In this paper, we study the spectrality and frame-spectrality of exponential systems of the type \(E(\Lambda ,\varphi ) = \{e^{2\pi i \lambda \cdot \varphi (x)}: \lambda \in \Lambda \}\) where the phase function \(\varphi \) is a Borel measurable which is not necessarily linear. A complete characterization of pairs \((\Lambda ,\varphi )\) for which \(E(\Lambda ,\varphi )\) is an orthogonal basis or a frame for \(L^{2}(\mu )\) is obtained. In particular, we show that the middle-third Cantor measures and the unit disc, each admits an orthogonal basis with a certain non-linear phase. Under a natural regularity condition on the phase functions, when \(\mu \) is the Lebesgue measure on [0, 1] and \(\Lambda = {{\mathbb {Z}}},\) we show that only the standard phase functions \(\varphi (x) = \pm x\) are the only possible functions that give rise to orthonormal bases. Surprisingly, however we prove that there exist a greater degree of flexibility, even for continuously differentiable phase functions in higher dimensions. For instance, we were able to describe a large class of functions \(\varphi \) defined on \({{\mathbb {R}}}^{d}\) such that the system \(E(\Lambda ,\varphi )\) is an orthonormal basis for \(L^{2}[0,1]^{d}\) when \(d\ge 2.\) Moreover, we discuss how our results apply to the discretization problem of unitary representations of locally compact groups for the construction of orthonormal bases. Finally, we conclude the paper by stating several open problems.

在本文中，我们研究了\（E（\ Lambda，\ varphi）= \ {e ^ {2 \ pi i \ lambda \ cdot \ varphi（x）}类型的指数系统的光谱和框架光谱：\ lambda \ in \ Lambda \} \）其中相位函数\（\ varphi \）是Borel可测量的，不一定是线性的。对\（（\ Lambda，\ varphi）\）的完整表征，其中\（E（\ Lambda，\ varphi）\）是\（L ^ {2}（\ mu）\ ）。特别是，我们证明了中间的三阶Cantor度量和单位圆盘都承认具有一定非线性相位的正交基础。在相位函数的自然规律条件下，当\（\ mu \）是在[0，1]和Lebesgue测度\（\ LAMBDA = {{\ mathbb {Z}}}，\） ，我们表明，只有标准相位函数\（\ varphi（X）= \时X \）是导致正交基数的唯一可能函数。但是，令人惊讶的是，我们证明了存在更大程度的灵活性，即使对于更高维的连续可微分的相位函数也是如此。例如，我们能够描述在\（{{\ mathbb {R}}} ^ {d} \）上定义的一大类函数\（\ varphi \），这样系统\（E（\ Lambda，\ varphi）\）是\（d ^ ge 2. \）时\（L ^ {2} [0,1] ^ {d} \）的正交基础此外，我们讨论了我们的结果如何应用于构造正交正交基的局部紧致群的统一表示的离散化问题。最后，我们通过陈述几个未解决的问题来结束本文。