Reconstruction of analytic signal in Sobolev space by framelet sampling approximation

Abstract

Based on dual framelets, we construct the sampling approximation for the whole Sobolev space where . In particular, the sampling system has adjustable shift parameters. By the B-spline sampling system, we construct the approximation of Hilbert transform of any function of . Combining the approximation of the function and that of its Hilbert transform, we establish a reconstruction method for the analytic signal. Particularly, the reconstruction series converges exponentially with respect to the scale level. Moreover, the numerical singularity emerging in computation of Hilbert transform can be removed by adjusting the shift parameters. That is, the method of reconstruction of analytic signal is numerically and -stable. Several numerical experiments are carried out to check the efficiency of our reconstruction method.

Keywords:

• Analytic signal
• dual framelets
• Sobolev space
• adjustable sampling system
• approximation order
• numerical singularity

AMS Subject Classifications:

• 42C40
• 65T60
• 94A20

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is supported by National Natural Science Foundation of China [grant number 61561006], [grant number 11501132] and Guangxi Natural Science Foundation [grant number 2016GXNSFAA380049]. The second author is supported by University Research Matching Grant FDCT079.