A Characterization of (Weak) Nonhomogeneous Wavelet Dual Frames and Mixed Oblique Principle in Sobolev Spaces on the Half Real Line

Abstract

Due to ${\mathbb{R}}_{+}=[0, \infty )$ being a group under Walsh addition ⊕, the wavelet frame theory on $L^2({\mathbb{R}}_{+})$ has been well developed in recent years. This paper addresses wavelet dual frames on Sobolev spaces defined on ${\mathbb{R}}_{+}.$ We introduce the Sobolev space ${\mathcal{H}}^s({\mathbb{R}}_{+})$ with $s\in \mathbb{R},$ characterize (weak) nonhomogeneous wavelet dual frames for $({\mathcal{H}}^s({\mathbb{R}}_{+}), {\mathcal{H}}^{-s}({\mathbb{R}}_{+})),$ and establish mixed oblique extension principles for constructing them.

Keywords:

• Nonhomogeneous wavelet system
• Walsh–Fourier transform
• weak dual frame

2020 MS Classification:

• 42C40
• 42C15

Acknowledgments

The authors would like to thank the reviewers for their comments.

Additional information

Funding

The first author was supported by Natural Science Foundation of Ningxia (Grant No. 2022AAC05037); National Natural Science Foundation of China (Grant No. 12061002); Ningxia Excellent Talents Support Program (Grant No. TJGC2019014); Construction Project of First-Class Disciplines in Ningxia Higher Education (NXYLXK2017B09). The second author was supported by National Natural Science Foundation of China (Grant No. 11971043).