Stack Overflow Question
Asked by srfp311t1 · Score: -4 Tags: analytic-number-theory, harmonic-analysis, modular-forms, interpolation source
Question
The Fourier interpolation theorem of Radchenko and Viazovska (Invent. Math. 2019) gives an interpolation formula involving the discrete sampling set: $$ \{\sqrt{n} : n \ge 0\}, $$ with interpolation kernels constructed using modular and quasimodular forms for $\mathrm{SL}2(\mathbb{Z})$. On the other hand, the Fourier interpolation theorem of Bondarenko, Radchenko, and Seip (Math. Ann. 2022) involves the spatial sampling set: $$ \{\log n : n \ge 1\}, $$ together with the ordinates of the nontrivial zeros of the Riemann zeta function. The two constructions have quite different appearances, but both involve Fourier duality and highly structured arithmetic sampling sets. Question. Is there a known transform or mathematical construction that explains the passage $$ \sqrt{n} \quad \longrightarrow \quad \log n $$ as a discrete-to-continuous degeneration of an underlying interpolation kernel or theta-type construction? More specifically, is there a natural family of kernels $K\alpha(x, t)$, or some deformation of the underlying automorphic data, whose interpolation/spectral data specializes to the Radchenko–Viazovska $\sqrt{n}$ system at one end and to the Bondarenko–Radchenko–Seip $(\log n, \gamma)$ system at another? Equivalently, is there a common framework in which the theta/modular structure underlying the $\sqrt{n}$ interpolation formula and the Mellin/zeta structure underlying the $\log n$ interpolation formula arise as different degenerations or specializations of the same object?
Comments
- Alex Elzenaar (3): I cannot find any paper co-authored by Radchenko and Viazovska in Inventiones in 2019. They did publish a paper in IHES that year though. It is your responsibility to check that the references spat out by your LLM actually correspond to real papers.
- Alex Elzenaar (2): Similarly, there is no paper indexed by Zentralblatt by Bondarenko, Radchenko, and Seip for any year before 2023, let alone in Math Ann in 2022.