$x(t)\rightarrow x(-t)$ and $x(t)\rightarrow x^\ast(-t)$ transforms

Question

I have to determine if these transforms are linear and the core of the transforms:

• $x(t)\rightarrow x(-t), \quad t\in \mathbb{R}$
• $x(t)\rightarrow x^\ast(-t), \quad t\in \mathbb{R}$

With "the core of the transforms" I mean the filter function $h(t)$ that performs the transform if I convolve it with $x(t)$

Answer 1

Hint: Denote the Fourier transform by $\mathscr{F}$, you may show $$ \mathscr{F}^2 ( x(t)) = x(-t), \quad t \in \mathbb{R} $$ when it is defined.