## Abstract

Kramer's sampling theorem, which is a generalization of the Whittaker-Shannon-Kotel'nikov (WSK) sampling theorem, enables one to reconstruct functions that are integral transforms of types other than the Fourier one from their sampled values. In this paper, we generalize Kramer's theorem to*N* dimensions (*N* ≥ 1) and show how the kernel function and the sampling points in Kramer's theorem can be generated. We then investigate the relationship between this generalization of Kramer's theorem and*N*-dimensional versions of both the WSK theorem and the Paley-Wiener interpolation theorem for band-limited signals. It is shown that the sampling series associated with this generalization of Kramer's theorem is nothing more than an*N*-dimensional Lagrange-type interpolation series.

### Similar content being viewed by others

## References

P. Butzer, “A Survey of the Whittaker-Shannon Sampling Theorem and Some of Its Extensions,”

*J. Math. Res. Exposition*, vol. 3, 1983, pp. 185–212.P. Butzer, W. Splettstöβer, and R. Stens, “The Sampling Theorem and Linear Prediction in Signal Analysis,”

*Jber. Deutsch. Math. Verein. Bd.*, 1988, pp. 1–70.A. Papoulis,

*Signal Analysis*, McGraw-Hill, New York, 1977.P. Weiss, “Sampling Theorems Associated with Sturm-Liouville Systems,”

*Bull. Am. Math. Soc.*, vol. 63, 1957, p. 242.L. Campbell, “A Comparison of the Sampling Theorems of Kramer and Whittaker,”

*J. SIAM*, vol. 12, 1964, pp. 117–130.A. Jerri, “On the Equivalence of Kramer's and Shannon's Sampling Theorems,”

*IEEE Trans. Inform. Theory*, vol. IT-15, 1969, pp. 497–499.H. Kramer, “A Generalized Sampling Theorem,”

*J. Math. Phys.*, vol. 38, 1959, pp. 68–72.G. Watson,

*A Treatise on the Theory of Bessel Functions*, 2nd ed., Cambridge University Press, Cambridge, England, 1962.J. Higgins, “An Interpolation Series Associated with the Bessel-Hankel Transform,”

*J. London Math. Soc.*(2), vol. 5, 1972, pp. 707–714.A. Jerri, “A Note on Sampling Expansion for a Transform with Parabolic Cylindrical Kernal,”

*Inform. Sci.*, vol. 26, 1982, pp. 155–158.A. Jerri, “Sampling Expansion for Laguerre-

*L*^{α}_{ p }Transforms,”*J. Res. Nat. Bur. Standards, Sec. B*, vol. 80, 1976, pp. 415–418.A. Jerri, “On the Application of Some Interpolating Functions in Physics,”

*J. Res. Nat. Bur. Standards, Sec. B*, vol. 80, 1969, pp. 241–245.A. Jerri, “Some Applications for Kramer's Generalized Sampling Theorem,”

*J. Eng. Math.*, vol. 3, 1969, pp. 103–105.F. Mehta, “A General Sampling Expansion,”

*Inform. Sci.*, vol. 16, 1978, pp. 4–46.F. Mehta, “Sampling Expansion for Band-Limited Signals through Some Special Functions,”

*J. Cybernetics*, 1975, pp. 61–68.M. Rawn, “On Nonuniform Sampling Expansions Using Entire Interpolating Functions and on the Stability of Bessel-Type Sampling Expansions,”

*IEEE Trans. Inform. Theory*, vol. 35, 1989, pp. 549–557.A. Zayed, “Sampling Expansion for the Continuous Bessel Transform,”

*J. Appl. Anal.*, vol. 27, 1988, pp. 47–64.A. Papoulis,

*Systems and Transforms with Applications in Optics*, McGraw-Hill, New York, 1968.A. Jerri, “The Shannon Sampling Theorem—Its Various Extensions and Applications: A Tutorial Review,”

*Proc. IEEE*, vol. 65, 11, 1977, pp. 1565–1596.A. Zayed, G. Hinsen, and P. Butzer, “On Lagrange Interpolation and Kramer-Type Sampling Theorems Associated with Sturm-Liouville Problems,”

*SIAM J. Appl. Math.*, vol. 50, 1990, pp. 893–909.A. Zayed, “On Kramer's Sampling Theorem Associated with General Sturm-Liouville Problems and Lagrange Interpolation,”

*SIAM J. Appl. Math.*, vol. 5, 1991, pp. 575–604.R. Paley and N. Wiener,

*Fourier Transforms in the Complex Domain*, Colloq. Publ., vol. 19, Am. Math. Soc., Providence, RI, 1934.N. Levinson,

*Gap and Density Theorems*, Am. Math. Soc. Colloq. Publ., vol. 26, Procidence, RI, 1940.R. Gosselin, “Singular Integrals and Cardinal Series,”

*Studia Math.*, vol. 44, 1972, pp. 39–45.R. Gosselin, “On the

*L*^{p}Theory of Cardinal Series,”*Ann. of Math.*, vol. 78, 1963, pp. 567–581.J. Higgins, “A Sampling Theorem for Irregularly Spaced Sample Points,”

*IEEE Trans. Inform. Theory*, vol. IT-22, 1976, pp. 621–622.K. Yao and J. Thomas, “On Some Stability and Interpolatory Properties of Nonuniform Sampling Expansions,”

*IEEE Trans. Circuit Theory*, vol. CT-14, 1967, pp. 404–408.J. Yen, “On Nonuniform Sampling of Bandwidth-Limited Signals,”

*IRE Trans. Circuit Theory*, CT-3, 1956, pp. 251–257.E. Parzen, “A Simple Proof and Some Extensions of Sampling Theorems,”

*Tech. Rep.*, vol. 7, Stanford University, Stanford, 1956.P. Butzer and G. Hinsen, “Two-Dimensional Nonuniform Sampling Expansions—An Iterative Approach,” preprint.

R. Butzer and G. Hinsen, “Reconstruction of Bounded Signals from Pseudo-Periodic, Irregularly Spaced Samples,” preprint.

J. Clark, M. Palmer, and P. Lawrence, “A Transformation Method for the Reconstruction of Functions from Nonuniformly Spaced Samples,”

*IEEE Trans. Acoust. Speech, Signal Processing*, vol. ASSP-33, 1985, pp. 1151–1165.R. Mersereau, “The Processing of Hexagonally Sampled Two Dimensional Signals,”

*Proc. IEEE*, vol. 67, 1979, pp. 930–949.R. Mersereau and T. Speake, “The Processing of Periodically Sampled Multidimensional Signals,”

*IEEE Trans. Acoust. Speech Signal Process.*, vol. ASSP-32, 1983, pp. 188–194.W. Montgomery, “

*K*-Order Sampling of*N*-Dimensional Band-Limited Functions,”*Int. J. Contr.*, vol. 1, 1965, pp. 7–12.D. Mugler and W. Splettstö∐er, “Reconstruction of Two Dimensional Signals from Irregularly Spaced Samples,”

*Proceedings 6 Aachner Symposium für Signaltheorie, Informatik Fachberichte*153, Springer-Verlag, New York, 1987, pp. 41–44.D. Petersen and D. Middleton, “Sampling and Reconstruction of Wave Number-Limited Functions in

*N*-Dimensional Euclidean Spaces,”*Inform. and Control*, vol. 5, 1962, pp. 279–323.R. Proesser, “A Multidimensional Sampling Theorem,”

*J. Math. Anal. Appl.*, vol. 16, 1966, pp. 574–584.W. Splettstö∐er, “Sampling Approximation of Continuous Functions with Multidimensional Domain,”

*IEEE Trans. Inform. Theory*, vol. IT-28, 1982, pp. 809–814.J. Higgins, “Five Short Stories about the Cardinal Series,”

*Bull. Am. Math. Soc.*, vol. 12, 1985, pp. 45–89.B. Sharma and F. Mehta, “Generalized Bandpass Sampling Theorem,”

*Math. Balkanica*, vol. 6, 1976, pp. 204–217.B. Levitan and I. Sargsjan,

*Introduction to Spectral Theory: Self-Adjoint Ordinary Differential Operators*, Transl. Math. Monographs, vol. 39, Am. Math. Soc., Providence, RI, 1975.E. Titchmarsh,

*Eigenfunction Expansions Associated with Second Order Differential Equations*, Part 1, 2nd ed., Clarendon Press, Oxford, 1962.E. Titchmarsh,

*Eigenfunctions Expansions Associated with Second Order Differential Equations*, Part II, Clarendon Press, Oxford, 1958.F. Reza,

*An Introduction to Information Theory*, McGraw-Hill, New York, 1961.

## Author information

### Authors and Affiliations

## Rights and permissions

## About this article

### Cite this article

Zayed, A.I. Kramer's sampling theorem for multidimensional signals and its relationship with Lagrange-type interpolations.
*Multidim Syst Sign Process* **3**, 323–340 (1992). https://doi.org/10.1007/BF01940228

Received:

Revised:

Issue Date:

DOI: https://doi.org/10.1007/BF01940228