Macdonald formula for spherical functions on affine buildings

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.

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Spherical functions for Gelfand pairs associated with compact groupoids

Afrika Matematika ◽

10.1007/s13370-015-0359-y ◽

2015 ◽

Vol 27 (3-4) ◽

pp. 573-582

Author(s):

Ibrahima Toure ◽

Kinvi Kangni

Keyword(s):

Spherical Functions ◽

Gelfand Pairs

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Spherical functions and local densities on the space of p-adic quaternion Hermitian forms

International Journal of Number Theory ◽

10.1142/s1793042122500324 ◽

2021 ◽

pp. 1-38

Author(s):

Yumiko Hironaka

Keyword(s):

Laurent Polynomial ◽

Spherical Functions ◽

Plancherel Formula ◽

Explicit Formulas ◽

Hermitian Forms ◽

Spherical Fourier Transform ◽

Local Densities ◽

Laurent Polynomial Ring ◽

Injective Map ◽

Residual Characteristic

We introduce the space [Formula: see text] of quaternion Hermitian forms of size [Formula: see text] on a [Formula: see text]-adic field with odd residual characteristic, and define typical spherical functions [Formula: see text] on [Formula: see text] and give their induction formula on sizes by using local densities of quaternion Hermitian forms. Then, we give functional equation of spherical functions with respect to [Formula: see text], and define a spherical Fourier transform on the Schwartz space [Formula: see text] which is Hecke algebra [Formula: see text]-injective map into the symmetric Laurent polynomial ring of size [Formula: see text]. Then, we determine the explicit formulas of [Formula: see text] by a method of the author’s former result. In the last section, we give precise generators of [Formula: see text] and determine all the spherical functions for [Formula: see text], and give the Plancherel formula for [Formula: see text].

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