Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and $$|f(z)|\sim |z|^{-a},\qquad |z|\to \infty,\qquad z\in S.$$ Let $g(x):=f(e^{i \theta} x)$. Then $x\mapsto g(|x|)$ is obviously continuous and bounded on $\mathbb{R}^{d}$, and consequently has a Fourier transform in the sense of tempered distributions.
Is it true that
$$|\mathcal{F}\left(g(|\cdot|)\right)(\xi)|\sim |\xi|^{a-d},\qquad \text{as}\quad |\xi|\to 0$$
