In this study, we explore biorthogonal nonuniform multiresolution analysis within Sobolev spaces, utilizing the framework of spectral pairs. For each \( j \in \mathbb {Z} \) , we consider the closed linear spans \( V_j \) and \( \tilde{V}_j \) , which are generated by the sequences \( \phi ^{(j)}_{j,\gamma } \) and \( \tilde{\phi }^{(j)}_{j,\gamma } \) , respectively. The index \( \gamma \) is taken from the spectrum \( \Lambda _{r,N} = \left\{ 0, \frac{r}{N} \right\} + 2\mathbb {Z}, \) where \( N \) is a positive integer and \( r \) is an odd integer satisfying \( 1 \le r \le 2N - 1 \) , with the constraint that \( r \) and \( N \) are coprime. We establish the pre-Riesz condition for scaling functions within Sobolev spaces and provide the required and sufficient criteria for the translations of a single scaling function to construct Riesz bases. Furthermore, we demonstrate that, under mild constraints, the associated wavelets can form Riesz bases. Lastly, we examine the frame condition for nonuniform wavelets in Sobolev spaces, ensuring a rigorous mathematical foundation for their application.
