Value of the Fourier transform of the Euclidean ball in $d$ dimensions at the origin

Question

Let $B$ stand for the centered Euclidean ball with radius $\frac{1}{2}$ in $d$ dimensions. What is the value of the Fourier transform of the indicator function of B evaluated at the origin?

Answer 1

Let $B_r$ denote the unit ball with radius $r$, and let $\chi_S$ denote the characteristic function of the set $S$. This post considers the ball with radius $1$. To apply it to our situation, note that $$ \chi_{B_{1/2}}(\mathbf x) = \chi_{B_1}(2\mathbf x). $$ With the "time-scaling property" of the Fourier transform (or equivalently after applying a substitution in the integral), we have $\hat{\chi}_{B_{1/2}}(\xi) = \frac 1{2^d} \hat\chi_{B_1}(\xi/2)$.

Answer 2

$\hat{1}_B(0) = \int 1_B(x) dx= {1 \over 2^d} m B(0,1)$.