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The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula $$ \int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...

Let us consider the three groups $(\mathbb{R},+)$, $(\mathbb{Z}/2\mathbb{Z},+)$ and $(\mathbb{R}^\times,\cdot)$ (where $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$). We endow $\mathbb{R}$ with ...

Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...

I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral. Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...