Sampling approximation by framelets in Sobolev space and its application in modifying interpolating error

Motivated by [B. Han, Z. Shen, Dual wavelet frames and Riesz bases in Sobolev spaces, Constr. Approx. 29 (2009) 369–406], we establish a sampling theorem in Hs(Rd)Hs(Rd), s>1/2,d≥1s>1/2,d≥1, using a special pair of dual frames. The converging rate of the corresponding sampling series is investigated, and then sampling approximation to a signal in Sobolev space is established. If a signal to be reconstructed satisfies a mild condition in Fourier transform, then its sampling series converges exponentially fast. As an application, the sampling approximation is used to modify interpolating error, which arises when using an interpolating refinable function to reconstruct f∈Hs(Rd)f∈Hs(Rd), such that ff can be arbitrarily approximated by its samples.