Research On Methods For Spheircal Scattered-Data Interpolation And Approximation Error

Thisthesisconsistsoftwoparts. Thefirstpartstudiesthescat-tered data approximation on the sphere, where we propose some methodsfor the error estimates. The “native space barrier” problem is discussed inthe first part as well as the multiscale approximation algorithm. While thesecond part focuses on the case of spherical cap, where we study the errorestimates locally and approximation theorems for Jackson-type operatorson the spherical cap.Chapter2discusses Lp-error estimates for hybrid interpolation on thesphere by using spherical Duchon framework, which combines sphericalpolynomialstogetherwithsphericalradialbasisfunctions(SBFs)construct-ed by a strictly positive definite zonal kernel. The discussion is first carriedout in the native space associated with the kernel, and then refined errorestimates for a target function in a still smaller space are established. Inaddition, the smooth radial basis functions are embedded in a larger nativespace generated by a less smooth kernel, and the error estimates for hybridinterpolation to a target function from the larger native space are given. Ina sense, the results of this chapter show that hybrid interpolation associatedwith the smooth kernel enjoys the same order of error estimate as hybridinterpolation associated with the less smooth kernel for a target functionfrom the rough native spaceChapter3derives the convergence results for multiscale interpolationin Sobolev norm and Lpnorm, as well as in uniform (supremum) norm. Inaddition,weestablishaBernsteininequality,bywhichwederiveaninversetheorem for the multiscale interpolation and approximation. Finally, wegive numerical experiments to illustrate the theoretical results.Chapter4focuses on the multiscale moving least squares approxima- tion scheme, where the scale depends on the current evaluation points. Thescheme is constructed by using a sequence of scaled weight functions, andis a little different from the classical moving least squares approximationon the sphere. More precisely, a multiscale moving least squares (MMLS)algorithm, in which the corresponding scale is changing with the associ-ated given point set, is proposed. In addition, the convergence analysisfor the multiscale scheme and some numerical experiments to illustrate thetheoretical results are given.Chapter5discusses local uniform error estimates for spherical basisfunctions (SBFs) interpolation, where error bounds for target functions arerestricted on spherical cap. The discussion is first carried out in the nativespace associated with the smooth SBFs, which is generated by a strictlypositive definite zonal kernel. Then the smooth SBFs are embedded ina larger space that is generated by a less smooth kernel, and for the targetfunctionsoutsidetheoriginalnativespacethelocaluniformerrorestimatesare established. Finally, some numerical experiments are given to illustratethe theoretical results.Chapter6discussestheapproximationonthesphericalcapbyJackson-type operators, where an equivalent theorem is established.