Are there any Landau poles in wavelet-based quantum field theory?

Following previous work by one of the authors [M.V.Altaisky, Unifying renormalization group and the continuous wavelet transform, Phys. Rev. D 93, 105043 (2016).], we develop a new approach to the renormalization group, where the effective action functional ${\mathrm{\Gamma }}_A[\phi ]$ is a sum of all fluctuations of scales from the size of the system $L$ down to the scale of observation $A$. It is shown that the renormalization flow equation of the type $\frac{\mathrm{\partial }{\mathrm{\Gamma }}_A}{\mathrm{\partial }\ln A}=-Y(A)$ is a limiting case of such consideration, when the running coupling constant is assumed to be a differentiable function of scale. In this approximation, the running coupling constant, calculated at one-loop level, suffers from the Landau pole. In general case, when the scale-dependent coupling constant is a non-differentiable function of scale, the Feynman loop expansion results in a difference equation. This keeps the coupling constant finite for any finite value of scale $A$. As an example we consider Euclidean ${\phi }^4$ field theory.