Fourier-Bessel transforms and generalized uniform Lipschitz classes

Abstract

Let $\nu >-1/2$, $\mathrm{d}{\mu }_{\nu }$ is defined on ${\mathbb{R}}_{+}=[0,+\mathrm{\infty })$ by $\mathrm{d}{\mu }_{\nu }(x)=[2^{\nu }\mathrm{\Gamma }(\nu +1){]}^{-1}{x}^{2\nu +1}\mathrm{d}x$. For f integrable on ${\mathbb{R}}_{+}$ with respect to $\mathrm{d}{\mu }_{\nu }(x)$ together with its Fourier-Bessel transform of order ν we give necessary and sufficient conditions to belong to the generalized Lipschitz classes $H_{\nu }^{\omega ,m}$ and $h_{\nu }^{\omega ,m}$. Also a condition for generalized Bessel differentiability of a function is proved.

Keywords:

• Fourier-Bessel transform
• generalized Lipschitz spaces
• generalized Bessel differentiability

Mathematics Subject Classification (2010):

• 44A15
• 42B35

Acknowledgments

The author would like to thank the referee for his/her valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work of author was supported by the Ministry of Science and Higher Education of the Russian Federation in the framework of the state assignment (project no. FSRR-2020-006).