Fractal interpolation functions capture the irregularity of some data very effectively in comparison with the classical interpolants. They yield a new technique for fitting experimental data sampled from real world signals, which are usually difficult to represent using the classical approaches. The affine fractal interpolants constitute a generalization of the broken line interpolation, which appears as a particular case of the linear self-affine functions for specific values of the scale parameters. We study the [Formula: see text] convergence of this type of interpolants for [Formula: see text] extending in this way the results available in the literature. In the second part, the affine approximants are defined in higher dimensions via product of interpolation spaces, considering rectangular grids in the product intervals. The associate operator of projection is considered. Some properties of the new functions are established and the aforementioned operator on the space of continuous functions defined on a multidimensional compact rectangle is studied.

中文翻译：

---

多元仿射分形插值

与经典插值相比，分形插值函数非常有效地捕获了一些数据的不规则性。它们产生了一种新技术，用于拟合从现实世界信号中采样的实验数据，这些信号通常难以使​​用经典方法来表示。仿射分形插值构成了折线插值的推广，它表现为针对特定尺度参数值的线性自仿射函数的特例。我们研究了[公式：见文本]这种类型的插值的收敛性，用于[公式：见文本]以这种方式扩展文献中可用的结果。在第二部分中，仿射近似值通过插值空间的乘积在更高维度上定义，考虑乘积间隔中的矩形网格。考虑投影的关联算子。建立了新函数的一些性质，研究了上述定义在多维紧致矩形上的连续函数空间上的算子。