Spectral synthesis in the space of functions of exponential growth on a finitely generated abelian group (Q2849194)

Let \(G \) be a finitely generated abelian group. Let \(\{v_1,\cdots, v_n\}\) be a generator set of \(G \) and define for \(x=x_1 v_1+\cdots+x_n v_n\in G\) the quasinorm \(| x|:=|{x_1}|+\cdots+| {x_n}| \). An exponential monomial on \(G \) is a function of the form \(f(x)=P(x)z^x \), where \(P\in \mathbb C[x_1,\cdots, x_n] \) and where \(z^x=z_1^{x_1}\cdots z_n^{x_n} \) for \(z=(z_1,\cdots, z_n)\in\mathbb C_*^n \;(\mathbb C_*:=\mathbb C\setminus \{0\})\) and an exponential polynomial is a linear combination of exponential monomials.NEWLINENEWLINE A topological vector space of functions \(\mathcal F \) on \(G \) is said to be invariant if the translation operators \(\tau_y: f\mapsto \tau_y f(x)=f(x+y) \) leave \(\mathcal F \) invariant and are continuous on \(\mathcal F \). A closed translation invariant subspace \(H \) of \(\mathcal F \) is said to admit spectral synthesis if \(H \) is the \(\mathcal F \)-closure of all exponential polynomials belonging to \(H \). Spectral synthesis occurs if every closed invariant subspace of \(\mathcal{F} \) admits spectral synthesis. For every \(k>0\in \mathbb N \) let \(C_k(G) \) be the (invariant) space of complex-valued functions \(f \) on \(G \) that satisfy \(| f(x)| e^{-k| x|}\to 0 \) as \(| x| \to \infty\) and let \(C_*(G)=\bigcup_{k>0}C_{k}(G) \). The spaces \(C_k(G) \) are Banach spaces with the norms \(\| f_k\|:=\sup_{x\in G}| {f(x)}| e^{-k| x|} \) and \(C_*(G) \) is an invariant topological vector space with the topology of inductive limit of the sequence \((C_k(G))_k \).NEWLINENEWLINEThe author considers first \(G=\mathbb Z^n \) and shows that the space dual of \(C_*(G) \) can be identified with the space of functions \(C'_*(G) \) on \(G \) consisting of all \(g:G\to \mathbb C \) which satisfy \(|{g(x)}| e^{k| x |}\to 0 \) as \(| x|\to \infty \), which is Fréchet algebra under convolution.NEWLINENEWLINE Furthermore the Fourier transform \(\Lambda: g\mapsto \hat g(z):=\sum_{g\in G}g(x)z^x, z\in \mathbb C_*^n\), defines an isomorphism of topological algebras of \(C'_*(G) \) onto the space \(\mathcal A \) of holomorphic functions on \(\mathbb C_*^n \) endowed with the topology of uniform convergence on compact subsets of \(\mathbb C_*^n \). For an element \(\beta\in \mathbb C^n \) let \(\mathcal F_\beta \) be the ring of all formal power series in \(n \) complex variables centered at \(\beta \) endowed with the topology of coefficient-wise convergence. Let \(\mathcal A_\beta \) be the ring of germs of holomorphic functions at the point \(\beta \). There exists a natural embedding of \(\mathcal{A}_\beta \) into \(\mathcal F_\beta \).NEWLINENEWLINE The author shows that every element \(F\in \mathcal F'_\beta \) can be represented in the form \(F=P(\mathcal D)\delta_\beta \) with some polynomial \(P\in\mathbb C[x_1,\cdots, x_n] \). Let now \(H \) be a closed invariant subspace of \(C_*(\mathbb Z^n) \), let \(H^\perp \) be its annihilator in \( C'_*(\mathbb Z^n)\) and let \(I=\Lambda(H^\perp) \) and for any \(\beta \) let \(I_\beta \) be the ideal generated by \(I \) in \(\mathcal A_\beta \) and \(\mathcal E_\beta \) be the ideal in \(\mathcal F_\beta \) generated by \(I \).NEWLINENEWLINE The author proves that an exponential monomial \(f(x)=P(x)\beta^x \) belongs to \(H \) if and only if \(P(\mathcal D)\delta_\beta \) belongs to \(\mathcal E_\beta^\perp (\subset \mathcal F'_\beta) \). This assertion allows to prove the following theorem: A closed invariant subspace \(H \) of \(C_*(\mathbb Z^n) \) admits spectral synthesis if and only if the corresponding closed ideal \(I=\Lambda(H^\perp) \) is localized, i.e. is the intersection of all \(I_\beta, \beta \in \mathbb C_*^n \). Since \(\mathbb C_*^n \) is a Stein domain, it follows that every closed ideal in \(\mathcal A \) has this property of localization and so spectral synthesis holds in \(C_*(\mathbb Z^n) \). Finally, since every finitely generated abelian group \(G \) is the quotient of some \(\mathbb Z^n \), it follows that we have a canonical mapping from \(C_*(G) \) into \(C_*(\mathbb Z^n) \) which allows to prove that spectral synthesis occurs also in \(C_*(G) \).