Annihilable systems of trigonometric systems (Q1059247)

The system \(\{e^{-i\alpha}n^ t\}\) \((\alpha_ n\in {\mathbb{C}})\) in the space \(L^ p=L^ p(-\pi,\pi)\), \(1\leq p<\infty\) \((C=C[-\pi,\pi])\) is said to be annihilable in the class \(L^ p\) (C) if there exists a function \(g\in L^ q\) \((\in BV[-\pi,\pi])\) such that \(g(z)=\int^{\pi}_{-\pi}e^{-izt}g(t)dt\) \((=\int^{\pi}_{- \pi}e^{-izt}dg(t))\) vanishes only at the points \(\alpha_ n\) and the \(\alpha_ n's\) are all simple zeros. The author proves a characterization for annihilability in \(L^ p\) and C for subsystems of the trigonometric system, i.e. for systems \(\{e^{-int}\}\), \(n\in {\mathbb{Z}}_{\infty}\setminus \Lambda\), \(\Lambda\) \(\subset {\mathbb{Z}}\) and applies the result to the case \(\Lambda =\{[k^{\alpha}]\}^{\infty}_{k=0}\), \(\alpha >1\).