Translation--Modulation Identities, Ergodic Log-Products and a Conditional Obstruction for Schwartz Functions

Abstract

We study very smooth functions on the real line, namely Schwartz functions, that satisfy a finite identity relating their translates and a single modulation. Concretely, we assume there is a nontrivial linear combination of translates of the function that equals a fixed frequency shift of the same function. Passing to the Fourier transform turns this into a multiplicative transfer relation: the value of the Fourier transform at one point is obtained by multiplying its value at another point by a trigonometric polynomial. Iterating this relation expresses the Fourier transform along an arithmetic progression as a product of such trigonometric factors times a fixed initial value. We then recast this product in an ergodic theoretic framework by viewing it as a Birkhoff sum for a continuous observable on a compact abelian group generated by a diagonal unitary matrix. The key quantity controlling the growth or decay of the Fourier transform along the progression is a space average, namely the integral over the compact group of the logarithm of the absolute value of the associated trigonometric polynomial. The main rigorously proved statement is the following. If, for some frequency where the Fourier transform does not vanish along the entire progression, this space average is nonzero, or if the average is zero but a certain recurrence set, in the sense of Atkinson's theorem, intersects a distinguished one-parameter subgroup, then the assumed translation--modulation identity forces exponential growth of the Fourier transform along some sequence of points. This contradicts the rapid decay required of a Schwartz function.