This paper is devoted to a hierarchical approach of constructing a class of fractal interpolants with trigonometric basis functions and to preserve the geometric behavior of given univariate data set by these fractal interpolants. In this paper, we propose a new family of ${\mathcal{C}}^1$-rational cubic trigonometric fractal interpolation functions (RCTFIFs) that are the generalized fractal version of the classical rational cubic trigonometric polynomial spline of the form $p_i\left(\theta \right)∕q_i\left(\theta \right)$, where $p_i\left(\theta \right)$ and $q_i\left(\theta \right)$ are cubic trigonometric polynomials with four shape parameters in each sub-interval. The convergence of the RCTFIF towards the original function in ${\mathcal{C}}^3$ is studied. We deduce the simple data dependent sufficient conditions on the scaling factors and shape parameters associated with the ${\mathcal{C}}^1$-RCTFIF so that the proposed RCTFIF lies above a straight line when the interpolation data set is constrained by the same condition. The first derivative of the proposed RCTFIF is irregular in a finite or dense subset of the interpolation interval and matches with the first derivative of the classical rational trigonometric cubic interpolation function whenever all scaling factors are zero. The positive shape preservation is a particular case of the constrained interpolation. We derive sufficient conditions on the trigonometric IFS parameters so that the proposed RCTFIF preserves the monotone or comonotone feature of prescribed data.

本文致力于构建一类具有三角基函数的分形插值的分层方法，并通过这些分形插值保留给定单变量数据集的几何行为。在本文中，我们提出了一个新的家族${\mathcal{C}}^1$- 有理三次三角分形插值函数 (RCTFIF)，它是形式的经典有理三次三角多项式样条的广义分形版本 ${磷}_{\mathrm{一世}}\left(\theta \right)∕q_{\mathrm{一世}}\left(\theta \right)$， 在哪里 ${磷}_{\mathrm{一世}}\left(\theta \right)$ 和 $q_{\mathrm{一世}}\left(\theta \right)$是三次三角多项式，每个子区间有四个形状参数。RCTFIF 向原始函数的收敛${\mathcal{C}}^3$被研究。我们推导出简单数据依赖于与相关的缩放因子和形状参数的充分条件${\mathcal{C}}^1$-RCTFIF 使得当插值数据集受相同条件约束时，建议的 RCTFIF 位于直线上方。所提出的 RCTFIF 的一阶导数在插值区间的有限或密集子集中是不规则的，并且只要所有缩放因子为零，就与经典有理三角三次插值函数的一阶导数匹配。正形状保留是约束插值的一个特例。我们推导出三角 IFS 参数的充分条件，以便建议的 RCTFIF 保留规定数据的单调或共调特征。