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

在本文中，我们通过假设分数多分辨率分析的核心子空间中单个函数的分数平移形成了Riesz基础而不是正交基础，说明了如何根据Riesz基础构建正交基础。在分数多分辨率分析的定义中，我们表明相交平凡性条件遵循其他条件。此外，我们表明，在缩放函数的分数阶傅立叶变换在0处连续的假设下，也遵循联合密度条件。在顶点时，我们提供了与分数多分辨率分析相关的缩放函数的完整表征。