Some Basic Properties in Harmonic Analysis

The article starts with the convergence of the Fourier series, discusses point-wise convergence, convergence almost everywhere, convergence in the L-p space, and transitions to Riemann-Lebesgue Lemma. Then the article starts from the summability in norms of Fourier series, then introduces three important kernels in Fourier analysis: Dirichlet kernel, Fejer kernel, and Poisson kernel, and explores whether they are Summability kernels. Closely related, the article explores Several properties of convolution. The last part of the article discusses some important inequalities in the L-p space and uses non-classical methods called the tensor power trick to prove some interpolation theorems, such as the Riesz-Thorin Interpolation theorem and the Hausdorff-Young Inequality.