Projected Inner-Function Dynamics and Crystalline Measures of Meyer-Blaschke type

We generalize a construction of Yves Meyer of sparse crystalline measures arising from powers of a Blaschke factor. Starting from a recursion $f_n=\theta^n f_0$ on the unit circle where $\theta$ is an inner function, we project the Fourier coefficient array $\widehat{f_n}(k)$ to the real line by placing its entries at the frequencies $k+\alpha n$. We identify the role of model spaces in this construction: in Meyer's one-factor Blaschke recursion, the requirement that the coefficient array $\widehat{f_n}(k)$ vanish whenever $kn<0$ is equivalent to $f_0\in K_{zb_\lambda}$, and for general inner functions the condition $f_0\in K_{z\theta}$ yields a purely atomic Radon measure with locally finite support and polynomial growth on the Fourier side. We also show that, when $f_0$ is holomorphic in an annulus containing the unit circle, exponential Fourier decay is sufficient to obtain a purely atomic Radon measure of polynomial growth, though not necessarily locally finite support. For finite Blaschke products, the coefficient recursion gives an explicit annihilating exponential polynomial whose zero set controls the support and separation of the inverse Fourier transform. This yields Meyer-Blaschke-type crystalline measures and Poisson identities with sampling and finite-truncation consequences.