Convergence of Biorthogonal Series in the System of Contractions and Translations of Functions in the Spaces $L^p[0,1]$

We obtain conditions for the convergence in the spaces $L^p[0,1]$, $1\le p<\infty$, of biorthogonal series of the form
$$f=\sum_{n=0}^\infty(f,\psi_n)\varphi_n$$
in the system $\{\varphi_n\}_{n\ge 0}$ of contractions and translations of a function $\varphi$. The proposed conditions are stated with regard to the fact that the functions belong to the space $\mathfrak L^p$ of absolutely bundle-convergent Fourier–Haar series with norm
$$\|f\|_p^\ast=|(f,\chi_0)| +\sum_{k=0}^\infty 2^{k(1/2-1/p)} \biggl(\mspace{2mu}\sum_{n=2^k}^{2^{k+1}-1} |(f,\chi_n)|^p\biggr)^{1/p},$$
where $(f,\chi_n)$, $n=0,1,\dots$, are the Fourier coefficients of a function $f\in L^p[0,1]$ in the Haar system $\{\chi_n\}_{n\ge 0}$. In particular, we present conditions for the system $\{\varphi_n\}_{n\ge 0}$ of contractions and translations of a function $\varphi$ to be a basis for the spaces $L^p[0,1]$ and $\mathfrak L^p$.