Nonuniform sampling and approximation in Sobolev space from perturbation of the framelet system
- PDF / 342,461 Bytes
- 22 Pages / 612 x 792 pts (letter) Page_size
- 69 Downloads / 180 Views
. ARTICLES .
https://doi.org/10.1007/s11425-018-1604-9
Nonuniform sampling and approximation in Sobolev space from perturbation of the framelet system Youfa Li1,∗ , Deguang Han2 , Shouzhi Yang3 & Ganji Huang1 1College
of Mathematics and Information Science, Guangxi University, Nanning 530004, China; of Mathematics, University of Central Florida, Orlando, FL 32816, USA; 3Department of Mathematics, University of Shantou, Shantou 535063, China
2Department
Email: [email protected], [email protected], [email protected], [email protected] Received November 1, 2018; accepted October 7, 2019
Abstract
The Sobolev space H ς (Rd ), where ς > d/2, is an important function space that has many appli-
cations in various areas of research. Attributed to the inertia of a measurement instrument, it is desirable in sampling theory to recover a function by its nonuniform sampling. In the present paper, based on dual framelet systems for the Sobolev space pair (H s (Rd ), H −s (Rd )), where d/2 < s < ς, we investigate the problem of constructing the approximations to all the functions in H ς (Rd ) by nonuniform sampling. We first establish the convergence rate of the framelet series in (H s (Rd ), H −s (Rd )), and then construct the framelet approximation operator that acts on the entire space H ς (Rd ). We examine the stability property for the framelet approximation operator with respect to the perturbations of shift parameters, and obtain an estimate bound for the perturbation error. Our result shows that under the condition d/2 < s < ς, the approximation operator is robust to shift perturbations. Motivated by Hamm (2015)’s work on nonuniform sampling and approximation in the Sobolev space, we do not require the perturbation sequence to be in ℓα (Zd ). Our results allow us to establish the approximation for every function in H ς (Rd ) by nonuniform sampling. In particular, the approximation error is robust to the jittering of the samples. Keywords
Sobolev space, framelet series, truncation error, perturbation error, nonuniform sampling and
approximation MSC(2010)
42C40, 65T60, 94A20
Citation: Li Y F, Han D G, Yang S Z, et al. Nonuniform sampling and approximation in Sobolev space from perturbation of the framelet system. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-0181604-9
1
Introduction
Sampling is a fundamental tool for the conversion between an analogue signal and its digital form (A/D). The most classical sampling theory is the Whittaker-Kotelnikov-Shannon (WKS) sampling theorem [43], which states that a bandlimited signal can be perfectly reconstructed if it is sampled at a rate greater than its Nyquist frequency. The WKS sampling theorem holds only for bandlimited signals. In order to extend the sampling theorem to non-bandlimited signals, researchers have established various sampling theorems for many other function spaces. Such examples include the sampling theory for shift-invariant subspaces [1–3, 12, 31, 32, 49, 51, 52], reproducing kernel subspaces of L2 (Rd ) [9, 20, 28, 41] and subspa
Data Loading...
