Relation between the correlation coefficients $\widehat F(w) = C(f(a), w^ta)$ and $(-1)^{f(a)}$ In the paper Correlation Matrices by Joan Daemen et. al., the authors state (on page 3) that if the correlation coefficients $C(f(a),w^ta)$ of a Boolean function $f$ are denoted by $\widehat F(w)$ then $$\widehat f(a) = \sum_w \widehat F(w) (-1)^{w^t a},$$
with $\widehat f(a) = (-1)^{f(a)}$.

Can someone explain why this formula is correct?
Kodlu already gave a hint, but here are a bit more details (since I don't think this is homework). By definition (or equation 3 in your reference),

$$\widehat F(w)= C(f(a), w^t a) = 2^{-n} \sum_{x \in \mathbb{F}_2^n} (-1)^{f(x) + w^t x}.$$

Hence

$$\sum_{w \in \mathbb{F}_2^n} \widehat F(w) (-1)^{w^t a} = 2^{-n} \sum_{x \in \mathbb{F}_2^n} (-1)^{f(x)} \left(\sum_{w \in \mathbb{F}_2^n} (-1)^{w^t (x + a)}\right).$$

The inner sum is zero whenever $x \neq a$ and equal to $2^n$ otherwise. Hence

$$\sum_{w \in \mathbb{F}_2^n} \widehat F(w) (-1)^{w^t a} = \sum_{x \in \mathbb{F}_2^n} (-1)^{f(x)} \delta_{x, a} = (-1)^{f(a)} = \widehat f(a).$$

I've used the same notation as in your reference. If you want a more general understanding of this result, you should read about Fourier transformations on (locally compact) groups.