Let $\delta_{\min} (\cdot)$ and $\delta_{\max}(\cdot)$ stand for the smallest and largest eigen-values of a matrix.
Given matrix $A(w)$ and $B(w_1, w_2)$ and $$0 < c_1 \leq \delta_{\min}(A(w)) \leq \delta_{\max}(A(w)) \leq c_2 < \infty,$$ $$\delta_{\min}\left(\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}B(w_1, w_2) \: dw_1 \, dw_2\right) \geq \xi_1 > 0,$$ $$\delta_{\max}\left(\int_{-\pi}^\pi \int_{-\pi}^\pi B(w_1, w_2) \: dw_1 \, dw_2\right) \leq \xi_2 < \infty.$$
Prove the following is positive definite. $$\int_{-\pi}^\pi \int_{-\pi}^\pi A(w_1)B(w_1, w_2)A^T(w_2)\:dw_1 \, dw_2$$
Any help would be much appreciated!
You cannot prove it. It is false. Let $a>0$ and consider the step functions \begin{aligned} A(t)&=\begin{cases}aI&\text{if }\ 0<t\le\pi,\\ I&\text{if }\ -\pi\le t\le0,\end{cases}\\ B(s,t)&=\begin{cases}-I&\text{if }\ (s,t)\in(0,\pi]^2,\\ I&\text{if }\ (s,t)\in[-\pi,\pi]^2\setminus(0,\pi]^2.\end{cases} \end{aligned} Then $\int_{-\pi}^\pi\int_{-\pi}^\pi B(s,t)\,ds\,dt=2\pi^2I$ is positive definite, but $\int_{-\pi}^\pi\int_{-\pi}^\pi A(t)B(s,t)A(t)^\top\,ds\,dt=(3-a^2)\pi^2I$ is negative definite when $a$ is large. You can also get a counterexample with smooth functions by slightly modifying $A$ and $B$ in the above.