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Biorthogonalité et théorie des opérateurs. (Biorthogonality and operator theory). (French) Zbl 0734.42008

The purpose of the paper under review is to study couples of unconditional biorthogonal bases of the standard Hilbert space \(L^ 2({\mathbb{R}})\) under the aspect of applications to the symbolic calculus of Calderón-Zygmund operators. In contrast to Hilbert bases, however, these couples can behave quite differently. The author shows that one basis can be formed by regular functions which have compact support whereas the other one may include functions with singularities at some points. It follows that the bases of \(L^ 2({\mathbb{R}})\) which are constructed by discretizing the wavelet transform do not necessarily form bases of the spaces \(L^ p({\mathbb{R}})\), \(H^ 1({\mathbb{R}})\), and BMO(\({\mathbb{R}})\).
Reviewer: W.Schempp (Siegen)

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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