# Construction of An-isotropic Covariance-Functions Using Sums of Riesz-Representers

## Abstract

We regard a reproducing kernel Hilbert space (RKHS) of functions harmonic in the set outside a sphere with radius R0, having a reproducing kernel *K* _{0} *(P,Q)*. (*P, Q,* and later *P* _{ n } being points in the set of harmonicity). The degree-variances of this kernel will be denoted σo_{n}.

The set of Riesz-representers associated with the evaluation functionals (or gravity functionals) related to distinct points *P* _{n} *,n* = 1,..., *N*, on a 2D-surface surrounding the bounding sphere will be linear independent. These functions are used to define a new N-dimensional RKHS with. kernel {fy1|90-1}

If the points all are located on a concentric sphere with radius *R* ^{ 1 } > Po, and form an є-net covering the sphere, and *a* _{ n } are suitable area-elements (depending on *N*) then this kernel will converge towards an isotropic kernel with degree-variances σ_{n} \(
\mathop \sigma \nolimits_n^2 = (2n + 1)*\sigma _{0n}^2 *\left( {\frac{{R_0 }}
{{R_1 }}} \right)\left( {2n + 2} \right)*(cons\tan t{\text{)}}
\)

Consequently, if we want *K* _{ N }(P,Q) to represent an isotropic covariance function of the Earth’s gravity potential, *COV*(P,Q), we can select σo_{n} so that σ_{n} becomes equal to the empirical degree-variances.

If the points are chosen at varying radial distances *R* _{ n } > *R* _{ o } then we have constructed an anisotropic kernel, or equivalent covariance function representation.

If the points are located in a bounded region, the kernel may be used to modify the original kernel, *CON* _{N} *(P,Q)*=*CON(P,Q)*+*K* _{N} *(P,Q)*

Values of an-isotropic covariance functions constructed based on these ideas have been calculated, and some first ideas are presented on how to select the points *P* _{ n }.