The Hadamard Product on Open Sets in the Extended Plane

Abstract

For two open sets 1 , 2 in the extended complex plane, we define a Hadamard product as an operator from H ( 1 ) × H ( 2 ) to H ( 1 * 2 ), where 1 * 2 is the so-called star product. Moreover, we study properties of this product and give applications.

Holomorphic cohomological convolution and Hadamard product

“…In his thesis [12] (see also [11]), Timo Pohlen introduced the more general notion of Hadamard product for holomorphic functions defined on open subsets of the Riemann sphere P = C ∪{∞} which do not necessarily contain the origin. This new definition led to interesting applications, (e.g.…”

Hadamard Multipliers on Spaces of Holomorphic Functions

“…(cf. [6,11,[16][17][18], as well as the survey article [20]) For any function f holomorphic around 0 let G f stand for the maximal starlike domain to which f is analytically continuable. Following Hadamard define the Hadamard multiplication on the space of germs of holomorphic functions around the origin: * : H({0}) × H({0}) → H({0}), n a n z n * n b n z n := n a n b n z n .…”

“…[4, Chapter 1.4], or [19,Chapter 6.3]; for the extended one see [18]). However, it says nothing about the possible holomorphic extension of f * g beyond its star product.…”

“…Ωg ∪ {0, ∞} , a finite union of closed curves such that γ separatesĈ\Ω f and z Ĉ \ 1 Ωg , and ind γ satisfies certain conditions (see [18,Theorem 2.4]). Define …”

The $*$-product of domains in several complex variables

“…It was extensively studied in various aspects: as a bilinear form, as a linear operator with one factor fixed (see the survey [19] and, for instance, the papers [1], [2], [3], [4], [12], [14], [15], [16], [17], [18], [20], [21], [22], [25]), and also as a map acting on spaces of real analytic functions (see [6], [7], [8], [9], [10], [11]).…”