Expansion of a bundle of fourth-order differential operators in a part of its eigenfunctions

Abstract

A bundle of differential operators $$\mathcal{L}(\lambda ),\lambda \in \mathbb{C}:\mathcal{L}(\lambda )y(x) = y^{(4)} (x) - 2\lambda ^2 y^{(2)} (x) + \lambda ^4 y(x),|x| \leqslant 1,y( \pm 1) = y\prime ( \pm 1) = 0,$$ is considered. In various function spaces, we establish the facts about the expansions of a pair of functions $f(x)$ and $g(x)$ in the system $\{y_k (x),\; iλ_k y_k (x)|}_{k=1}^{ ∞}$, where $y_k(x), k = 1,2,...,$ are the eigenfunctions of the bundle $L (λ)$ corresponding to the eigenvalues $λ_k$, with $\Im λ_k > 0$.