Application of Haar functions with cyclic translations in the search for cosmic strings

Cosmic strings are theoretically predicted linear structures of a cosmological size. The search for cosmic strings in optical and radio observations has been an important area of research in recent years. One of the most effective methods of searching for these objects is the analysis of the anisotropy of cosmic microwave background radiation that could be induced by cosmic strings. For this purpose, the authors propose a method of signal expansion in modified Haar functions with cyclic translations. However, for a system of these functions to be used correctly, one needs to prove its orthonormality and completeness, which is the purpose of this paper.