Recovery of Riesz transform of functions in Sobolev space by wavelet multilevel sampling. (3rd May 2018) - Li, Youfa; Yang, Honglei; Shang, Jing; Yang, Shouzhi
Abstract : The explicit formula of the Riesz transform of the box splineB 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 fromB 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 fromB 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.