Convergence of biorthogonal series in the system of contractions and translations of functions in the spaces Lp[0,1]

We obtain conditions for the convergence in the spaces L(p)[0, 1], 1 <= p < infinity, of biorthogonal series of the form [GRAPHICS] in the system {phi(n)} n >= 0 of contractions and translations of a function.. The proposed conditions are stated with regard to the fact that the functions belong to the space (p) pound of absolutely bundleconvergent Fourier - Haar series with norm [GRAPHICS] where (f, X(n)), n = 0, 1,..., are the Fourier coefficients of a function f is an element of L(p)[0, 1] in the Haar system {X(n)}n >= 0. In particular, we present conditions for the system {phi(n)}n >= 0 of contractions and translations of a function phi to be a basis for the spaces L(p)[0, 1] and (p) pound .