Solid hulls and cores of classes of weighted entire functions defined in terms of associated weight functions
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Solid hulls and cores of classes of weighted entire functions defined in terms of associated weight functions Gerhard Schindl1 Received: 8 May 2020 / Accepted: 13 July 2020 © The Author(s) 2020
Abstract In the spirit of very recent articles by J. Bonet, W. Lusky and J. Taskinen we are studying the so-called solid hulls and cores of spaces of weighted entire functions when the weights are given in terms of associated weight functions coming from weight sequences. These sequences are required to satisfy certain (standard) growth and regularity properties which are frequently arising and used in the theory of ultradifferentiable and ultraholomorphic function classes (where also the associated weight function plays a prominent role). Thanks to this additional information we are able to see which growth behavior the so-called ”Luskynumbers”, arising in the representations of the solid hulls and cores, have to satisfy resp. if such numbers can exist. Keywords Weighted spaces of entire functions · Weight sequences and weight functions · Solid hulls and solid cores Mathematics Subject Classification 30D15 · 30D60 · 46E05 · 46E15
1 Introduction Spaces of weighted entire functions are defined as follows Hv∞ (C) := { f ∈ H (C) : f v := sup | f (z)|v(|z|) < +∞}, z∈C
and the weight v : [0, +∞) → (0, +∞) is usually assumed to be continuous, non-increasing and rapidly decreasing, i.e. limr →+∞ r k v(r ) = 0 for all k ≥ 0. In the recent publications [6] and [3] the authors have studied solid hulls and solid cores of such spaces, using the so-called j with its sequence of Taylor-coefficients (a ) the identification of f (z) = +∞ a z j j j∈N . For j=0 more references and historical background we refer to the introductions of these papers.
G. Schindl is supported by FWF-Projects J 3948-N35 and P32905.
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Gerhard Schindl [email protected] Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria 0123456789().: V,-vol
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It has turned out that the so-called regularity condition (b) on the weight v, see [6, (2.2)] and [3, Definition 2.1] and (4.10) in the present work, plays the key-role for a more concrete description of the solid hulls and cores of Hv∞ (C). It is weaker than condition (B) introduced on [16, p. 20], see [6, Rem. 2.7], and the arising expressions in (b) have already been studied in [16] as well. Verifying this condition for concrete given examples requires quite technical computations and might be challenging: The expressions under consideration are involving the extremal points of the functions r → r k v(r ) and one has to find and compute a strictly increasing sequence of positive real numbers, the so-called Lusky numbers. The goal of this paper is to study the situation when v(r ) = exp(−ω M (r )), with ω M denoting the so-called associated weight function (see (2.1)), and M = (M p ) p∈N a given sequence of positive real numbers satisfying some basic regularity and growth properties. First, this question has been motivated by r
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