Fractional multiresolution analysis and associated scaling functions in $$L^{2}({\mathbb {R}})$$

In this paper, we show how to construct an orthonormal basis from Riesz by assuming that the fractional translates of a single function in core subspace multiresolution analysis form instead basis. definition analysis, intersection triviality condition follows other conditions. Furthermore, union density also under assumption Fourier transform scaling is continuous at 0. At culmination, provide complete characterization functions associated with analysis.