# Proof of Theorems¶

We briefly prove the two major theorems used in this work

## The DCT Least Squares Approximation Theorem¶

Given a set of $N$ samples of a signal $X = \{x_0, ..., x_N\}$, let $Y = \{y_0,...,y_n\}$ be the DCT coefficients of $X$. Then for any $1 \leq m \leq N$, the approximation

$$p_m(t) = \frac{1}{\sqrt{n}}y_0 + \sqrt{\frac{2}{n}}\sum_{k=1}^m y_k \cos\left(\frac{k(2t+1)\pi}{2n}\right)$$

of $X$ minimizes the least squared error