Let $\varphi:\mathbb{R}^2\to\mathbb{R}$ be a continuous function. Moreover, consider that $f:\mathbb{R}\to\mathbb{R}$ is a schwarzian function, i.e. $f\in C^{\infty}$ and $\lim\limits_{x\to\pm\infty} |x^n f^{(m)}(x)|=0,\ \forall m,n\in\mathbb{N}$. (they are called rapidly decreasing function). Is it true that the following integral exists?
$$\widehat{f}(x)=\int_{-\infty}^{\infty}f(t)e^{i\varphi (x,t)}dt $$
Since $\varphi$ is real valued $|e^{i\varphi(x,t)}|=1$ and $$ \Bigl|\int_{-\infty}^{\infty}f(t)\,e^{i\varphi (x,t)}\,dt \Bigr|\le\int_{-\infty}^{\infty}|f(t)|\,dt=\|f\|_1. $$ For the existence of the integral it is ehough that $f\in L^1$.