New Findings on the Bank–Sauer Approach in Oscillation Theory

In 1988, S. Bank showed that if { z n } is a sparse sequence in the complex plane, with convergence exponent zero, then there exists a transcendental entire A ( z ) of order zero such that f ″+ A ( z ) f =0 possesses a solution having { z n } as its zeros. Further, Bank constructed an example of a zero sequence { z n } violating the sparseness condition, in which case the corresponding coefficient A ( z ) is of infinite order. In 1997, A. Sauer introduced a condition for the density of the points in the zero sequence { z n } of finite convergence exponent such that the corresponding coefficient A ( z ) is of finite order. In 2010, the second author proposed a unit disc analog of Bank’s first result. In the analog, { z n } is a sparse Blaschke sequence and A ( z ) belongs to the Korenblum space. The aim of the present paper is to introduce unit disc analogs of the two remaining results due to Bank and Sauer.