A multidimensional Wiener-Wintner theorem and spectrum estimation

Abstract:

Sufficient conditions are given for a bounded positive measure $\mu$ on ${\mathbb {R}^d}$ to be the power spectrum of a function $\varphi$. Applications to spectrum estimation are made for the cases in which a signal $\varphi$ is known or its autocorrelation ${P_\phi }$ is known. In the first case, it is shown that \[ \int {|\hat f(\gamma )|^2}d{\mu _\phi }(\gamma )= \lim \limits _{R \to \infty } \frac {1}{|B(R )|} \int _{B(R)} |f \ast \varphi (t)|^2\;dt,\] where ${\hat P}_{\varphi }= {\mu _\varphi }$, $B(R)$ is the $d$-dimensional ball of radius $R$, and $f$ ranges through a prescribed function space. In the second case, an example, which is a variant of the Tomas-Stein restriction theorem, is \[ \forall f \in {L^1}({\mathbb {R}^d}) \cap {L^p}({\mathbb {R}^d}), \\ \left (\int _{\sum _{d - 1}} | \hat f(\theta )|^2 d {\mu _{d - 1}}(\theta ) \right )^{1/2} \leq \left (\frac {1}{2}\parallel \mu _{d - 1}^{\vee } \parallel _{p’}^{1/2} \right )\;\left (\parallel f {\parallel _{1}} + \parallel f{\parallel _{p}} \right ),\] where $1 \leq p < 2d/(d + 1)$ and the power spectrum ${\mu _{d - 1}}$ is the compactly supported restriction of surface measure to the unit sphere $\sum \nolimits _{d - 1} { \subseteq } \;{{\hat {\mathbb {R}}}^d}$.