Exponential type bases on a finite union of certain disjoint intervals of equal length

Abstract

Let $\Lambda$ be a sequence of distinct real numbers and $\mathsf {T}=\{\epsilon _0,\dots ,$ $\epsilon _{\nu -1}: \epsilon _i<\epsilon _{i+1}\}$ be a finite set of nonnegative integers. To each $(\Lambda ,\mathsf {T})$, we associate a system of exponentials

$$\begin{aligned} \mathcal {E}(\Lambda ,\mathsf {T}):=\left\{ x^{\epsilon _0}{\rm e}^{2\pi i\lambda x},x^{\epsilon _1}{\rm e}^{2\pi i\lambda x},\dots ,x^{\epsilon _{\nu -1}}{\rm e}^{2\pi i\lambda x}:\lambda \in \Lambda \right\} . \end{aligned}$$

For a disconnected set $\Omega$, the construction of a Riesz basis of exponential system $\mathcal {E}(\Lambda ,\mathsf {T})$ on $L^2(\Omega )$ is generally a difficult problem. In the literature, the exponential Riesz basis problem is solved for certain disconnected sets $\Omega$ for $\nu =1$ and $\epsilon _0=0$. It is well-known that this problem is equivalent to the construction of a complete interpolation set in the Paley–Wiener space $\mathcal{PW}\mathcal{}(\Omega )$. In this paper, we construct a complete interpolation set involving derivative samples in the Paley–Wiener space of a finite union of certain disjoint intervals of equal length. Furthermore, we provide an explicit sampling formula in this case.