Spatial and frequency centres and variances of an $L^2$ function with finite second moments

Assume countable choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $n\ge 1$ and let $f\in L^2({\mathbb{R}}^n;\mathbb{C})$ (The space $L^p(\mu )$ as the quotient by null functions) be nonzero with finite second moments, ${\int }_{{\mathbb{R}}^n}\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{\mid }f(x){\mathrm{\mid }}^{2}dx<\mathrm{\infty },{\int }_{{\mathbb{R}}^n}\mathrm{\mid }\xi {\mathrm{\mid }}^2\mathrm{\mid }\widehat{f}(\xi ){\mathrm{\mid }}^{2}d\xi <\mathrm{\infty },$ where $\widehat{f}$ is the Plancherel transform of Plancherel theorem. Define the spatial mean $a=(a_1,\dots ,a_n)$ and spatial variance $V_x(f)$, and the frequency mean $b=(b_1,\dots ,b_n)$ and frequency variance $V_{\xi }(f)$, by $a_j:=\frac{1}{\mathrm{\parallel }f{\mathrm{\parallel }}_2^2}{\int }_{{\mathbb{R}}^n}{x}_j\mathrm{\mid }f(x){\mathrm{\mid }}^{2}dx,V_x(f):=\frac{1}{\mathrm{\parallel }f{\mathrm{\parallel }}_2^2}{\int }_{{\mathbb{R}}^n}\mathrm{\mid }x-a{\mathrm{\mid }}^2\mathrm{\mid }f(x){\mathrm{\mid }}^{2}dx,$ $b_j:=\frac{1}{\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_2^2}{\int }_{{\mathbb{R}}^n}{\xi }_j\mathrm{\mid }\widehat{f}(\xi ){\mathrm{\mid }}^{2}d\xi ,V_{\xi }(f):=\frac{1}{\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_2^2}{\int }_{{\mathbb{R}}^n}\mathrm{\mid }\xi -b{\mathrm{\mid }}^2\mathrm{\mid }\widehat{f}(\xi ){\mathrm{\mid }}^{2}d\xi \mathrm{.}$ These are the probability-normalised means and variances of the measures $\mathrm{\mid }f(x){\mathrm{\mid }}^{2}dx\mathrm{/}\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$ and $\mathrm{\mid }\widehat{f}(\xi ){\mathrm{\mid }}^{2}d\xi \mathrm{/}\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_2^2$. The integrals are read in the componentwise convention of Integrable real and complex functions, and their integrals, with $\mathrm{\mid }f{\mathrm{\mid }}^2=f\bar{f}$ and $x_j,{\xi }_j$ the real coordinate functions; the frequencies are measured in the $e^{-2\pi ix\cdot \xi }$ convention of Plancherel theorem and Complex Lp classes and Euclidean test-function conventions. No centring is asserted here: the mean is subtracted in the variance but the transformation property of the pair $(V_x,V_{\xi })$ is proved separately in Centring by translation and modulation preserves the variance product.

Since $f\mathrm{\ne }0$ in $L^2$, we have $\mathrm{\parallel }f{\mathrm{\parallel }}_2>0$. By the $L^2$ Cauchy–Schwarz inequality and the pairing convention of Complex completeness, density, and inner product: the consumer interface, applied to the functions $x\mapsto \mathrm{\mid }{x}_j\mathrm{\mid }\mathrm{\mid }f(x)\mathrm{\mid }$ and $\mathrm{\mid }f\mathrm{\mid }$, ${\int }_{{\mathbb{R}}^n}\mathrm{\mid }{x}_j\mathrm{\mid }\mathrm{\mid }f(x){\mathrm{\mid }}^{2}dx\le \parallel \mathrm{\mid }{x}_j\mathrm{\mid }f{\parallel }_2\mathrm{\parallel }f{\mathrm{\parallel }}_2\le \parallel \mathrm{\mid }x\mathrm{\mid }f{\parallel }_2\mathrm{\parallel }f{\mathrm{\parallel }}_2<\mathrm{\infty },$ where $\mathrm{\mid }{x}_j\mathrm{\mid }\le \mathrm{\mid }x\mathrm{\mid }$ and the hypothesis on the second moment bound the first factor. Each numerator $\mathrm{\mid }\int x_j\mathrm{\mid }f{\mathrm{\mid }}^2\mathrm{\mid }$ is therefore finite, and the same estimate with $\mathrm{\mid }x-a{\mathrm{\mid }}^2\le 2\mathrm{\mid }x{\mathrm{\mid }}^2+2\mathrm{\mid }a{\mathrm{\mid }}^2$ makes the spatial variance numerator finite. On the frequency side Plancherel theorem gives $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_2=\mathrm{\parallel }f{\mathrm{\parallel }}_2>0$ and, together with the assumed second moment of $\widehat{f}$, the same Cauchy–Schwarz estimate makes both frequency numerators finite. Hence all four quantities are well-defined finite real numbers and $a,b\in {\mathbb{R}}^n$.