$\lim_{a \rightarrow \infty}\int_0^{a} t^k \hat{f}(-t)dt$

Question

Let $f \in C^\infty(\mathbb{R})$ be a smooth function with compact support. Does this limit exist :

$$\lim_{a \to\infty}\int_0^{a} t^k \hat{f}(-t)dt$$

Where $k$ is a fixed natural number and $\hat{f}$ denotes the Fourier transform of $f$?

Answer 1

Since $f$ belongs to the Schwarz space so does $\hat f$. Hence, $t^{k}\hat f(t)$ is integrable and the limit exists.