2026/07/21 by Oleg Szehr, Rachid Zarouf
Mathematics · #math.CA
We generalize a construction of Yves Meyer of sparse crystalline measures arising from powers of a Blaschke factor. Starting from a recursion fn=θn f0 on the unit circle where θ is an inner function, we project the Fourier coefficient array \widehatfn(k) to the real line by placing its entries at the frequencies k+αn. We identify the role of model spaces in this construction: in Meyer's one-factor Blaschke recursion, the requirement that the coefficient array \widehatfn(k) vanish whenever kn<0 is equivalent to f0∈ Kzbλ, and for general inner functions the condition f0∈ Kzθ yields a purely atomic Radon measure with locally finite support and polynomial growth on the Fourier side. We also show that, when f0 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.