Abstract
The paper is related to the lower and upper estimates of the norm for Mercer kernel matrices. We first give a presentation of the Lagrange interpolating operators from the view of reproducing kernel space. Then, we modify the Lagrange interpolating operators to make them bounded in the space of continuous function and be of the de la Vallée Poussin type. The order of approximation by the reproducing kernel spaces for the continuous functions is thus obtained, from which the lower and upper bounds of the Rayleigh entropy and the l 2-norm for some general Mercer kernel matrices are provided. As an example, we give the l 2-norm estimate for the Mercer kernel matrix presented by the Jacobi algebraic polynomials. The discussions indicate that the l 2-norm of the Mercer kernel matrices may be estimated with discrete orthogonal transforms.
Similar content being viewed by others
References
F. J. Narcowich and J. D. Ward, Norms estimates for the inverses of a general class of scattered data radial function interpolation matrices, J. Approx. Theory, 69 (1992), 84–109.
F. J. Narcowich, N. Sivakumar and J. D. Ward, On condition numbers associated with radial function interpolation, J. Math. Anal. and Appl., 186 (1994), 457–485.
K. Jetter, J. Stǒckler and J. D. Ward, Error estimates for scattered data interpolation on spheres, Math. Comput., 68(226) (1999), 733–747.
R. Schaback, Lower bounds for norms of inverses interpolation matrices for radial basis functions, J. Approx. Theory, 79 (1994), 287–306.
H. Wendland, Scattered Data Approximation, Cambridge University Press (Cambridge, 2005).
M. D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press (Cambridge, 2003).
F. J. Narcowich, N. Sivakumar and J. D. Ward, Stability results for scattered data interpolation on Euclidean spheres, Advances in Computational Mathematics, 8 (1998), 137–168.
D. X. Zhou, The covering number in learning theory, J. Complexity, 18 (2002), 739–767.
D. X. Zhou, Capacity of reproducing kernel spaces in learning theory, IEEE Trans. Inform. Theory, 49 (2003), 1743–1752.
N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc., 68 (1950), 337–404.
F. Cucker and S. Smale, On the mathematical foundations of learning, Bull. Amer. Math. Soc., 39 (2001), 1–49.
F. Cucker and D. X. Zhou, Learning theory: an approximation theory viewpoint, Cambridge University (New York, 2007).
F. Cucker and S. Smale, Best choices for regularization parameters in learning theory: on the bias-variance problem, Found. Comput. Math., 2 (2002), 413–428.
S. Smale and D. X. Zhou, Shannon sampling and function reconstruction from point values, Bull. Amer. Math. Soc., 41 (2004), 279–305.
K. Ball, Eigenvalues of Euclidean distance matrices, J. Approx. Theory, 68 (1992), 74–82.
L. Szili, On the summability of trigonometric interpolation process, Acta Math. Hungar., 91 (2001), 131–158.
L. Szili, On the summability of weighted Lagrange interpolation on the roots of Jacobi polynomials, Acta Math. Hungar., 99 (2003), 209–231.
L. Szili and P. Vértesi, On summability of weighted Lagrange interpolation. I (General weights), Acta Math. Hungar., 101 (2003), 323–344.
L. Szili and P. Vértesi, On summability of weighted Lagrange interpolation. II (Freud-type weights), Acta Math. Hungar., 103 (2004), 1–17.
L. Szili and P. Vértesi, On summability of weighted Lagrange interpolation. III (Jacobi weights), Acta Math. Hungar., 104 (2004), 39–62.
R. Askey and S. Wainger, A convolution structure for Jacobi series, Amer. J. Math., 91 (1969), 463–485.
V. S. Pawelke, Ein Satz vom Jacksonschen Typ für algebraische Polynome, Acta Sci. Math. (Szeged), 33 (1972), 323–336.
G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. (1976).
J. Szabados and P. Vértesi, Interpolation of Functions, World Scientific Publishing Co. Pte. Ltd. (1990).
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the national NSF (No: 10871226) of P.R. China.
Rights and permissions
About this article
Cite this article
Sheng, B. Estimates of the norm of the Mercer kernel matrices with discrete orthogonal transforms. Acta Math Hung 122, 339–355 (2009). https://doi.org/10.1007/s10474-008-8037-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-008-8037-2