Boundary values of scaling functions and wavelets

Wavelets on the interval 147

6. 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 e ϕ 00 see 35 do not vanish at zero. Thus, the idea is to biorthogonalize first the functions in the sets 8 ∗ = {ϕ 0k : k = 1, . . . , e L − 1} and e 8 ∗ = e ϕ 0k : k = 1, . . . , e L − 1 , so that the resulting sys- tems contain functions which all vanish at zero. To do this, we need to check the non-singularity of the Gramiam matrix hϕ 0k , e ϕ 0k ′ i k,k ′ =1,... ,e 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 12 and also [16]. From now on we suppose this property is verified and biorthogonalize 8 ∗ and e 8 ∗ . 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 invari- ant the functions in 8 ∗ and e 8 ∗ . Precisely, we have the following general result. P ROPERTY 1. Let 8 ∗ and e 8 ∗ be the two biorthogonal systems described above. Consider 8 = {ϕ 00 } ∪ 8 ∗ , e 8 = {e ϕ 00 } ∪ e 8 ∗ and suppose that the matrix hϕ 0k , e ϕ 0k ′ i, k, k ′ = 0, . . . , e L − 1, is non-singular. Then, it is possible to construct new biorthogonal systems spanning the same sets as 8 and e 8 , respectively, in which only the two functions ϕ 00 , e ϕ 00 have been modified. Proof. Let us set ϕ 00 = e L−1 X k=1 α k ϕ 0k + α ϕ 00 , e ϕ 00 = e L−1 X l=1 β l e ϕ 0l + β e ϕ 00 . We want to prove that we can find α k , β l , k, l = 0, . . . , e L − 1 so that α β 6= 0, hϕ 00 , e ϕ 0l i = hϕ 0k , e ϕ 00 i = 0 , k, l = 1, . . . , e L − 1 , 79 and hϕ 00 , e ϕ 00 i = 1 . 80 Imposing the conditions 79 and using the biorthogonality of the systems 8 ∗ , e 8 ∗ , we get α k = −α hϕ 00 , e ϕ 0k i and β l = −β he ϕ 0l , e ϕ 00 i , k, l = 1, . . . , e L − 1 . Substituting these relations in 80, we end up with the identity K α β = 1 where K = ϕ 00 , e ϕ 00 − e L−1 X k=1 he ϕ 00 , ϕ 0k ie ϕ 0k . 148 L. Levaggi – A. Tabacco To conclude the proof it is enough to show that K 6= 0. Indeed, if K = 0, the function η = e ϕ 00 − P e L−1 k=1 he ϕ 00 , ϕ 0k ie ϕ 0k ∈ span 8 ∗ ⊥ , would also be orthogonal to ϕ 00 ; moreover η ∈ span e 8 and the systems 8 and e 8 are biorthogonalizable, i.e., span 8 ⊥ ∩ span e 8 = {0}; this would mean η = 0, contradicting the linear independence of the functions in e 8 . Next we consider the wavelet case. Suppose we have constructed biorthogonal wavelets {ψ j k } k≥0 , e ψ j k k≥0 starting from biorthogonal scaling systems {ϕ j k } k≥0 , e ϕ j k k≥0 such that ϕ j 0 e ϕ j 0 6= 0 and ϕ j k = e ϕ j k = 0 , ∀k ≥ 1 , ∀ j ≥ 0 . 81 Recalling that ϕ j +1,k = T 1 ϕ j k , one has ϕ j +1,0 = 2 1 p ϕ j 0 0 , e ϕ j +1,0 = 2 1 p ′ e ϕ j 0 0 . 82 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, e S be two spaces of functions defined on some set  with biorthogonal bases E = {η l } l∈ ✄ and e E = { ˜η l } l∈ ✄ here ✁ is some set of indices with respect to some bilinear form S h·, ·i e S on S × e S. Let F = {ν l } l∈ ✄ and e F = {˜ν l } l∈ ✄ be two other bases such that ν l = K lm η m , ˜ν l = e K lm ˜η m , where K and e K are suitable generalized matrices. It is easy to check that, to preserve the biorthogonality, we must have e K = K −T . L EMMA 1. With the previous notation, suppose the elements of E and e E are continuous functions, then the quantity X l∈ ✄ η l x ˜η l x, ∀x ∈  is invariant under any change of biorthogonal basis. Proof. Let us denote by ex the vector η l x l∈ ✄ , and similarly for e ex, f x, e f x. Note that f x = K ex and e f x = K −T e ex; thus X l∈ ✄ ν l x ˜ν l x = f x T · e f x = ex T K T · K −T e ex = ex T · e ex = X l∈ ✄ η l x ˜η l x . We will apply this result to our biorthogonal wavelets. C OROLLARY 2. Suppose the scaling functions ϕ, e ϕ are continuous on , then X k≥0 ψ 0k 0e ψ 0k 6= 0 . 83 Proof. Let S = V 1 + , e S = e V 1 + . Using the relations V 1 + = V + ⊕ W + and e V 1 + = e V + ⊕ e W + , we have two couples of biorthogonal bases on S and e S: E = {ϕ 1k } k≥0 , e E = {e ϕ 1k } k≥0 and F = {ϕ 0k } k≥0 ∪ {ψ 0k } k≥0 , e F = {e ϕ 0k } k≥0 ∪ e ψ 0k k≥0 . Using the previous Lemma, 81 and 82, we have ϕ 10 e ϕ 10 = 2ϕ 00 e ϕ 00 = ϕ 00 e ϕ 00 + X k≥0 ψ 0k 0e ψ 0k 0 . Wavelets on the interval 149 Thus X k≥0 ψ 0k 0e ψ 0k = ϕ 00 e ϕ 00 6= 0 . This implies that we can always find k ≥ 0 such that ψ 0k 0e ψ 0k 6= 0. Without loss of generality, we can suppose k = 0. Let us define, for k ≥ 1, ψ ∗ 0k x : = ψ 0k x − ψ 0k ψ 00 ψ 00 x =: ψ 0k x − c k ψ 00 x 84 and e ψ ∗ 0k x : = e ψ 0k x − e ψ 0k e ψ 00 e ψ 00 x =: e ψ 0k x − ˜c k e ψ 00 x . 85 Observe that ψ ∗ 0k = e ψ ∗ 0k = 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 k≥1 and e ψ ∗ 0k k≥1 . Indeed L EMMA 2. The matrix Y ∗ = ψ ∗ 0k , e ψ ∗ 0l m ∗ −1 k,l=1 is non-singular. Proof. Let us set M = m ∗ − 1; for k, l = 1, . . . , M we have ψ ∗ 0k , e ψ ∗ 0l = 1 + c k ˜c k if k = l c k ˜c l if k 6= l . It is easy to see that Y ∗ has only two different eigenvalues: λ 1 = 1, with multiplicity M − 1 and λ 2 = 1 + P M k=1 c k ˜c k , with multiplicity 1. Thus, by Corollary 2, det Y ∗ = 1 + M X k=1 c k ˜c k = P k≥0 ψ 0k 0e ψ 0k ψ 00 0e ψ 00 6= 0 . Finally we construct our wavelet systems as described in Property 1.

7. Characterization of Besov spaces B