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- W1992904014 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 _{pâ}^{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 $sum nolimits _{d - 1} { subseteq } ;{{hat {mathbb {R}}}^d}$." @default.
- W1992904014 created "2016-06-24" @default.
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- W1992904014 date "1991-02-01" @default.
- W1992904014 modified "2023-09-26" @default.
- W1992904014 title "A multidimensional Wiener-Wintner theorem and spectrum estimation" @default.
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- W1992904014 doi "https://doi.org/10.1090/s0002-9947-1991-1013327-9" @default.
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