论文信息 - On the Derivatives of Radial Positive Definite Functions

On the Derivatives of Radial Positive Definite Functions

Abstract A radial positive definite function ϕ(‖ · ‖) on R d which is 2k times differentiable at zero has 2k + [(d − 1)/2] continuous derivatives away from zero, and the estimate is sharp. Furthermore, ϕ(iv)(0) ≥ 3d|ϕ″(0)|2/(d + 2) if the derivatives exist; this proves a conjecture of Paul Switzer and has implications on the smoothness of isotropic random fields and their local averages. The techniques used have evolved in geostatistics, and side results refer to Matheron's turning bands operator and the montee.

Tilmann Gneiting | T. Gneiting

[1] The smoothness of Hankel transforms , 1969 .

[2] C. Micchelli. Interpolation of scattered data: Distance matrices and conditionally positive definite functions , 1986 .

[3] S. Cohn,et al. Ooce Note Series on Global Modeling and Data Assimilation Construction of Correlation Functions in Two and Three Dimensions and Convolution Covariance Functions , 2022 .

[4] G. Matheron. The intrinsic random functions and their applications , 1973, Advances in Applied Probability.

[5] Paul Switzer. Geometrical measures of the smoothness of random functions , 1976 .

[6] A. Yaglom. Correlation Theory of Stationary and Related Random Functions I: Basic Results , 1987 .

[7] I. J. Schoenberg. Metric spaces and completely monotone functions , 1938 .

[8] Holger Wendland,et al. On the smoothness of positive definite and radial functions , 1999 .

[9] S. Kotz,et al. Symmetric Multivariate and Related Distributions , 1989 .

[10] W. Trebels. SOME NECESSARY CONDITIONS FOR RADIAL FOURIER MULTIPLIERS , 1976 .

[11] B. Golubov. On Abel—Poisson type and Riesz means , 1981 .

[12] Holger Wendland,et al. Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree , 1995, Adv. Comput. Math..

[13] Irene A. Stegun,et al. Handbook of Mathematical Functions. , 1966 .