Abstract
In this paper we prove general sampling theorems for functions belonging to a reproducing kernel Hilbert space (RKHS) which is also a closed subspace of a particular Sobolev space. We present details of this approach as applied to the standard sampling theory and its extension to nonuniform sampling. The general theory for orthogonal sampling sequences and nonorthogonal sampling sequences is developed. Our approach includes as concrete cases many recent extensions, for example, those based on the Sturm-Liouville transforms, Jacobi transforms, Laguerre transforms, Hankel transforms, prolate spherical transforms, etc., finite-order sampling theorems, as well as new sampling theorems obtained by specific choices of the RKHS. In particular, our setting includes nonorthogonal sampling sequences based on the theory of frames. The setting and approach enable us to consider various types of errors (truncation, aliasing, jitter, and amplitude error) in the same general context.
Similar content being viewed by others
References
[A] N. Aronszajn, Theory of reproducing kernels,Trans. Amer. Math. Soc.,68 (1950), 337–404.
[B1] L. de Branges,Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood Cliffs, NJ, 1968.
[B2] P. L. Butzer, A survey of the Whittaker-Shannon sampling theorem and some of its extensions,J. Math. Res. Exposition,3 (1983), 185–212.
[BSS] P. L. Butzer, W. Splettstößer, and R. L. Stens, The sampling theorem and linear predictions in signal analysis,Jahresber. Deutsch. Math.-Verein,90 (1988), 1–60.
[C] L. L. Campbell, A comparison of the sampling theorems of Kramer and Whittaker,J. SIAM,12 (1964) 117–130.
[E] A. Erdelyiet al. (Bateman manuscript project),Higher Transcendental Functions, Vol. I, McGraw-Hill, New York, 1953.
[GS] I. M. Gelfand, and G. E. Shilo,Generalized Functions, Vol. 1 and 2, Academic Press, New York, 1964.
[G] H. Glaeske, The Laguerre transform of some elementary functions,Z. Anal,3 (1984), 237–244.
[H1] J. R. Higgins, Five short stories about the cardinal series,Bull. Amer. Math. Soc.,12 (1985), 45–89.
[H2] E. Hille, Introduction to the general theory of reproducing kernels,Rocky Mountain J. Math.,2 (1972), 321–368.
[HP] R. F. Hoskins and deSousa Pinto, Sampling expansions for functions bandlimited in the distributional sense,SIAM J. Appl. Math.,44 (1984), 605–610.
[J1] A. J. Jerri, On the application of some interpolating functions in physics,J. Res. Nat. Bur. Standards, Sect. B,73B (1969), 241–245.
[J2] A. J. Jerri, Sampling expansion for Laguerre-L a v transforms,J. Res. Nat. Bur. Standards, Sect. B,80B (1976), 415–418.
[J3] A. J. Jerri, The Shannon sampling theorem—its various extensions and applications: A turorial review,Proc. IEEE,65 (1977), 1565–1596.
[KW] T. Koornwinder and G. G. Walter, The finite continuous Jacobi transform and its inverse,J. Approx. Theory,60 (1990), 83–100.
[K] H. P. Kramer, A generalized sampling theorem,J. Math. Phys.,38 (1959), 68–72.
[NW] M. Z. Nashed and G. Wahba, Convergence rates of approximate least squares solutions of linear integral and operator equations of the first kind,Math. Comp.,28, (1974), 69–80.
[PW] R. E. A. C. Paley and N. Wiener,Fourier Transforms in the Complex Domain, Colloquium Publications, Vol. 29, American Mathematical Society, Providence, RI, 1934.
[P] A. Papoulis,Signal Analysis, McGraw-Hill, New York, 1977.
[R1] M. D. Rawn, Generalized sampling theorems for Bessel-type transforms of bandlimited functions and distributions,SIAM J. Appl. Math.,49 (1989), 638–649.
[R2] W. Rudin,Functional Analysis, McGraw-Hill, New York, 1973.
[S1] H. S. Shapiro,Topics in Approximation Theory, Springer-Verlag, New York, 1981.
[S2] G. Szeğo,Orthogonal Polynomials, Colloquium Publications, Vol. 23, American Mathematical Society, Providence, RI, 1939.
[T] E. C. Titchmarsh,Eigenfunction Expansions Associated with Second-Order Differential Equations, Clarendon Press, Oxford, 1953.
[W] H. W. Weinert (ed.),Reproducing Kernel Hilbert Spaces. Applications in Statistical Signal Processing, Hutchinson Ross, Stroudsburg, PA, 1982.
[Y1] K. Yao, Applications of reproducing kernel Hilbert spaces—bandlimited signal models,Inform. and Control,11 (1967), 429–444.
[Y2] R. M. Young,An Introduction to Non-Harmonic Fourier Series, Academic Press, New York, 1980.
[Z1] M. Zakai, Band-limited functions and the sampling theorem,Inform. and Control,8 (1965), 143–158.
[ZHB] A. Zayed, G. Hinsen, and P. L. Butzer, On Lagrange interpolation and Kramer-type sampling theorems associated with Sturm-Liouville problems,SIAM J. Appl. Math.,50 (1990), 893–909.
[Z2] A. Zygmund,Trigonometric Series, Cambridge University Press, New York, 1957.
Author information
Authors and Affiliations
Additional information
This work was supported in part by the National Science Foundation under Grant DMS-901526.
Rights and permissions
About this article
Cite this article
Nashed, M.Z., Walter, G.G. General sampling theorems for functions in reproducing kernel Hilbert spaces. Math. Control Signal Systems 4, 363–390 (1991). https://doi.org/10.1007/BF02570568
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02570568