Maybe this is a trivial question, but I couldn't find an answer yet. Is there some discrete wavelet transformation which works with overlapping wavelets (half on half)? In other words, is there some smooth $f:\mathbf{R}\to\mathbf{C}$ with support in $[-1,1]$ such that such that $$\{t\mapsto\sum_{i=\infty}^\infty f(2^n(x-2i)-k)\}_{n<N,k<2^n}$$ is a basis for all discrete functions of period 2, sampled at $k2^{-N}$?

Some illustration of what I have in mind:



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