Let $\alpha > 1$, $\xi \in\mathbb{R}$. and $\chi_A$ be the characteristic function of the set $A$.
Are there some known ways of computing (or estimating in terms of $\xi$) of this kind of integrals? $$ \int_{\mathbb{R}} \cos(\xi x)\cos( (1-x)^{\alpha} ) \chi_{[0,1]}(x) dx$$
Integrating by parts we obtain $$ \int_{\mathbb{R}}\cos(x\,\xi)\cos((1-x)^{\alpha})\chi_{[0,1]}(x)\,dx=\frac{\sin\xi}{\xi}-\frac{\alpha}{\xi}\int_0^1\sin(x\,\xi)\sin((1-x)^{\alpha})(1-x)^{\alpha-1}\,dx. $$ Since $\alpha-1>0$, the last integral converges to $0$ as $\xi\to\infty$. Then $$ \int_{\mathbb{R}}\cos(x\,\xi)\cos((1-x)^{\alpha})\chi_{[0,1]}(x)\,dx=\frac{\sin\xi}{\xi}+O(\xi^{-1}),\quad\xi\to\infty. $$ Depending on the value of $\alpha$, the integration by parts can be done more times, obtaining better asymptotics.