- The paper introduces a novel interpolation formula for even Schwartz functions on the real line using discrete sampling points and basis functions derived from modular forms.
- A key finding demonstrates that if an even Schwartz function and its Fourier transform vanish at specific discrete points, the function itself must vanish identically.
- This interpolation formula offers a new tool for analysis and signal processing, overcoming limitations of classical methods like Whittaker-Shannon and suggesting applications in broader contexts.
Fourier Interpolation on the Real Line: A Comprehensive Analysis
The paper "Fourier Interpolation on the Real Line" by Danylo Radchenko and Maryna Viazovska introduces a novel interpolation formula applicable to Schwartz functions on the real line. This work extends classical interpolation results, notably addressing cases where the Fourier transform does not exhibit bounded support, thereby overcoming the limitations of the Whittaker-Shannon interpolation formula.
Theoretical Contributions
The core contribution of this paper lies in establishing an interpolation formula valid for even Schwartz functions, given by Theorem 1. The formula expresses the function's value at any real point as a combination of its values (and those of its Fourier transform) at the discrete set of points {0, ±√n | n ∈ ℕ}. The authors derive a basis for these interpolations using properties of weakly holomorphic modular forms, formulated in terms of an intrinsic kernel function, K(τ, z).
Explicit Construction and Methodology
The interpolation basis functions, {an(x)}, are crafted via integrals over semicircles in the upper half-plane and leverage certain weakly holomorphic modular forms of weight 3/2. The theoretical foundations are deeply rooted in the modular properties of these forms. For instance, the paper demonstrates the relation between these forms and the classical theta series, thus situating the work in a broader context of modular form theory.
The paper provides rigorous proofs of the convergence and decomposition properties of the interpolation formula. The conceptual framework includes the deployment of an isomorphism between the Schwartz functions and specific rapidly decreasing sequences, as demonstrated by Theorem 2. This isomorphism allows for a comprehensive understanding of how Schwartz functions and their Fourier transforms manifest at roots of non-negative integers.
Results and Numerical Validation
An intriguing consequence of Theorem 1, presented in Corollary 1, is that an even Schwartz function and its Fourier transform vanish jointly at these points, implying the function itself must vanish identically. This highlights the robustness and sensitivity of the interpolation method at detecting function properties through discrete sampling.
The growth estimates for coefficients in the interpolation series, addressed by Theorem 4, assert that these coefficients exhibit polynomial growth, facilitating their practical computation and application in numerical algorithms. These foundational results are extended to odd functions in Section 7, where analogous interpolation theorems are developed. The careful treatment of both even and odd cases allows for a unified theoretical structure.
Implications and Future Work
This interpolation formula opens new doors for tasks in signal processing or numerical analysis that previously relied on bounded support assumptions. The approach might reveal more intricate structures in other functional spaces, suggesting potential research into generalized spaces beyond Schwartz functions.
Future research may explore how these interpolation techniques can be integrated with numerical algorithms for solving partial differential equations or other complex systems, where the explicit basis might offer computational advantages. Additionally, further investigation into the applicability of these interpolations in higher dimensions or other non-Euclidean settings appears promising.
Overall, Radchenko and Viazovska's work enriches the toolkit for analyzing Schwartz functions and expands the mathematical landscape for applied Fourier analysis. This development stands as a significant stride in melding classical harmonic analysis with modern theories of modular forms.
