Geometric conditions for interpolation in weighted spaces of entire functions

Abstract

ABSTRACT. We use L 2 estimates for the∂ equation to find geometric conditions on discrete interpolating varieties for weighted spaces A p (C) of entire functions such that |f (z)| ≤ Ae Bp(z) for some A, B > 0. In particular, we give a characterization when p(z) = e |z| and more generally when ln p(e r ) is convex and ln p(r) is concave.

The Range of the Restriction Map for a Multiplicity Variety in Hörmander Algebras of Entire Functions

“…Berenstein, Li and Vidras [4]). An easier proof was presented by Ounaïes [17] (see also [11]). These geometric characterizations were formulated in terms of the counting function and the integrated counting function of the multiplicity variety V , that are defined as follows: For z ∈ C and r > 0, we set…”

“…This corresponds to spaces of type A p (C) for non radial weights p. The case of Roumieu ultradistributions was studied by Zio lo [21]. The arguments of Berenstein and Li [3] were simplified by Hartmann and Massaneda [11] and by Ounaïes [17] using Hörmander's L 2 estimates for the ∂ equation, treating also weights that are radial but not doubling. In [18] Ounaïes uses divided differences to characterize those sequences that are in the range R V (A p (C)) of the restriction map R V when the multiplicity variety satisfies the assumption (a) (i) in Theorem 1.1.…”

Representation of mean-periodic functions in series of exponential polynomials

“…To any discrete doubly indexed sequence a = {a k,l } k∈N,0≤l<m k of complex numbers, we associate the sequence of divided differences Ψ(a) = {b k,l } k∈N,0≤l<m k . We recall that they are the coefficients of the Newton polynomials, (11) Q q (ξ) =…”

“…In order to find an explicit formula for the coefficients c n,l , consider the elements B k,l of B θ (C) defined by (13) and observe that, by the definition of the Newton polynomials (see (11)) with respect to the coefficients of B k,l , for all q ≥ k, we have…”

Expansion in series of exponential polynomials of mean-periodic functions

“…In order to find an explicit formula for the coefficients c n,l , consider the elements B k,l of B θ (C) defined by (15) and observe that, by the definition of the Newton polynomials (see (13)) with respect to the coefficients of B k,l , for all q ≥ k, we have…”