Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A multidimensional Wiener-Wintner theorem and spectrum estimation


Author: John J. Benedetto
Journal: Trans. Amer. Math. Soc. 327 (1991), 833-852
MSC: Primary 42B10
DOI: https://doi.org/10.1090/S0002-9947-1991-1013327-9
MathSciNet review: 1013327
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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}$.

References [Enhancements On Off] (What's this?)

  • [Ba] J. Bass, Fonctions de corrélation fonctions pseudo-aléatoires et applications, Masson, Paris, 1984.
  • [B] J. Benedetto, Spectral synthesis, Academic Press, New York, 1975. MR 0622040 (58:29850b)
  • [BH] J. Benedetto and H. Heinig, Fourier transform inequalities with measure weights, Adv. in Math. (to appear). MR 1196988 (93m:42004)
  • [Be] J.-P. Bertrandias, Espaces de fonctions continues et bornées en moyenne asymptotique d'ordre $ p$, Mém. Soc. Math. France 5 (1966). MR 0196411 (33:4598)
  • [Bo] N. Bourbaki, Intégration, Livre VI, Hermann, Paris, 1952.
  • [CD] Y.-C. Chang and K. Davis, Lectures on Bochner-Riesz means, London Math. Soc. Lecture Note, Ser. 114, Cambridge Univ. Press, 1987. MR 921849 (88m:42031)
  • [Ma] P. Malliavin, Intégration et probabilités, analyse de Fourier et analyse spectrale, Masson, Paris, 1982. MR 686271 (85c:00003)
  • [Me] Y. Meyer, Le spectre de Wiener, Studia Math. 27 (1966), 189-201. MR 0203355 (34:3208)
  • [S] E. Stein, Oscillatory integrals in Fourier analysis, Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., no. 112, Princeton Univ. Press, 1986. MR 864375 (88g:42022)
  • [T] P. Tomas, Restriction theorems for the Fourier transform in harmonic analysis in Euclidean spaces, Proc. Sympos. Pure Math., vol. 35, Part 1, Amer. Math. Soc., Providence, R. I., 1979, pp. 111-114. MR 545245 (81d:42029)
  • [W] N. Wiener, Collected works, Vol. II, P. Masani, Ed., The MIT Press, 1979. MR 0532698 (58:27161)
  • [WW] N. Wiener and A. Wintner, On singular distributions, J. Math. Phys. 17 (1939), 233-246 (Collected Works, Vol. II, P. Masani, Ed.).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42B10

Retrieve articles in all journals with MSC: 42B10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-1013327-9
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society