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A remark on the div-curl lemma

Pierre Gilles Lemarié-Rieusset — 2012

Studia Mathematica

We prove the div-curl lemma for a general class of function spaces, stable under the action of Calderón-Zygmund operators. The proof is based on a variant of the renormalization of the product introduced by S. Dobyinsky, and on the use of divergence-free wavelet bases.

Ondelettes generalisées et fonctions d'échelle à support compact.

Pierre-Gilles Lemarié-Rieusset — 1993

Revista Matemática Iberoamericana

We show that to any multi-resolution analysis of L(R) with multiplicity d, dilation factor A (where A is an integer ≥ 2) and with compactly supported scaling functions we may associate compactly supported wavelets. Conversely, if (Ψ = A Ψ (Ax - k)), 1 ≤ ε ≤ E and j, k ∈ Z, is a Hilbertian basis of L(R) with continuous compactly supported mother functions Ψ, then it is provided by a multi-resolution analysis with dilation factor A, multiplicity d = E / (A - 1) and with compactly supported scaling...

Analyses multi-résolutions non orthogonales, commutation entre projecteurs et derivation et ondelettes vecteurs à divergence nulle.

Pierre Gilles Lemarie-Rieusset — 1992

Revista Matemática Iberoamericana

The notion of non-orthogonal multi-resolution analysis and its compatibility with differentiation (as expressed by the commutation formula) lead us to the construction of a multi-resolution analysis of L(R) which is well adapted to the approximation of divergence-free vector functions. Thus, we obtain unconditional bases of compactly supported divergence-free vector wavelets.

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