Exercise 19, § 1

Let $n\in \mathbf{N}$ and let $f\in L^1(\mathbf{R}^n)$ (resp. $L^2(\mathbf{R}^n)$) be such that $f\circ \sigma =f$ for every element $\sigma$ of the special orthogonal group SO($n,\mathbf{R}$). Then the Fourier transform $\widehat{f}$ of $f$ satisfies $\widehat{f}(\sigma (y)) =\widehat{f}(y)$ for every $\sigma \in$ SO($n,\mathbf{R}$) and every $y\in \mathbf{R}^n$ (resp. almost every $y\in \mathbf{R}^n$).