Convergence of Fourier series and integrals is the most fundamental question in classical harmonic analysis from its beginning. In one dimension convergence in Lebesgue spaces is fairly well understood. However in higher dimensions the problem becomes more intriguing since there is no canonical way to sum (and integrate) Fourier series (and integrals, respectively), and convergence of the multidimensional Fourier series and integrals is related to complicated phenomena which can not be understood in perspective of convergence in one dimension. The Bochner-Riesz conjecture may be regarded as an attempt to understand multidimensional Fourier series and integrals. Even though the problem is settled in two dimensions, it remains open in higher dimensions. In this talk we review developments in the Bochner-Riesz conjecture and discuss its connection to the related problems such as the restriction and Kakeya conjectures.
