Let $x=(x_{1},..., x_n)\in \mathbb R^n$, $\gamma, \beta$ is multi-index of $n-$tuple of nonnegative integers. Put $x^{\gamma }= x_1^{\gamma_1}\cdots x_{n}^{\gamma_n}.$ Let $f\in \mathcal{S}(\mathbb R^n)$ (Schwartz space).
Let $$\displaystyle I_{\gamma}(x):=\int_{\mathbb R^n} x^{\gamma}e^{it\cdot x}t^{\beta}f(t)\ dt. $$
Can we say $$\displaystyle I_{\gamma}(x)=\int_{\mathbb R^n}e^{it\cdot x}\sum_{\alpha\leqslant \gamma}\binom{\gamma}{\alpha}\partial^{\alpha}f(t)t^{\beta-\alpha}\frac{\beta !}{(\beta-\alpha)!}\ dt?$$
(I guess, I should use Integrating by parts and using Leibniz formula, but I do not how do this explicitly in this situation)
Motivation: I am trying to understand this answer