# Recasting the Proof of Parseval's Identity

## TL;DR

This paper extends Fourier analysis to measurable subsets of b^n, providing new proofs of classical inequalities and a generalized Parseval's identity, with applications to integral inequalities.

## Contribution

It introduces a new proof of the Integral Cauchy-Schwarz Inequality and generalizes Parseval's Identity for measurable subsets of b^n, enabling new integral inequality criteria.

## Key findings

- New proof of the Integral Cauchy-Schwarz Inequality
- Generalized Parseval's Identity for b^n
- Criteria for additional integral inequalities

## 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.