Abstract
A spectral synthesis property is obtained for closed shift-invariant subspaces of vector-valued functions on the lattice Zd. This result generalises Marcel Lefranc's 1958 theorem for scalar-valued functions. Applications are given to homogeneous systems of multivariable vector-valued discrete difference equations and to the first-order flexibility of crystallographic bar-joint frameworks.
Highlights
It is the 60th anniversary of Marcel Lefranc’s proof [21] that the discrete group Zd admits spectral synthesis
Lefranc showed that every proper closed translation invariant space of complexvalued functions on Zd is the closed linear span of a set of exponential monomials
In this context an exponential monomial is a multi-sequence of the form k → h(k)ωk, k ∈ Zd, where h is a multivariable polynomial in d indeterminates and ω = (ω1, . . . , ωd) belongs to (C\{0})d
Summary
It is the 60th anniversary of Marcel Lefranc’s proof [21] that the discrete group Zd admits spectral synthesis. Building on particular classical results of Schwartz, Malgrange, Ehrenpreis and Kahane, independent determinations of spectral synthesis were obtained in the setting of general locally compact abelian groups, by Elliott [12], [13], in 1965, and by Gilbert [15], in 1966. The former articles have some incorrect claims in the case of infinite discrete rank groups, while Gilbert examines restricted contexts. We have found few biographical details of Marcel Lefranc beyond the fact that he was a professor of Mathematics at the University of Montpellier II, and that before this, in 1957, he was a lecturer there in mathematics and astronomy
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