On the Solutions of Certain Functional Equations in Classes of Functions Analytic in Convex Domains

Coefficients of Exponential Series for Analytic Functions and the Pommiez Operator

“…Let a be an entire function of exponential type in C whose adjoint diagram is contained in a convex compact K ⊂ C. We denote by a(D) the convolution operator with the characteristic function a: [34,38]). Note that for any p ∈ N ∪ {0} and μ ∈ C, the following equality holds:…”

“…By Theorem 2.2 (for m(λ) = p(λ), λ ∈ V (L)), there exists an absolutely representing system of quasipolynomials v jq ∈ E j in H(G + K), 0 ≤ q < q j , j ∈ N. Following [43], we take an entire function a in C of completely regular growth (of the order 1) with the indicator H K . According to [34], the convolution operator a(D) : H(G + K) → H(G) is surjective. Therefore, the system w jq := a(D)(w jq ) ∈ E j , 0 ≤ q < q j , j ∈ N, is an absolutely representing system in H(G).…”

Generalized spherical mean value operators on Euclidean space

“…Such equations have been studied by many authors on various situations, for example, on spaces of holomorphic functions on convex domains, those with growth conditions near the boundary, hyperfunctions, Fourier hyperfunctions, etc. See Hörmander [7], Korobeȋnik [11], Kawai [12], Ishimura-Okada [10], Abanin-Ishimura-Khoi [1], Langenbruch [13] and the references therein. Necessary or sufficient conditions for solvability are often written in terms of the Fourier (or Laplace) transform of the kernel.…”

Convolution operators on spaces of real analytic functions

“…Since then convolution operators on holomorphic functions or germs on subsets of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {C}$\end{document} or \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {C}^d$\end{document} have been intensively studied. A selection of corresponding papers is contained in the references (see Albanin et al 1, Korobeinik 18, 19, 21, Krivosheev and Grantsev 23, the survey of Krivosheev and Napalkov 24 including the references, Mal'tsev 33, 34 and Okada 49).…”