I. Daubechies [49]. The resulting functions are not piecewise polynomials, indeed their analytic expression is not even known (it is possible to compute their values on the dense set of dyadic points, using recursive algorithm). Moreover, they are not symmetric with respect to the center of their support, as in the case of spline wavelets. The function m (ξ) = m (ξ) has the form ( ) + e iξ L m (ξ) = r (ξ), where r (ξ) is a suitable trigonometric polynomial (see [49] or [5] for more details). We report some numerical values of τ and σ corresponding to a given L. L τ σ For the plots of the Daubechies scaling functions and wavelets, see Figure.9. 5

35 ϕ(x), L = ϕ(x) ψ(x) ψ(x) L = L = L = L = Figure.6: Biorthogonal splines for L = and L =, 4, 6, 8: primal scaling function (top panel), dual scaling function (left column), primal wavelet (middle column) and dual wavelet (right column). 5

36 .8 ϕ(x), L = ϕ(x) ψ(x) ψ(x).5 L = L = L = L = Figure.7: SBiorthogonal splines for L = 3 and L = 5, 7, 9,. 5

37 .8 ϕ(x), L = ϕ(x) ψ(x) ψ(x) 4.5 L = L = L = L = Figure.8: Biorthogonal splines for L = 4, L = 8,,, 4. 53

38 .5 ϕ(x).5 ψ(x).5 L = L = L = L = L = Figure.9: Daubechies scaling functions (left column) and wavelets (right column) for different values of L ranging from (top) to 6 (bottom). 54

39 Chapter Multilevel decompositions on the unit interval. Introduction One of the key properties of wavelets is translation invariance. This leads to the construction of multilevel decompositions naturally defined on the full real line, or on bounded domains with periodic boundary conditions. However, most of the applications require the solution of PDE s within a bounded domain with possibly complex boundary conditions. Therefore, if wavelets are to be used, special care must be taken in the construction of wavelet bases inherently defined on bounded domains. The simplest case is represented by wavelet bases on the unit interval (, ). These can also be used as a building block to construct wavelet bases on multidimensional domains of complex shape [9,, 45]. We concentrate our attention to the construction of biorthogonal wavelet bases on the half-line (, + ) and, subsequently on the unit interval. The construction of such bases originates naturally by wavelets defined on the whole line by introducing suitable modifications to account for the edges of the domain. We focus on biorthogonal systems instead of orthogonal ones, like the famous Daubechies compactly supported orthogonal wavelets [5], because the former can be choosen to have useful properties, like e.g. compact support and central symmetry. As an example we can cite the biorthogonal B-splines wavelets [33]. All orthogonal systems are a particular case of the more general biorthogonal setting. Many constructions of orthogonal and/or biorthogonal wavelets on the unit interval or half-line can be found in the literature, see, e.g., [3, 34, 4, 74]. Generally, these constructions do not uniquely determine one particular biorthogonal system, but leave the freedom to insure additional properties of the wavelet bases. Here, we follow [74, 6] and we will exploit this possibility and we will propose a different definition of the border wavelets which seems to be better adapted to the applications. The key advantages of this new construction are a good localization of the modified border wavelets and a short length of the corresponding filters. These two features are of great importance in the construction of adaptive schemes for the solution of PDE s. 55

40 Let Ω denote either the half-line (, + ) or the unit interval (, ). In this chapter, we will use the Besov spaces Bpq(Ω) s as defined in [89] (see also [58]) and the Sobolev spaces W s,p (Ω)(= Bpp(Ω), s unless p and s IN), and H s (Ω) = B s (Ω). For the sake of completeness, we recall the definition of Besov space Bpq(Ω) s which extends the definition given in the previous chapter. For v L p (Ω), < p < +, let us denote by r h the difference of order r IN \ {} and step h IR, defined as r h v(x) = r ( ) r ( ) r+ v(x + h), = x Ω rh = {x Ω : x + rh Ω}. For t >, let ω (r) p be the modulus of smoothness of order r defined as ω p (r) (v, t) = sup r h v L p (Ω rh ). h t For s > and < p, q < +, we say v B s pq(ω) whenever the semi-norm v B s pq (Ω) = ( + [t s ω p (r) (v, t)] q dt t )/q is finite, provided r is any integer > s (different values of r give equivalent semi-norm). B s pq(ω) is a Banach space endowed with the norm v B s pq (Ω) := v L p (Ω) + v B s pq (Ω). Moreover, we recall the following real interpolation result: (L p (Ω), B s s pq(ω)) s,q = Bs pq(ω) (..) where < p, q < + and < s < s. Finally, we will be interested in spaces of functions satisfying homogeneous boundary value conditions. To this end, let C (Ω) denote the space of C (Ω)-functions whose support is a compact subset of Ω. We consider the Besov spaces Bpq, s (Ω) (s, < p, q < + ) defined as the completion of C (Ω) in Bs pq(ω). One can show that, if s p, Bpq, s (Ω) = Bs pq(ω), while, if s > p, Bs pq, (Ω) is strictly contained in Bs pq(ω). In this second case one also has B s pq,(ω) = {v B s pq(ω) : d v dx = on Ω if < s }. (..) p Moreover, for s, < p, q < +, let B s pq,(ω) = {v B s pq(ir) : supp v Ω}; (..3) then, if s p / IN, B s pq,(ω) = B s pq,(ω). (..4) 56

41 For these spaces, we have the real interpolation result: (L p (Ω), B s pq, (Ω)) s /s,p = B s pq, (Ω), (..5) where < p, q < + and < s < s. In particular, if < p < +, < s < s, we have where W s,p (L p (Ω), W s,p (Ω)) s /s,p = W s,p B s pq, (Ω) W s,p (Ω) if s p IN, if s IN, (..6) (Ω) if s, s p / IN, s,p (Ω), W (Ω) are defined similarly to the Besov spaces (..) and (..3). We will also consider negative values of s. For s <, < p, q < +, let us denote by p and q the conugate index of p and q respectively (i.e. p + p = q + q = ). Then, we set B s pq(ω) = (B s p q, (Ω)).. Preliminaries In this section we review those aspects of the construction of scaling functions and wavelets that will be extensively used in the forthcoming sections. We follow [74, 7, 6], where the formal proofs can be found, (see also [3, 4]). Finally, we draw some considerations on the modified border scaling functions filters, which will be important for the subsequent construction of the border wavelets... Biorthogonal decomposition in IR In view of the subsequent constructions, we summarize the key elements of a biorthogonal decomposition on the real line, and we explicitly write the refinement and reconstruction equations. Scaling functions. Let us consider two compactly supported scaling functions ϕ, ϕ L (IR) satisfying the following refinement equations with finite real filters h and h, ϕ(x) = n n=n h n ϕ(x n), ϕ(x) = ñ n=ñ hn ϕ(x n). (..7) Without loss of generality we will assume ñ n n ñ so that supp ϕ = [n, n ] supp ϕ = [ñ, ñ ]. If this is not the case, the role of the primal and the dual functions can be exchanged. From now on, we will only give details for the primal setting. The dual construction, herewith indicated with a tilde, follows by analogy. Setting for, k Z, ϕ k (x) = / ϕ( x k), the biorthogonality relations ϕ k, ϕ k IR := IR ϕ k (x) ϕ k (x)dx = δ kk,, k, k Z (..8) 57