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Recovery of Riesz transform of functions in Sobolev space by wavelet multilevel sampling
Author(s) -
Li Youfa,
Yang Honglei,
Shang Jing,
Yang Shouzhi
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4886
Subject(s) - mathematics , sobolev space , singularity , mathematical analysis , wavelet , riesz potential , riesz transform , wavelet transform , artificial intelligence , computer science
The explicit formula of the Riesz transform of the box spline B 2 ( x , y ) : = B 2 ( x ) B 2 ( y ) is given, where B 2 is the cardinal B‐spline of order 2. By using the wavelet multilevel method, a sampling recovery scheme derived from B 2 is established to recover the Riesz transform of the functions in Sobolev space H s ( R 2 ) with s > 1. For any fixed level, our recovery is involved with a finite sum series. Since the Riesz transforms of some functions are continuous but R B 2 has numerical singularity at some points, it is necessary to eliminate the numerical singularity. We first establish the shift‐perturbation error estimate of the multilevel sampling approximation, derived from B 2 , to the functions in H s ( R 2 ) . By the perturbed approximation system, we give a method to eliminate the numerical singularity. Numerical simulations are conducted to test the recovery efficiency.