Difference equations over locally compact abelian groups

Abstract

A homogeneous linear difference equation with constant coefficients over a locally compact abelian group G is an equation of the form Σ j = 1 n c j f ( t j x ) = 0 \Sigma _{j\, = \,1}^n {{c_j}f({t_j}x)\, = \,0} which holds for all x ∈ G x\, \in \,G where c 1 , … , c n {c_1}, \ldots ,\,{c_n} are nonzero complex scalars, t 1 , … , t n {t_1},\, \ldots \,,\,{t_n} are distinct elements of G, and f is a complex-valued function on G. A function f has linearly independent translates precisely when it does not satisfy any nontrivial linear difference equation. The locally compact abelian groups without nontrivial compact subgroups are exactly the locally compact abelian groups such that all nonzero f ∈ L p ( G ) f\, \in \,{L_p}(G) with 1 ⩽ p ⩽ 2 1\, \leqslant \,p\, \leqslant \,2 have linearly independent translates. Moreover, if G is the real line or, more generally, if G is R n {R^n} and the difference equation has a characteristic trigonometric polynomial with a locally linear zero set, then the difference equation has no nonzero solutions in C 0 ( G ) {C_0}(G) and no nonzero solutions in L p ( G ) {L_p}(G) for 1 ⩽ p > ∞ 1\, \leqslant \,p\, > \,\infty . But if G is some R n {R^n} for n ⩾ 2 n\, \geqslant \,2 and the difference equation has a characteristic trigonometric polynomial with a curvilinear portion of its zero set, then there will be nonzero C 0 ( R n ) {C_0}({R^n}) solutions and even nonzero L p ( R n ) {L_p}({R^n}) solutions for p > 2 n / ( n − 1 ) p\, > \,2n/(n - 1) . These examples are the best possible because if 1 ⩽ p > 2 n / ( n − 1 ) 1\, \leqslant p\, > \,2n/(n - 1) , then any nonzero function in L p ( R n ) {L_p}({R^n}) has linearly independent translates. Also, the solutions to linear difference equations over the circle group can be simply described in a fashion which an example shows cannot be extended to all compact abelian groups.