A Sampling Principle Associated with Saitoh’s Fundamental Theory of Linear Transformations
Chapter
Abstract
A more general form of a sampling theorem due to Saitoh is given, to the effect that members of certain reproducing kernel Hilbert function spaces are recoverable from samples via a sampling series. This series can be viewed as a discrete form of reproducing equation. The ‘information loss error’ associated with this series expansion is discussed.
The sampling principle of Kramer is generalized in this context, and several examples are given where the kernels arise from polynomials of Meixner type.
Keywords
Hilbert Space Orthogonal Basis Sampling Theorem Sampling Theory Reconstruction Function
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
References
- 1.M.H. Annaby and G. Freiling, Sampling expansions associated with Kamke problems,submitted.Google Scholar
- 2.S. Azizi and D. Cochran, Reproducing kernel structure and sampling on time-warped spaces with application to warped wavelets,IEEE Trans. Info. Theory, submittedGoogle Scholar
- 3.T.S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New York, 1978.MATHGoogle Scholar
- 4.P.J. Davis, Interpolation and approximation, Blaisdell, New York, 1963.MATHGoogle Scholar
- 5.W.N. Everitt and G. Nasri-Roudsari, Sturm-Liouville problems with coupled boundary conditions and Lagrange interpolation series, J. Comput. Anal. Appl. 1, pp. 319–347, 1999.MathSciNetMATHGoogle Scholar
- 6.C.D. Godsil, Algebraic combinatorics, Chapman and Hall Mathematics, New York, 1993.MATHGoogle Scholar
- 7.J.R. Higgins, Sampling theory in Fourier and signal analysis: foundations, Clarendon Press, Oxford, 1996.MATHGoogle Scholar
- 8.J.R. Higgins and R.L. Stens, Eds., Sampling theory in Fourier and signal analysis: advanced topics, Clarendon Press, Oxford, 1999.MATHGoogle Scholar
- 9.W. Magnus, F. Oberhettinger and R.P. Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition, Springer-Verlag, Berlin, 1966.Google Scholar
- 10.J.T. Marti, Introduction to the theory of bases, Springer-Verlag New York Inc., Berlin Heidelberg, 1969.Google Scholar
- 11.M.Z. Nashed and G.G. Walter, General sampling theorems for functions in reproducing kernel Hilbert spaces, Mathematics of Control, Signals and Systems 4, pp. 363–390, 1991.MathSciNetMATHCrossRefGoogle Scholar
- 12.M. Nees, Interpolation on R of Hermite polynomials and its application to sampling, Presented at Sampta’97, the 1997 International Workshop on Sampling Theory and Applications, Universidade de Aveiro, Portugal. Unpublished.Google Scholar
- 13.S. Saitoh, Theory of reproducing kernels and its applications, Longman Scientific and Technical, Harlow, 1988.MATHGoogle Scholar
- 14.S. Saitoh, Integral transforms, reproducing kernels and their applications, Longman, Harlow, 1997.MATHGoogle Scholar
- 15.E. Schmidt, Über die Charlier—Jordansche Entwicklung einer willkürlichen Funktion nach der Poissonschen Funktion und ihren Ableitungen, Z. Angew. Math. Mech. 13, pp. 139–142, 1933.CrossRefGoogle Scholar
- 16.G. Szegö, Orthogonal polynomials, American Math. Soc., Providence, R.I., 1959.Google Scholar
- 17.A.I. Zayed, On Kramer’s sampling theorem associated with general Sturm—Liouville problems and Lagrange interpolation, SIAM J. Appl. Math. 51, pp. 575–604, 1991.MathSciNetMATHGoogle Scholar
- 18.A.I. Zayed, Advances in Shannon’s sampling theory, CRC Press, Boca Raton, FL., 1993.Google Scholar
Copyright information
© Springer Science+Business Media Dordrecht 2001