Jackson-type inequalities for spherical neural networks with doubling weights

Recently, the spherical data processing has emerged in many applications and attracted a lot of attention. Among all the methods for dealing with the spherical data, the spherical neural networks (SNNs) method has been recognized as a very efficient tool due to SNNs possess both good approximation capability and spacial localization property. For better localized approximant, weighted approximation should be considered since different areas of the sphere may play different roles in the approximation process. In this paper, using the minimal Riesz energy points and the spherical cap average operator, we first construct a class of well-localized SNNs with a bounded sigmoidal activation function, and then study their approximation capabilities. More specifically, we establish a Jackson-type error estimate for the weighted SNNs approximation in the metric of LpLp space for the well developed doubling weights.