A modification of the uniqueness criterion for the solution of the Watson problem for a half-plane

Abstract

It is proved that a known theorem yielding the solution of the Watson problem for a half-plane in terms of the Ostrovskii function remains valid if the Ostrovskii function$T (r) = \mathop {\sup }\limits_{n \geqslant 0} r^n /m_n$ is replaced by the function$\tilde T (r) = \mathop {\sup }\limits_{r \geqslant x > 0} r^x /m (x)$, where for x ε [n, n+1) the function m(x)=m_n, or by the function$T*(r) = \mathop {\sup }\limits_{r \geqslant n \geqslant 0} r^n /m_n$.