Boas and Titchmarsh Type Theorems for Generalized Lipschitz Classes and $q$-Bessel Fourier Transform

Necessary and sufficient conditions for a function $f$ to belong to the generalized Lipschitz classes $H^{m,\omega}_{q,\nu}$ and $h^{m,\omega}_{q,\nu}$ for fractional $m$ are given in terms of its $q$-Bessel–Fourier transform $\mathcal F_{q,\nu}(f)$. Dual results are considered as well. An analog of the Titchmarsh theorem for fractional-order differences is proved.