Absolute and uniform convergence of expansions in the root vector functions of the Dirac operator (Q1930847)

Consider the one-dimensional Dirac operator \[ Du= B{{du}\over{dx}}+P(x)u,\quad u=(u^1,u^2)^{\text{T}}, \] defined on \(G=(0,\pi),\) and where \(B=(b_{ij})^2_{i,j=1}\), \(b_{i,3-i}=(-1)^{i-1}\), \(b_{ii}=0,\) and \(P(x)=\text{diag\,}(p(x),q(x))\) is a complex-valued function. The authors analyze the absolute and uniform convergence of the biorthogonal expansion of a vector function from the class \(W^1_p(G),\) \(p\geq1,\) in the root vector functions of \(D\). Sufficient conditions for absolute and uniform convergence are established and the rate of the uniform convergence on \([0,\pi]\) is estimated.