Truncation error estimates for two-dimensional sampling series associated with the linear canonical transform

The linear canonical transform (LCT) provides a more general framework for many well-known linear integral transforms, such as Fourier transform, fractional Fourier transform, Fresnel transform, and so forth, in digital signal processing and optics. There has been a great deal of research on sampling expansions of bandlimited signals in the LCT domain. However, results on error estimation and convergence analysis appear to be relatively rare, especially for multi-dimensional sampling series associated with the LCT. In this paper, we present error estimation and convergence analysis for two-dimensional (2D) sampling series in the LCT domain. Specifically, we first prove the absolute convergence and uniform convergence of 2D LCT sampling series. Then, we analyze truncation error estimates for uniformly sampling 2D bandlimited signals in the LCT domain, as well as deriving three truncation error bounds. Finally, our theoretical results are demonstrated by numerical examples.

Automated summary

This paper presents error estimation and convergence analysis for two-dimensional (2D) sampling series in the LCT domain. The absolute convergence and uniform convergence of 2D LCT sampling series are first proven, followed by an analysis of truncation error estimates for uniformly sampling 2D bandlimited signals in the LCT domain and derivation of three truncation error bounds. The theoretical results are then demonstrated through numerical examples.