We set :

• $\phi_1, \phi_2 : \mathbb{R} \mapsto \mathbb{R}^M$ with compact support
• $\theta \in \mathbb{R}^M$ non-zero, $\theta_{p-1} = (\theta_i^{p-1})_{1 \leq i \leq M} \in \mathbb{R}^M$ with $p \geq 2$.

We assume that the matrix $$ \int_\mathbb{R} \phi_1 \phi_1^T - \int_\mathbb{R} \phi_2 \phi_2^T $$ is positive definite. I then want to show that (if it is true) : $$ \int_\mathbb{R} \theta_{p-1}^T \phi_1(t) \phi_2(t)^T \theta < \int_\mathbb{R} \theta_{p-1}^T \phi_1(t) \phi_1(t)^T \theta $$ If $p=2$ the problem is easy with Cauchy-Schwarz on the second term + assumption. I struggle to prove the general form (which seems to be true according to some code but I am not sure).