Concrete and discrete operator reproducing formulae for abstract Paley–Wiener space

John R. Higgins (1)

1 IHP - Institut Henri Poincaré

The classical Paley–Wiener space possesses two reproducing formulae; a ‘concrete' reproducing equation and a ‘discrete' analogue, or sampling series, and there is a striking comparison between them. It is shown that such analogies persist in the setting of Paley–Wiener spaces that are more general than the classical case. In fact, there are ‘operator' versions of the reproducing equation and of the sampling series that are also comparable, not ‘exactly' but nearly so. Reproducing kernel theory and abstract harmonic analysis are brought together to achieve this, then the special case of multiplier operators with respect to the Fourier transform is considered. The Riesz transforms provide a two-dimensional example, with possibilities of extension to higher dimensions and to further classes of operators.

Domaines: Analyse classique [math.CA]

John R. Higgins. Concrete and discrete operator reproducing formulae for abstract Paley–Wiener space. SAMPTA'09, May 2009, Marseille, France. pp.General session. ⟨hal-00453459⟩