In this paper, we construct explicit exponential bases of unions of segments of total measure one. Our construction applies to finite or infinite unions of segments, with some conditions on the gaps between them. We also construct exponential bases on finite or infinite unions of cubes in \mathbb {R}^d$\mathbb {R}^d$ and prove a stability result for unions of segments that generalize Kadec’s \frac 14$\frac 14$-theorem.

We consider three special and significant cases of the following problem. Let D\subset \mathbb{R}^{d}$D\subset \mathbb{R}^{d}$ be a (possibly unbounded) set of finite Lebesgue measure. Let E(\mathbb{Z}^{d})=\{e^{2\unicode[STIX]{x1D70B}ix\cdot n}\}\text{}_{n\in \mathbb{Z}^{d}}$E(\mathbb{Z}^{d})=\{e^{2\unicode[STIX]{x1D70B}ix\cdot n}\}\text{}_{n\in \mathbb{Z}^{d}}$ be the standard exponential basis on the unit cube of \mathbb{R}^{d}$\mathbb{R}^{d}$. Find conditions on D$D$ for which E(\mathbb{Z}^{d})$E(\mathbb{Z}^{d})$ is a frame, a Riesz sequence, or a Riesz basis for L^{2}(D)$L^{2}(D)$.

We give a necessary and sufficient condition for a sequence to be a localization set for a determining average sampler.

Using Local Residues and the Duality Principle a multidimensional variation of the completeness theorems by T. Carleman and A. F. Leontiev is proven for the space of holomorphic functions defined on a suitable open strip {{T}_{\alpha }}\,\subset \,{{\mathbf{C}}^{2}}${{T}_{\alpha }}\,\subset \,{{\mathbf{C}}^{2}}$. The completeness theorem is a direct consequence of the Cauchy Residue Theorem in a torus. With suitable modifications the same result holds in {{\mathbf{C}}^{n}}${{\mathbf{C}}^{n}}$.