Abstract
Certain spaces of functions which arise in the process of interpolation by Hankel translates of a basis function, as developed by the authors elsewhere, are defined with respect to a seminorm which is given in terms of the Hankel transform of each function. This kind of seminorm is called an indirect one. Here we discuss essentially two cases in which the seminorm can be rewritten in direct form, that is, in terms of the function itself rather than its Hankel transform. This is expected to lead to better estimates of the interpolation error.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions, with Formulas, Graphs and Mathematical Tables, 9th printing. National Bureau of Standards, Applied Mathematics Series, vol. 55 (1970)
Arteaga, C., Marrero, I.: A scheme for interpolation by Hankel translates of a basis function. J. Approx. Theory 164, 1540–1576 (2012)
Betancor, J.J., Marrero, I.: Multipliers of Hankel transformable generalized functions. Comment. Math. Univ. Carolin. 33(3), 389–401 (1992)
Betancor, J.J., Marrero, I.: The Hankel convolution and the Zemanian spaces β μ and \(\beta '_{\mu }\). Math. Nachr. 160, 277–298 (1993)
Bozzini, M., Rossini, M., Schaback, R.: Generalized Whittle-Matérn and polyharmonic kernels. Adv. Comput. Math. (2012). doi:10.1007/s10444-012-9277-9. (In press, published on line 22 August 2012)
Cholewinski, F.M.: A Hankel convolution complex inversion theory. Mem. Amer. Math. Soc. 58 (1965)
Corrada, H., Leeb, K., Klein, B., Klein, R., Iyengarc, S., Wahbad, G.: Examining the relative influence of familial, genetic, and environmental covariate information in flexible risk models. Proc. Natl. Acad. Sci. USA 106(20), 8128–8133 (2009)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Tables of integral transforms, vol. 2. McGraw-Hill, New York (1954)
Gigante, G.: Transference for hypergroups. Collect. Math. 52(2), 127–155 (2001)
Haimo, D.T.: Integral equations associated with Hankel convolutions. Trans. Amer. Math. Soc. 116, 330–375 (1965)
Hirschman, Jr, I.I.: Variation diminishing Hankel transforms. J. Anal. Math. 8, 307–336 (1960/61)
Levesley, J., Light, W.: Direct form seminorms arising in the theory of interpolation by translates of a basis function. Adv. Comput. Math. 11, 161–182 (1999)
Marrero, I., Betancor, J.J.: Hankel convolution of generalized functions. Rend. Mat. Appl. (7) 15(3), 351–380 (1995)
de Sousa Pinto, J.: A generalised Hankel convolution. SIAM J. Math. Anal. 16(6), 1335–1346 (1985)
Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton (1971)
Zemanian, A.H.: A distributional Hankel transform. SIAM J. Appl. Math. 14, 561–576 (1966)
Zemanian, A.H.: The Hankel transformation of certain distributions of rapid growth. SIAM J. Appl. Math. 14, 678–690 (1966)
Zemanian, A.H.: Generalized integral transformations. Interscience Publishers, New York (1968)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Leslie Greengard
Dedicated to Professor Fernando Pérez-González on the occasion of his sixtieth birthday.
Rights and permissions
About this article
Cite this article
Arteaga, C., Marrero, I. Direct form seminorms arising in the theory of interpolation by Hankel translates of a basis function. Adv Comput Math 40, 167–183 (2014). https://doi.org/10.1007/s10444-013-9303-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-013-9303-6
Keywords
- Basis function
- Bessel operator
- Direct form seminorm
- Generalized surface spline
- Generalized thin-plate spline
- Hankel convolution
- Indirect form seminorm
- Minimal norm interpolant