Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 014, 13 pages      arXiv:1410.4125      https://doi.org/10.3842/SIGMA.2015.014

Generalized Convolution Roots of Positive Definite Kernels on Complex Spheres

Victor S. Barbosa and Valdir A. Menegatto
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Caixa Postal 668, 13560-970, São Carlos - SP, Brasil

Received October 16, 2014, in final form February 10, 2015; Published online February 13, 2015

Abstract
Convolution is an important tool in the construction of positive definite kernels on a manifold. This contribution provides conditions on an $L^2$-positive definite and zonal kernel on the unit sphere of $\mathbb{C}^q$ in order that the kernel can be recovered as a generalized convolution root of an equally positive definite and zonal kernel.

Key words: positive definiteness; zonal kernels; recovery formula; convolution roots; Zernike or disc polynomials.

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