Recasting the Proof of Parseval's Identity

Abstract

We generalize aspects of Fourier Analysis from intervals on $\mathbb{R}$ to bounded and measurable subsets of ${\mathbb{R}}^n$. In doing so, we obtain a few interesting results. The first is a new proof of the famous Integral Cauchy-Schwarz Inequality. The second is a restatement of Parseval's Identity that doubles as a representation of integrating bounded and measurable functions over bounded and measurable subsets of ${\mathbb{R}}^n$. Finally, we apply these first two results to develop some sufficient criteria for additional integral inequalities that are elementary in nature.

10 pages, 0 figures