The Study Of Approximation Method For Scattered Data

This thesis aims at putting forward new models and meth-ods for spherical scattered data approximation, which mainly focus on theinterpolation and quasi-interpolation tools on the sphere.Based on the fact that spherical basis functions and spherical polyno-mial are both effective tools in dealing with scattered data approximationon the sphere, in Chapter2, we consider a hybrid interpolation model com-bining spherical polynomial with SBFs and solve the model by using themethod of moving least squares. For this hybrid interpolation model, first-ly, wegivea Bernsteininequalityin aBessel-potentialSobolevspace. Sec-ondly, a direct theorem for approximation error is established. Finally, wefocus on the stability for the hybrid interpolation scheme.Chapter3discusses the multivariate Bernstein polynomial on simplexwhere we extend the regular nodes to the case of scattered data. Particular-ly, the Bernstein polynomial acts as an sampling operator. The discussionisconductedintermsoftheregularcaseaswellastherandomcase. Ononehand, we establish the Chebyshev-type error estimates, on the other hand,by combining star discrepancy theory with uniform distribution property,the approximation order for the operator is obtained as well as the conver-gence in LPnorm.Chapter4constructs a quasi-interpolation operator in terms of a classof scattered centers by using multivariate spline function. Then we focuson the polynomial reproducing property with the help of commutator. Theassociated approximation order based on the mesh norm is obtained in thischapter.Chapter5proposes a radial quasi-interpolation operator in terms ofscattereddataonthespherebyusingsphericalgeneralFix-strangcondition and SBFs. The spherical quadrature forluma is discussed by which wecan derive the approximation capacity of the quasi-interpolation operatordealing with different function.