In this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products. In the proof, we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.

中文翻译：

---

三角多项式小波的边缘检测

在本文中，我们证明了某些三角多项式小波是方向性deValléePoussin型小波的特例，它们能够检测沿周期特征函数边界曲线的不连续性。受二维离散离散小波的最新结果的推动，我们对相应内积的大小提供了较低和较高的估计。在证明中，我们使用三角多项式小波的时域和频域定位特性，以及某些菲涅尔积分的边界。此外，我们给出了数值示例，突显了理论结果。