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.

On non-separated zero sequences of solutions of a linear differential equation

“…The following statement shows that the corollary is sharp (cf. [9]). Theorem E. For arbitrary ρ > 0 there exists a sequence of distinct numbers {z n } in D with the following properties:…”

On non-normal solutions of linear differential equations

“…is analytic, since the interpolation property Lemma 4(i) guarantees that A has a removable singularity at each point ζ n for n ∈ N. We also have A ∈ H ∞ 2 , since {ζ n } ∞ n=1 is uniformly separated and g ∈ BMOA; see [7] for more details. Since f is a solution of (1) with A ∈ H ∞ 2 , and lim sup…”

On the Complexity of Finding a Necessary and Sufficient Condition for Blaschke-Oscillatory Equations

“…If {z n } ∞ n=1 is a union of two exponential sequences [4, p. 156] approaching pairwise to one another exponentially, then it is proved in [8,Theorem 5] that A(z) cannot belong to the Korenblum space A −∞ [9, p. 110]. In particular, both of the two component sequences are uniformly separated, and yet all solutions of (2) are "far away" from the class N as being of infinite order of growth.…”