Constructions And Universalities Of The Reproducing Kernels On The Sphere

With the growth in the amount of data,we have entered the era of big data.The primary key for solving many scientific problems is to get more valuable information from the huge amount of data.Effective data processing method has become an urgent demand for the development of science and technology.Data processing methods based on reproducing kernels have been widely investigated and achieved great success.In recent years,environmental problems such as global warming and air pollution have received more and more attention.Enormous amount of spherical data needs to be collected and analyzed,based on which the experimental results in turn helps improve the environment.Because of the large demand of processing spherical data in practical applications,the development of the theory of reproducing kernels on the sphere and kernel based methods for processing spherical data will have theoretical significance and application prospect.This paper is mainly divided into three parts.First of all,we summarize briefly the concepts and theories of reproducing kernels and reproducing kernel spaces,including scalar-valued reproducing kernels,vector-valued reproducing kernels and reproducing kernels in Banach spaces.Secondly,we propose several approaches for constructing reproducing kernels on the sphere by making use of feature maps,spherical harmonic functions and positive definite functions.We also discuss the relationships among these methods.Finally,we review the concept of universal reproducing kernels and different characterizations of the universality of such kernels.Based on that,the universality of spherical reproducing kernels is discussed.