On generalization of Paley-Wiener theorem for weighted Hardy spaces

Abstract

We consider the Hardy space (C +) in the half-plane with an exponential weight. In this space we study the analytic continuation from the boundary. In the previous works for the case ∈ (1, 2] a result on analytic continuation from the imaginary axis was obtained, and it was a generalization of Paley-Wiener theorem. But for many applications the case = 1 is more interesting. For this case in the paper we obtain estimates for a function satisfying certain standard conditions.

On splitting functions in Paley-Wiener space

“…As in the proof the first part of the lemma we show that F 1 ∈ L 1 ( π σ ; +∞) and F 2 ∈ L 1 ( π σ ; +∞) without using (10). Therefore F 3 ∈ L 1 ( π σ ; +∞).…”

“…This problem is generated by studies of completeness of functions ( [10]) and considered in [5]. Above problem is interesting in the theory of integral operators and the shift operator.…”

Splitting of some spaces of analytic functions

“…, the belonging of sequence of the coefficients in expansion (2) for function 2 to space 1 we obtain condition (7).…”

“…For the further advancing in this direction, namely, for the description of all subspaces in space 2 (C + ) translation invariant w.r.t. the shift operator, a key role is played by the following statement [7], whose proof we know only in the case ∈ (1; 2].…”

“…We denote by (C + ), 1 < +∞, 0, the space of analytic in C + = { : Re > 0} functions obeying the condition (− 2 ; 2 ) < +∞. Certain properties of this space were studied by B.V. Vinnitskii and the author (see [6], [7], [18]). By (C + ), 1 < +∞, we indicate the Hardy space of analytic in C + functions satisfying…”