Abstract
Periodization and sampling operators are defined, and the Fourier transform of periodization is uniform sampling in a well-defined sense. Implementing this point of view, Poisson Summation Formulas are proved in several spaces including integrable functions of bounded variation (where the result is known) and elements of mixed norm spaces. These Poisson Summation Formulas can be used to prove corresponding sampling theorems.
The sampling operators used to understand and prove the aforementioned Poisson Summation Formulas lead to the introduction of spaces of continuous linear operators which commute with integer translations. Operators L of this type are appropriately called sampling multipliers. For a given function f, they give rise to new sampling formulas, whose sampling coefficients are of the form Lf. In practice, Lf can be used to model noisy data or data where point values are not available. By representation theorems of the second named author, some of these operator spaces are proved to be mixed norm spaces.
The approach and results of this paper were developed in the context of Duffin and Schaeffer’s theory of frames. In particular, sampling multipliers L are related to the Bessel map used by Duffin and Schaeffer in their definition of the frame operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Benedek, A. and Panzone, R., (1961). The spacesL P, with mixed norm.Duke Math. J. 28, 301–324.
Benedetto, J.J., (1975).Spectral Synthesis. Academic Press, New York.
Benedetto, J.J., (1992). Irregular Sampling and Frames, inWavelets: A Tutorial in Theory and Applications. Chui, C., Ed. Academic Press, Boston, pp. 445–507.
Benedetto, J.J., (1996).Harmonic Analysis and Applications. CRC Press Inc., Boca Raton, FL.
Benedetto, J.J. and Frazier, M., Eds., (1994).Wavelets: Mathematics and Applications. CRC Press Inc., Boca Raton, FL.
Butzer, P., Splettstösser, W. and Stens, R., (1988). The sampling theorem and linear prediction in signal analysis.Jahresber. Deutsch. Math.-Verein. 90, 1–70.
Carleson, L., (1966). On convergence and growth of partial sums of Fourier series.Acta Math. 116, 135–157.
Duffin, R.J. and Schaeffer, A.C., (1952). A class of nonharmonic Fourier series.Trans. Amer. Math. Soc. 72, 341–366.
Duffin, R.J. and Weinberger, H., On dualizing a multivariable Poisson Summation Formula.J. Fourier Analysis and Appl. 3, 487–497.
Fournier, J.J. and Stewart, J., (1985). Amalgams ofL p andl q.Bull. Amer. Math. Soc. (N.S.) 13, 1–21.
Herz, C.S., (1971). The theory ofp-spaces with an application to convolution operators.Trans. Amer. Math. Soc. 154, 69–82.
Higgins, J., (1985). Five short stories about the cardinal series.Bull. Amer. Math. Soc. (N.S.) 12, 54–89.
Hörmander, L., (1983). The Analysis of Linear Partial Differential Operators, Grundlehren der mathematischen Wissenschaften Bd. 256, Springer Verlag, Berlin.
Hunt, R.A., (1968). On the convergence of Fourier series,in Proceedings of the Conference on Orthogonal Expansions and Their Continuous Analogues, Southern Illinois University Press, pp. 234–255.
Jia, R.Q. and Micchelli, C.A., (1991). Using the refinement equation for the construction of pre-wavelets II: Powers of two, inCurves and Surfaces. Laurent, P.J. et al., Eds., Academic Press, Boston, pp. 209–246.
Kahane, J.-P. and Lemarié-Rieusset, P.-G., (1994). Remarques sur la formule sommatoire de Poisson.Studia Mathematica 109, 303–316.
Katznelson, Y., (1976).An Introduction to Harmonic Analysis. Dover Publications, New York.
Larsen, R., (1971).An Introduction to the Theory of Multipliers. Grundlehren der mathematischen Wissenschaften Bd. 175, Springer Verlag, Berlin.
Marti, J.T., (1969).Introduction to the Theory of Bases. Springer Tracts in Natural Philosophy 18, Springer Verlag, New York.
Schwartz, L., (1966).Théorie des Distributions. Hermann, Paris.
Stein, E.M. and Weiss, G., (1971).Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton.
Wiener, N., (1958).The Fourier Integral and Certain of its Applications. Dover Publications, New York.
Zimmermann, G., (1994).Projective Multiresolution Analysis and Generalized Sampling. Ph.D. Thesis, University of Maryland, College Park, MD.
Zimmermann, G., Double projective approximation and projective decomposition in the construction of wavelet bases. Submitted.
Zimmermann, G., Representation theorems for generalized sampling operators, quasiinterpolation, and attenuation factors. In preparation.
Zygmund, A., (1959).Trigonometric Series. Vol. I & II, Cambridge University Press, Cambridge.
Author information
Authors and Affiliations
Additional information
The first named author was supported in part by AFOSR contract F49620-96-1-0193.
The second named author was supported by the Cusanuswerk.
Rights and permissions
About this article
Cite this article
Benedetto, J.J., Zimmermann, G. Sampling multipliers and the Poisson Summation Formula. The Journal of Fourier Analysis and Applications 3, 505–523 (1997). https://doi.org/10.1007/BF02648881
Received:
Issue date:
DOI: https://doi.org/10.1007/BF02648881
