Uniform boundedness of the derivatives of meromorphic inner functions on the real line

Abstract

Inner functions are an important and popular object of study in the field of complex function theory. We look at meromorphic inner functions with a given spectrum and provide sufficient conditions for them to have uniformly bounded derivative on the real line. This question was first studied by Louis de Branges in 1968 and was later revived by Anton Baranov in 2011.

Restricted interpolation by meromorphic inner functions

“…In fact, the counterexample that Baranov had constructed was more general and served to show that even within this class, it was possible to have sequences which do not satisfy the required property. This was communicated to others working in this field and his work greatly influenced the work of the second author in [30], where spectra of MIFs with bounded derivatives were further studied.…”

“…can be shown to be strongly 1-regular but nonetheless any MIF with this sequence as spectrum will have unbounded derivative on R. We refer the reader to [30] for the proof. As for Theorem 4.2, fortunately for the experts in the area, the statement remains true and the proof can be fixed without major complications (we are not aware if a short correct proof is published at the moment).…”

“…In the case of a separated sequence ƒ it was shown by M. Mitkovski and the first author in [22] (see also [27]), that a necessary and sufficient condition for ƒ to support a measure with a spectral gap is that its interior Beurling-Malliavin density It is, therefore, natural to ask which sequences ƒ have the property as demanded by de Branges. In [30], the second author addresses this question and provides answers for a large class of sequences. It is known that for separated sequences whose corresponding gaps are bounded from above, there always exist MIFs with the required properties.…”

“…If ƒ C D N, then how sparse can ƒ be? In [30], it has been shown that ƒ can be at most exponentially growing, i.e., there is a c > 0 such that j n j . e cjnj for all n 2 ƒ :…”

“…It was used to obtain an extension of the Beurling-Malliavin theory ( [19], [20]) and to study classical problems of Fourier analysis ( [22], [23], [25], [26]). Some function theoretic questions arising from such applications were treated in [30].…”