Uniqueness theorems for Gaussian measures in \(\ell _ q\), \(1\leq q<\infty\) (Q1822401)

We prove that a Gaussian measure \(\mu\) in \(\ell _ q\) is uniquely determined by the function \[ y\to \int _{\ell _ q}\| x+y\| ^ p_ qd\mu (x),\quad y\in \ell _ q, \] iff \(p\neq q\). This follows from properties of the mapping \[ s\to \int _{\ell _ q}\| x+sy\| ^ p_ qd\mu (x),\quad s\in {\mathbb{R}}, \] where y is either a fixed unit vector of \(\ell _ q\) or the sum (or difference) of two such vectors. Besides \(q=1\) or \(q=2\) this function describes the distribution of the coordinate functional generated by this unit vector.