Functional Equations Related to Sine Type Functions

Abstract

Functional equations in three classes of entire functions related to sine type functions are considered. These functional equations occur in inverse Sturm–Liouville problems with eigenvalue parameter dependent boundary conditions. In the first class $\mathcal {A}$, which consists of certain pairs of entire functions (P, Q) with leading terms $-\lambda \sin \lambda a$ and $\cos \lambda a$, respectively, it is shown that, for each $\eta \in \mathbb {R}$, the functional equation $Q(\lambda )Y(\lambda )-P(\lambda )Z(\lambda )=\eta$ has a unique solution (Y, Z) in $\mathcal {A}$. As a consequence, the functional equation $Q(\lambda )S_1(\lambda )-P(\lambda )S_0(\lambda )=1$ has a unique solution $(S_1,S_0)$ in a class of functions with leading terms $\cos \lambda a$ and $\lambda ^{-1}\sin \lambda a$, respectively. A third class of families $\Phi _\alpha$, $\alpha >0$, $\alpha \not =1$, consists of entire functions with leading terms $-\lambda \sin \lambda a+i\lambda \alpha \cos \lambda a$. It is shown that for each $\phi \in \Phi _\alpha$ and $\Delta \in \mathbb {R}$, the functional equation $\phi (\lambda )X(-\lambda )-\phi (-\lambda )X(\lambda )=2i\alpha \lambda \Delta$ has a unique solution $X\in \Phi _\alpha$.