Linear algorithms of affine synthesis in the Lebesgue space $L^1[0,1]$

We prove that there are no linear algorithms of affine synthesis for affine systems in the Lebesgue space $L^1[0,1]$ with respect to the model space $\ell^1$, although the corresponding affine synthesis problem has a positive solution under the most general assumptions. At the same time, by imposing additional conditions on the generating function of the affine system, we can give an explicit linear algorithm of affine synthesis in the Lebesgue space when the model space is that of the coefficients of the system. This linear algorithm generalizes the Fourier–Haar expansion into orthogonal series.