Recovery Of Riesz Transform Of Functions In Sobolev Space By Wavelet Multilevel Sampling

Wavelet transform is a local transformation in the time domain and fre-quency domain.The multiscale refinement of functions through scaling and translation can simultaneously provide time and frequency information.In re-cent years,wavelet transforms have been used in mathematical fields such as numerical analysis and function approximation,as well as filtering and images.Signal processing,such as recognition and image compression,has been wide-ly used.The Riesz transformation is a singular integral,which is the gener-alization of Hilbert transform in high dimensionand.how to use time domain sampling to recover Hilbert transforms and Riesz transforms are problems that have theoretical and practical application value.At present,researches on the sampling recovery of Hilbert transforms have achieved a lot of results.Howev-er,researches on recovering Riesz transforms based on time domain sampling are rare.By using the box-spline and the wavelet multilevel method,a scheme is established to recover the Riesz transform of the functions in Sobolev space Hs(R2)with s>1.The Riesz transforms of some functions are continuous but it has numerical singularity at some points,then it is necessary to eliminate the numerical singularity.Its main contents are as follows:Firstly,since the box spline has an explicit expression,it has been widely used in numerical analysis.In addition,the second-order cardinal box spline B2 are refinement function.This paper will give an explicit expression of the Riesz transformation of B2,and based on the approximation formula of box spline,a scheme is established to recover the Riesz transform of the functions in Sobolev space Hs(R2)with s>1,and establish the recovery error estimate.Secondly,the Riesz transforms of some functions are continuous,but RB2 has numerical singularities at some points,eliminating numerical singularities is especially important.We first establish the shift-perturbation error estimate of the multilevel sampling approximation.By the perturbed approximation sys-tem,we give a method to eliminate the numerical singularity.In the last,numerical simulations are conducted to test the recovery effi-ciency on different functions.