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Transactions of the American Mathematical Society

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A multidimensional Wiener-Wintner theorem and spectrum estimation

Author: John J. Benedetto
Journal: Trans. Amer. Math. Soc. 327 (1991), 833-852
MSC: Primary 42B10
MathSciNet review: 1013327
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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

$\displaystyle \int {\vert\hat f(\gamma)\vert^2}d{\mu _\phi}(\gamma)= \mathop {\... ...}\,\frac{1}{\vert B(R )\vert}\,\int_{B(R)} \vert f \ast \varphi (t)\vert^2\;dt,$

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

$\displaystyle \forall f \in {L^1}({\mathbb{R}^d})\, \cap \,{L^p}({\mathbb{R}^d}... ...ght)\;\left(\parallel f {\parallel _{1}} + \parallel f{\parallel _{p}} \right),$

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}$.

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