Abstract
We characterize the Hilbert spaces H whose elements are distributions supported on the interval [0, 1] and which have the property that the system of exponentials {e2πinx}n∈Z forms a complete orthogonal system for H, generalizing in this way the classical situation where H=L2([0, 1]) and the system is actually orthonormal. This characterization is extended to the more general setting of spectral pairs and is used to obtain sampling results in various related spaces of functions, that generalize the classical Shannon sampling theorem.
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References
Campbell, L.L. (1968). Sampling theorem for the Fourier transform of a distribution with bounded support.SIAM J. Appl. Math.,16, 626–636.
Edwards, R.E. (1995).Functional Analysis, Theory and Applications, Dover Publishers, Inc., New York.
Fuglede, B. (1974). Commuting self-adjoint partial differential operators and a group theoretic problem.J. Funct. Anal.,16, 101–121.
Gabardo, J.-P. and Nashed, M.Z. (1998). Nonuniform multiresolution analyses and spectral pairs,J. Funct. Anal.,158, 209–241.
Gabardo, J.-P. and Nashed, M.Z. (1998). An analogue of Cohen's condition for nonuniform multiresolution analyses, in:Wavelets, Multiwavelets and Their Applications, Contemporary Mathematic, Vol. 216, Aldroubi, A. and Lin, E.B., Eds., American Mathematical Society, Providence, RI, 41–61.
Garciá, A.G., Hernández-Medina, M.Á., and Moro, J. (1998). On the distributional Fourier duality and its applications,J. Math. Anal. Appl.,227, 43–54.
Jorgensen, P.E.T. (1982). Spectral theory of finite volume domains inR n,Adv. Math.,44, 105–120.
Jorgensen, P.E.T. and Pedersen, S. (1987). Harmonic analysis on tori,Acta Appl. Math.,10, 87–99.
Jorgensen, P.E.T. and Pedersen, S. (1991). Sur un problème spectral algébrique,C. R. Acad. Sci. Paris Sér. I Math.,312, 495–498.
Jorgensen, P.E.T. and Pedersen, S. (1992). Spectral theory for Borels sets inR n of finite measure.J. Funct. Anal.,107, 72–104.
Jorgensen, P.E.T. and Pedersen, S. (1993). Group-theoretic and geometric properties of multivariable Fourier series,Exposition. Math.,11, 309–329.
Lee, A.J. (1976). Characterization of band-limited functions and processes,Inform. Control.,31, 258–271.
Pedersen, S. (1987). Spectral theory of commuting self-adjoint partial differential operators,J. Funct. Anal.,73, 122–134.
Pedersen, S. (1996). Spectral sets whose spectrum is a lattice with a base.J. Funct. Anal.,141, 496–509
Pfaffelhuber, E. (1971). Sampling series for band-limited generalized functions.IEEE Trans. Inform. Theory, IT 17, 650–654.
Schwartz, L. (1966).Théorie des Distributions, Hermann, Paris.
Zakai, M. (1965). Band-limited functions and the sampling theorem,Inform. Control,8, 143–158.
Zayed, A.I. (1993).Advances in Shannon's Sampling Theory, CRC Press, Boca Raton, FL.
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Communicated by Hans G. Feichtinger
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Gabardo, JP. Hilbert spaces of distributions having an orthogonal basis of exponentials. The Journal of Fourier Analysis and Applications 6, 277–298 (2000). https://doi.org/10.1007/BF02511156
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DOI: https://doi.org/10.1007/BF02511156