On the properties of systems of root functions of a second-order discontinuous operator
Let \(G=\cup^m_{l=1}G_l\), \(G_l=(\xi_{l-1},\xi_l)\), \(a=\xi_0<\xi_1<\dots <\xi_m=b\), where \(-\infty<a<b<+\infty\). By \(W^n_r(G)\), \(1\leq r\leq\infty\), is denoted the space of functions possessing the property that, if \(f\in W^n_r(G)\) then, for every \(G_l\), \(l=1,2,\dots,m\), there exists a function \(f_l\) from the Sobolev space \(W^n_r(G_l)\) such that \(f(x)=f_l(x)\) for \(\xi_{l-1}<x<\xi_l\). On \(G\), the formal differential operator \(Lu=u''+p(x)u'+q(x)u\) with complex-valued coefficients \(p\in L_r(G)\), \(1<r\leq 2\), and \(q\in L_1(G)\) is considered. A system of root functions of the operator \(L\) is an arbitrary system \(\{u_k\}_{k=1}^{\infty}\) of complex-valued identically nonzero functions from \(W_1^2(G)\) that satisfy a.e. in \(G\) the equation \(Lu_k+\lambda_ku_k=\theta_ku_{k-1}\), where \(\theta_k\) is either \(0\) (in this case \(u_k\) is called an eigenfunction) or \(1\) (in this case it is required that \(\lambda_k=\lambda_{k-1}\) and \(u_k\) is called an associated function of order \(j\), where \(\theta_k=\theta_{k-1}=\dots=\theta_{k-j+1}=1\) and \(\theta_{k-j}=0\)). The authors establish a necessary and sufficient condition for the Riesz basis property of a system of root functions of the operator \(L\) and then analyze the absolute and uniform convergence and the rate of uniform convergence of biorthogonal expansions of discontinuous functions in terms of root functions of the operator \(L\) (for \(p(x)\equiv 0\)). Finally, some concrete examples of boundary value operators are presented.
scientific article; zbMATH DE number 5666526