Algebraic spectral synthesis and crystal rigidity
A theorem of \textit{M. Lefranc} [C. R. Acad. Sci., Paris 246, 1951--1953 (1958; Zbl 0084.10302)] states that in the topology of pointwise convergence, every closed translation invariant linear space of complex-valued functions on \(\mathbb{Z}^d\) is the closed linear span of a set of exponential monomials \[ \mathbb{Z}^d \leftrightarrow \mathbb{C} , \quad (k_1,\ldots , k_d)\mapsto h(k_1, \ldots , k_d)\omega_1^{k_1}\cdots \omega_d^{k_d}, \] where \(h(k_1, \ldots , k_d)\) is a polynomial in \(d\) variables and \(\omega_i\), \(i=1, \ldots , d\), are nonzero complex numbers. In paper under review, the authors give a generalization of Lefranc's result for the closed invariant subspaces of \(C(\mathbb{Z}^d , \mathbb{C}^r)\), the space of vector-valued functions \(u \colon \mathbb{Z}^d \leftrightarrow \mathbb{C}^r\) with the topology of coordinatewise convergence. More precisely, let \(A\) be a closed invariant subspace of \(C(\mathbb{Z}^d, \mathbb{C}^r)\). Then there is a finite set of geometric indices such that \(A\) is the closed linear span of vectorial \(pg\)-sequences in \(A\) with geometric indices in this set (Theorem 3.10). In the last section of the paper the authors discuss connections of their results to difference equations and crystal flexibility. The paper contains a brief historical discussion of Lefranc's result.
scientific article; zbMATH DE number 7076026
scientific article