Sektion 9 - Reelle Analysis, Funktionalgleichungen

On covariant embeddings of a linear functional equation with respect to an analytic iteration group

Harald Fripertinger, Karl Franzens Universität Graz (Koautor: Ludwig Reich)

Let $ a(x)$, $ b(x)$, $ p(x)$ be formal power series in the indeterminate $ x$ over $ \mathbb{C}$ (i.e. elements of the ring $ \mathbb{C}\,[\![x]\!]$ of such series), such that $ \mathop{\rm ord}\nolimits a(x)=0$, $ \mathop{\rm ord}\nolimits p(x)=1$ and $ p(x)$ is embeddable into an analytic iteration group $ (\pi(s,x))_{s\in\mathbb{C}}$ in $ \mathbb{C}\,[\![x]\!]$. By a covariant embedding of the linear functional equation

$\displaystyle \varphi (p(x))=a(x)\varphi (x)+b (x), \eqno(L) $

(for the unknown series $ \varphi(x)\in\mathbb{C}\,[\![x]\!]$) with respect to $ (\pi(s,x))_{s\in\mathbb{C}}$ we understand families $ (\alpha(s,x))_{s\in\mathbb{C}}$ and $ (\beta(s,x))_{s\in\mathbb{C}}$ with entire coefficient functions in $ s$, such that the system of functional equations and boundary conditions

$\displaystyle \varphi (\pi (s,x))=\alpha (s,x)\varphi (x)+\beta (s,x) \eqno(Ls) $

$\displaystyle \alpha (t+s,x)= \alpha (s,x)\alpha (t,\pi (s,x)) \eqno(Co1) $

$\displaystyle \beta (t+s,x)= \beta (s,x)\alpha (t,\pi (s,x)) +\beta (t,\pi (s,x)) \eqno(Co2) $

$\displaystyle \alpha (0,x)=1 \qquad \beta (0,x)=0 \eqno(B1) $

$\displaystyle \alpha (1,x)=a(x) \qquad \beta (1,x)=b(x) \eqno(B2) $

holds for all solutions $ \varphi(x)$ of $ (L)$ and $ s,t\in\mathbb{C}$. We show how to solve the system $ ((Co1),(Co2))$ (of so called cocycle equations) completely, describe when and how the boundary conditions $ (B1)$ and $ (B2)$ can be satisfied and present a large class of equations $ (L)$ together with iteration groups $ (\pi(s,x))_{s\in\mathbb{C}}$ for which there exist covariant embeddings of $ (L)$ with respect to $ (\pi(s,x))_{s\in\mathbb{C}}$.

E-Mail: harald.fripertinger@kfunigraz.ac.at
Homepage: www-ang.kfunigraz.ac.at/~fripert/

Zurück zur Sektionsübersicht