This paper gives a characterization of the frequency domain of weakly translation invariant frame scaling functions with frequency domain \(\operatorname{supp} \widehat{\varphi}=G\). Based on this, it further characterizes convex and ball-shaped frequency domains of a bandlimited scaling function. When the frequency domain is convex and completely symmetric about the origin, then \(0 \in \operatorname{supp} \widehat{\varphi} \subset\left[-\frac{4}{3} \pi, \frac{4}{3} \pi\right]^d\) (this result cannot be improved). For the ball-shaped frequency domain in \(\mathbb{R}^d\) \((d>1)\), whether its center is the origin or not, its radius must satisfy: \[ r \leq \max \left\{\frac{4}{3} \pi,\left(\sqrt{2+\frac{1}{4 d}}-\sqrt{\frac{1}{4 d}}\right) \pi\right\}. \] More importantly, these frequency domain characters are uniquely owned by frame scaling functions and not by orthogonal scaling functions: there does not exist an orthogonal scaling function with a ball-shaped frequency domain. If the frequency domain of the orthogonal scaling function is convex and completely symmetric about the origin, it must contain \([-\pi, \pi]^d\).