New Findings on the Bank–Sauer Approach in Oscillation Theory

Abstract

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.