Summability of eigenfunction expansions of differential and integral operators (Q1337881)

Sufficient conditions are given for Abel summability with order \(\alpha\) of the generalized eigenfunction expansion associated with an ordinary differential operator \(L\) of \(n\)th order subject to decaying normalized boundary conditions. The operator \(L\) is given by \[ Lu(x)= u^{(n)}(x)+ \sum^{n- 2}_{k= 0} p_ k(x) u^{(k)}(x),\quad 0\leq x\leq 1,\tag{i} \] for \(u(x)\) such that \[ \begin{aligned} U_ j(u) &= u^{(s_ j)}(0)+ \sum^{s_ j- 1}_{k= 0} b_{jk} u^{(k)}(0)= 0, \qquad j= 1,\dots, n-p,\\ U_ j(u)&= u^{(s_ j)} (1)+ \sum^{s_ j- 1}_{k= 0} b_{jk} u^{(k)}(1)= 0, \qquad j= n-p+ 1,\dots, n,\end{aligned} \tag{ii} \] \(0\leq s_ 1<\cdots< s_{n- p}\leq n-1\), \(0\leq s_{n- p+1}<\cdots< s_ n\leq n- 1\). The author imposes sufficient conditions on \(p_ k(x)\), \(f(x)\) such that the eigenfunction expansion for \(f(x)\) is uniformly Abel summable on \([0,1]\) with order \(\alpha\) when \(\alpha\) satisfies \(1/n< \alpha< 1/| n-2p|\) and \(p\) is such that \(| n- 2p|< 4+ \min(p,n- p)\), \(n= 2j\), \(| n- 2p|< 3+ \min(p, n- p)\), \(n= 2j- 1\), \(j= 1,2,\dots\) The results extend ones due to \textit{A. P. Khromov} [Ordinary differential equations and Fourier series, CGU, Saratov, 29-41 (1968) (Russian)]. The result follows from a more general result on summability of the eigenfunction expansion associated with a Volterra integral operator which is a generalization of the operator inverse to \(L\).