In this article, we introduce the notion of nonuniform biorthogonal wavelets on
positive half line. We frst establish the characterizations for the translates of a single
function to form the Riesz bases for their closed linear span. We provide the complete
characterization for the biorthogonality of the translates of scaling functions of two
nonuniform multiresolution analysis and the associated biorthogonal wavelet families
in L2(R+). Furthermore, under the mild assumptions on the scaling functions and the
corresponding wavelets associated with nonuniform multiresolution analysis, we show
that the wavelets can generate Reisz bases.