Some remarks on the theory of spherical functions on symmetric Riemannian manifolds

We study homeomorphisms of Riemannian manifolds with unbounded characteristic such that the inverse mappings satisfy the Poletsky-type inequality. It is established that their families are equicontinuous if the function Q which is related to the Poletsky inequality and is responsible for a distortion of the modulus, is integrable in the given domain, here the original manifold is connected and the domain of definition and the range of values of mappings have compact closures.

Download Full-text

Correction to the article Relative spherical functions on℘-adic symmetric spaces (three cases)

Pacific Journal of Mathematics ◽

10.2140/pjm.2008.236.195 ◽

2008 ◽

Vol 236 (1) ◽

pp. 195-200 ◽

Cited By ~ 1

Author(s):

Omer Offen

Keyword(s):

Symmetric Spaces ◽

Spherical Functions

Download Full-text

Spheric radial basis functions in Mahalanobis measuring for displaying local field features

Geodesy and Cartography ◽

10.22389/0016-7126-2019-952-10-2-9 ◽

2019 ◽

Vol 952 (10) ◽

pp. 2-9

Author(s):

Yu.M. Neiman ◽

L.S. Sugaipova ◽

V.V. Popadyev

Keyword(s):

Gravitational Field ◽

Quadratic Form ◽

Local Field ◽

Euclidean Distance ◽

Basis Functions ◽

Spherical Functions ◽

Local Models ◽

Stationary Field ◽

Modeling Process ◽

The Earth

As we know the spherical functions are traditionally used in geodesy for modeling the gravitational field of the Earth. But the gravitational field is not stationary either in space or in time (but the latter is beyond the scope of this article) and can change quite strongly in various directions. By its nature, the spherical functions do not fully display the local features of the field. With this in mind it is advisable to use spatially localized basis functions. So it is convenient to divide the region under consideration into segments with a nearly stationary field. The complexity of the field in each segment can be characterized by means of an anisotropic matrix resulting from the covariance analysis of the field. If we approach the modeling in this way there can arise a problem of poor coherence of local models on segments’ borders. To solve the above mentioned problem it is proposed in this article to use new basis functions with Mahalanobis metric instead of the usual Euclidean distance. The Mahalanobis metric and the quadratic form generalizing this metric enables us to take into account the structure of the field when determining the distance between the points and to make the modeling process continuous.

Download Full-text