Separated sequences and interpolation
Volume 274 / 2024
Studia Mathematica 274 (2024), 79-99
MSC: Primary 26E10; Secondary 41A05, 46F10
DOI: 10.4064/sm230620-11-10
Published online: 3 January 2024
Abstract
We introduce the notion of $\nu $-separated increasing sequences $\{ x_{n}\} _{n=1}^{\infty }$. We establish that interpolation problems of the kind $\varphi ( x_{n}) =z_{n}$ have solutions $\varphi \in \mathcal {S}( \mathbb {R}) $ for all sequences $\{ z_{n}\} $ of rapid decay in the sense that $z_{n}=o( x_{n}^{-\alpha }) $ for all $\alpha \gt 0$ if and only if $\{ x_{n}\} $ is $\nu $-separated for some $\nu $. We also give several generalizations of this result.
