Contraction of functions in modulation spaces

Abstract

Let F be a contraction on $\mathbb{C}$ and $F\left(f\right)$ be the composition of functions F and f. Some sufficient conditions are obtained for nonlinearity $fF\left(f\right)$ and $|f|$ to be in $M^{1,1}\left(\mathbb{R}\right)$ whenever $f\in M^{1,1}\left(\mathbb{R}\right)$.

Keywords:

• Bargmann transform
• short-time Fourier transform
• nonlinear operation
• modulation spaces

2010 Mathematics Subject Classification:

• Primary: 44A15, 42B35
• Secondary: 42B37

Acknowledgements

The author wishes to thank Prof. P. K. Ratnakumar for suggesting to look at this problem, and the anonymous referees for valuable comments which helped to improve the manuscript. He also wishes to thank the Harish-Chandra Research Institute, the Dept. of Atomic Energy, Govt. of India, for providing excellent research facility.

Disclosure Statement

No potential conflict of interest was reported by the author.