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Approximation by shape preserving fractal functions with variable scalings
Calcolo ( IF 1.7 ) Pub Date : 2021-02-15 , DOI: 10.1007/s10092-021-00396-8
Sangita Jha , A. K. B. Chand , M. A. Navascués

The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions \(f^{\varvec{\alpha }}\in {\mathcal {C}}(I)\), where \(\varvec{\alpha }\) is a vector with functional components. The presence of scaling function in these fractal functions helps to get a wide variety of mappings for approximation problems. The current article explores the shape-preserving properties of the \(\varvec{\alpha }\)-fractal functions with variable scalings, where the optimal ranges of the scaling functions are derived for fundamental shapes of the germ f. We provide several examples to illustrate the shape preserving results and apply our fractal methodologies in approximation problems. Also, it is shown that the order of convergence of the \(\varvec{\alpha }\)-fractal polynomial to the original shaped function matches with that of polynomial approximation. Further, based on the shape preserving properties of the \(\varvec{\alpha }\)-fractal functions, we provide the fractal analogue of the Chebyshev alternation theorem. To the end, we deduce the fractal version of the classical full Müntz theorem in \({\mathcal {C}}[0,1]\).



中文翻译:

通过可变比例缩放的保形分形函数进行逼近

具有适当迭代函数系统的分形插值函数提供了一种在紧致区间I上扰动和逼近连续函数的方法。此方法在{\ mathcal {C}}(I)\)中生成一类函数\(f ^ {\ varvec {\ alpha}} \,其中\(\ varvec {\ alpha} \)是具有函数的向量成分。这些分形函数中缩放函数的存在有助于获得各种近似问题的映射。本文探讨了具有可变缩放比例的\(\ varvec {\ alpha} \)-分形函数的保形特性,其中针对细菌f的基本形状导出了缩放函数的最佳范围。我们提供了几个示例来说明形状保持结果,并将分形方法应用于近似问题。此外,还表明\(\ varvec {\ alpha} \)-分形多项式到原始形状函数的收敛顺序与多项式逼近的顺序匹配。此外,基于\(\ varvec {\ alpha} \)-分形函数的形状保持性质,我们提供了切比雪夫交替定理的分形类似物。最后,我们用\({\ mathcal {C}} [0,1] \)推论出经典的完整Müntz定理的分形形式。

更新日期:2021-02-16
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