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Wavelet transforms on Gelfand-Shilov spaces and concrete examples.
Naohiro Fukuda1, Tamotu Kinoshita2, Kazuhisa Yoshino2
1Matsue College, National Institute of Technology, Matsue, Shimane 690-8518 Japan.
Journal of Inequalities and Applications
|June 10, 2017
Summary
This study examines wavelet transform continuity in Gelfand-Shilov spaces using vanishing moments. Researchers computed Fourier and wavelet transforms for specific functions within these spaces.
Area of Science:
- Mathematical Analysis
- Harmonic Analysis
- Functional Analysis
Background:
- Gelfand-Shilov spaces are crucial in distribution theory and signal processing.
- Wavelet transforms offer time-frequency analysis, essential for signal and image processing.
- Continuity properties are fundamental for the theoretical underpinnings of transform methods.
Purpose of the Study:
- To investigate the continuity properties of wavelet transforms within Gelfand-Shilov spaces.
- To analyze the impact of vanishing moment conditions on these properties.
- To compute and demonstrate Fourier and wavelet transforms for concrete functions in these spaces.
Main Methods:
- Application of vanishing moment conditions to wavelet transforms.
- Analysis of function spaces, specifically Gelfand-Shilov spaces.
- Computation of Fourier transforms and wavelet transforms for selected functions.
Main Results:
- Established continuity properties of wavelet transforms in Gelfand-Shilov spaces.
- Demonstrated the role of vanishing moments in characterizing transform behavior.
- Provided explicit calculations of Fourier and wavelet transforms for functions within these specialized spaces.
Conclusions:
- The study confirms the applicability and behavior of wavelet transforms in Gelfand-Shilov spaces.
- Vanishing moment conditions are key factors influencing continuity.
- The computed examples validate the theoretical findings and offer practical insights.
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