Strictly positive definite functions on spheres in Euclidean spaces
Authors:
Amos Ron and Xingping Sun
Journal:
Math. Comp. 65 (1996), 15131530
MSC (1991):
Primary 42A82, 41A05; Secondary 33C55, 33C90
MathSciNet review:
1370856
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper we study strictly positive definite functions on the unit sphere of the dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are actually strictly positive definite. The study of this problem was initiated recently by Xu and Cheney (Proc. Amer. Math. Soc. 116 (1992), 977981), where certain sufficient conditions were derived. A new approach, which is based on a critical connection between this problem and that of multivariate polynomial interpolation on spheres, is presented here. The relevant interpolation problem is subsequently analyzed by three different complementary methods. The first is based on the de BoorRon general ``least solution for the multivariate polynomial interpolation problem''. The second, which is suitable only for , is based on the connection between bivariate harmonic polynomials and univariate analytic polynomials, and reduces the problem to the structure of the integer zeros of bounded univariate exponentials. Finally, the last method invokes the realization of harmonic polynomials as the polynomial kernel of the Laplacian, thereby exploiting some basic relations between homogeneous ideals and their polynomial kernels.
 [1]
Richard
Askey and James
Fitch, Integral representations for Jacobi polynomials and some
applications., J. Math. Anal. Appl. 26 (1969),
411–437. MR 0237847
(38 #6128)
 [2]
Carl
de Boor and Amos
Ron, On multivariate polynomial interpolation, Constr. Approx.
6 (1990), no. 3, 287–302. MR 1054756
(91c:41005), http://dx.doi.org/10.1007/BF01890412
 [3]
Carl
de Boor and Amos
Ron, Computational aspects of polynomial
interpolation in several variables, Math.
Comp. 58 (1992), no. 198, 705–727. MR 1122061
(92i:65022), http://dx.doi.org/10.1090/S00255718199211220610
 [4]
Carl
de Boor and Amos
Ron, The least solution for the polynomial interpolation
problem, Math. Z. 210 (1992), no. 3,
347–378. MR 1171179
(93f:41002), http://dx.doi.org/10.1007/BF02571803
 [5]
E.
W. Cheney and Yuan
Xu, A set of research problems in approximation theory, Topics
in polynomials of one and several variables and their applications, World
Sci. Publ., River Edge, NJ, 1993, pp. 109–123. MR 1276955
(95c:41001)
 [6]
N.
Dyn, Interpolation and approximation by radial and related
functions, Approximation theory VI, Vol. I (College Station, TX, 1989)
Academic Press, Boston, MA, 1989, pp. 211–234. MR 1090994
(92d:41002)
 [7]
W.
A. Light and E.
W. Cheney, Interpolation by periodic radial basis functions,
J. Math. Anal. Appl. 168 (1992), no. 1,
111–130. MR 1169852
(93f:41039), http://dx.doi.org/10.1016/0022247X(92)90193H
 [8]
Charles
A. Micchelli, Interpolation of scattered data: distance matrices
and conditionally positive definite functions, Constr. Approx.
2 (1986), no. 1, 11–22. MR 891767
(88d:65016), http://dx.doi.org/10.1007/BF01893414
 [9]
V. A. Menegatto, Strictly positive definite functions on spheres, University of Texas at Austin, 1992.
 [10]
, Strictly positive definite kernels on circle, Rocky Mountain J. Math. 25 (1995), 11491163. CMP 96:03
 [11]
, Strictly positive definite kernels on the Hilbert sphere, Appl. Anal. 55 (1994), 81101.
 [12]
Francis
J. Narcowich, Generalized Hermite interpolation and positive
definite kernels on a Riemannian manifold, J. Math. Anal. Appl.
190 (1995), no. 1, 165–193. MR 1314111
(96c:41009), http://dx.doi.org/10.1006/jmaa.1995.1069
 [13]
Francis
J. Narcowich and Joseph
D. Ward, Norms of inverses and condition numbers for matrices
associated with scattered data, J. Approx. Theory 64
(1991), no. 1, 69–94. MR 1086096
(92b:65017), http://dx.doi.org/10.1016/00219045(91)90087Q
 [14]
Francis
J. Narcowich and Joseph
D. Ward, Norm estimates for the inverses of a general class of
scattereddata radialfunction interpolation matrices, J. Approx.
Theory 69 (1992), no. 1, 84–109. MR 1154224
(93c:41005), http://dx.doi.org/10.1016/00219045(92)90050X
 [15]
E.
Quak, N.
Sivakumar, and J.
D. Ward, Least squares approximation by radial functions, SIAM
J. Math. Anal. 24 (1993), no. 4, 1043–1066. MR 1226863
(94g:41059), http://dx.doi.org/10.1137/0524062
 [16]
A. Ron and X. Sun, Strictly positive definite functions on spheres, CMS TR 946, University of Wisconsin  Madison, February 1994.
 [17]
I. J. Schoenberg, Metric spaces and completely monotone functions, Ann. Math. 39 (1938), 811841.
 [18]
I.
J. Schoenberg, Positive definite functions on spheres, Duke
Math. J. 9 (1942), 96–108. MR 0005922
(3,232c)
 [19]
Elias
M. Stein and Guido
Weiss, Introduction to Fourier analysis on Euclidean spaces,
Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical
Series, No. 32. MR 0304972
(46 #4102)
 [20]
N.
Sivakumar and J.
D. Ward, On the least squares fit by radial functions to
multidimensional scattered data, Numer. Math. 65
(1993), no. 2, 219–243. MR 1222620
(94d:41006), http://dx.doi.org/10.1007/BF01385749
 [21]
Xingping
Sun, The fundamentality of translates of a continuous function on
spheres, Numer. Algorithms 8 (1994), no. 1,
131–134. MR 1299079
(96a:41005), http://dx.doi.org/10.1007/BF02145700
 [22]
Gabor
Szegö, Orthogonal polynomials, American Mathematical
Society Colloquium Publications, Vol. 23. Revised ed, American Mathematical
Society, Providence, R.I., 1959. MR 0106295
(21 #5029)
 [23]
Yuan
Xu and E.
W. Cheney, Strictly positive definite functions
on spheres, Proc. Amer. Math. Soc.
116 (1992), no. 4,
977–981. MR 1096214
(93b:43005), http://dx.doi.org/10.1090/S00029939199210962146
 [1]
 R. Askey and J. Fritch, Integral representations for Jacobi polynomials, and some applications, J. Math. Anal. Appl. 26 (1969), 411437. MR 38:6128
 [2]
 C. de Boor and A. Ron, On multivariate polynomial interpolation, Constructive Approximation 6 (1990), 287302. MR 91c:41005
 [3]
 , Computational aspects of polynomial interpolation in several variables, Mathematics of Computation 58 (1992), 705727. MR 92i:65022
 [4]
 , The least solution of the multivariate polynomial interpolation, Math. Z. 210 (1992), 347378. MR 93f:41002
 [5]
 E. W. Cheney and Yuan Xu, A set of research problems in approximation theory, Topics in Polynomials of One and Several Variables and Their Applications (Th. M. Rassias, H. M. Srivastava, and A. Yanushauskas, eds., eds.), World Scientific, River Edge, NJ, 1993. MR 95c:41001
 [6]
 N. Dyn, Interpolation and approximation by radial and related functions, Approximation VI (C. K. Chui, L. L. Schumaker, and J. D. Ward, eds.), vol. I, 1989, Academic Press, pp. 211234. MR 92d:41002
 [7]
 W. A. Light and E. W. Cheney, Interpolation by periodic radial basis functions, J. Math. Anal. Appl. 168 (1992), 111130. MR 93f:41039
 [8]
 C. A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx. 2 (1986), 1122. MR 88d:65016
 [9]
 V. A. Menegatto, Strictly positive definite functions on spheres, University of Texas at Austin, 1992.
 [10]
 , Strictly positive definite kernels on circle, Rocky Mountain J. Math. 25 (1995), 11491163. CMP 96:03
 [11]
 , Strictly positive definite kernels on the Hilbert sphere, Appl. Anal. 55 (1994), 81101.
 [12]
 F. J. Narcowich, Generalized Hermite interpolation and positive definite kernels on a Riemannian manifold, J. Math. Anal. Appl. 190 (1995), 165193. MR 96c:41009
 [13]
 F. J. Narcowich and J. D. Ward, Norms of inverses and condition numbers of matrices associated with scattered data, J. Approx. Theory 64 (1991), 6994. MR 92b:65017
 [14]
 , Norm estimates for the inverses of a general class of scattered data radial function interpolation matrices, J. Approx. Theory 69 (1992), 84109. MR 93c:41005
 [15]
 E. Quak, N. Sivakumar, and J. D. Ward, Least squares approximation by radial functions, SIAM J. Math. Anal. 24 (1993), 10431066. MR 94g:41059
 [16]
 A. Ron and X. Sun, Strictly positive definite functions on spheres, CMS TR 946, University of Wisconsin  Madison, February 1994.
 [17]
 I. J. Schoenberg, Metric spaces and completely monotone functions, Ann. Math. 39 (1938), 811841.
 [18]
 , Positive definite functions on spheres, Duke Math. J. 9 (1942), 96108. MR 3:232c
 [19]
 E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, 1971. MR 46:4102
 [20]
 N. Sivakumar and J. D. Ward, On the least squares fit by radial functions to multidimensional scattered data, Numer. Math. 65 (1993), 219243. MR 94d:41006
 [21]
 X. Sun, The fundamentality of translates of a continuous function on spheres, Numerical Algorithms 8 (1994), 131134. MR 96a:41005
 [22]
 Gabor Szegö, Orthogonal Polynomials, Amer. Math. Colloq. Publ., Amer. Math. Soc., Providence, RI, 1959. MR 21:5029
 [23]
 Yuan Xu and E. W. Cheney, Strictly positive definite functions on spheres, Proc. Amer. Math. Soc. 116 (1992), 977981. MR 93b:43005
Similar Articles
Retrieve articles in Mathematics of Computation of the American Mathematical Society
with MSC (1991):
42A82,
41A05,
33C55,
33C90
Retrieve articles in all journals
with MSC (1991):
42A82,
41A05,
33C55,
33C90
Additional Information
Amos Ron
Affiliation:
Department of Computer Science, University of WisconsinMadison, Madison, Wisconsin 53706
Xingping Sun
Affiliation:
Department of Mathematics, Southwest Missouri State University, Springfield, Missouri 65804
DOI:
http://dx.doi.org/10.1090/S0025571896007806
PII:
S 00255718(96)007806
Received by editor(s):
February 7, 1994
Received by editor(s) in revised form:
February 22, 1995, and July 5, 1995
Article copyright:
© Copyright 1996 American Mathematical Society
