Bochner-Riesz Summability Below the Critical Index: Applications and Sharp Estimates

Abstract

This research paper investigates the realm of Bochner-Riesz summability in two dimensions, providing distinct results and insights on the convergence properties of Fourier series. More precisely, the study examines the convergence properties of Fourier series. This article establishes two significant theorems: Theorem 4.1.1 offers proof that the Bochner-Riesz operator in L²(R²) is bounded for a certain multiplier function. However, this claim is contingent upon a crucial criterion. Theorem 4.1.2 presents precise estimates that indicate the logarithmic dependence on certain parameters and highlight the intricate behavior of the operator. These theorems have potential applications in signal processing and imaging, and they also serve as a solid foundation for comprehending harmonic analysis and singular integral operators. The last portion of the paper has some suggestions for further investigation. The proposals include doing research on higher dimensions, investigating the impact of additional variables, and exploring practical applications in real-world scenarios.