Recursive expansions with respect to a chain of subspaces

In this work, recursive expansions in Hilbert space $H=L_2[0,1]$ are considered. We discuss a related notion of frames in finite-dimensional spaces. We also suggest a constructive approach to extend an arbitrary basis to obtain a tight frame. The algorithm of extending is applied to bases of a special form, whose Gram matrix is circulant. A construction of a chain of nested subspaces $\{V^n\}_{n=1}^\infty$ is given, and in its foundation lies an example of a function that can be expressed as a linear combination of its contractions and translations. The main result of the paper is the theorem that provides the uniform convergence of recursive Fourier series with respect to the chain $\{V^n\}_{n=1}^\infty$ for continuous functions.