南京理工大学学报（自然科学版）2026，Vol.50Issue(4)：472-480,9.DOI:10.14177/j.cnki.32-1397n.2026.50.04.010
广义分数阶微积分下的连续函数保维逼近
Dimension-preserving approximation of continuous functions under generalized fractional calculus
摘要
Abstract
To extend the theory of dimension-preserving approximation for integer-order calculus of continuous functions to the fractional-order case,this study adopts the Katugampola fractional calculus framework and conducts research from the perspective of fractal structures.By constructing a dense subset in the space of continuous functions and proving the commutativity between the limit operation and the Katugampola fractional calculus operator,the dimension-preserving property of continuous functions under the Katugampola fractional calculus operator is obtained,and the construction method of the corresponding continuous approximation sequence is also presented.The results show that for continuous functions with fractal dimensions between 1 and 2,there exists a sequence of continuous functions that converges uniformly to the original function,and the fractal dimensions of the Katugampola fractional calculus of the sequence and the original function remain consistent.Explicit approximation sequences with specific fractal dimension features can be constructed by combining Bernstein polynomials with Weierstrass-type fractal functions,achieving arbitrary relationships between the fractal dimensions of the approximating and approximated functions.This research enriches the theory of fractal function approximation and provides proof strategies for analyzing fractal dimension invariance,along with techniques for constructing approximation sequences.关键词
Katugampola分数阶微积分/保维逼近/分形维数/Hölder连续性/连续函数空间Key words
Katugampola fractional calculus/dimension-preserving approximation/fractal dimensions/Hölder continuity/space of continuous functions分类
数理科学引用本文复制引用
李美茜,俞斌琰,梁永顺..广义分数阶微积分下的连续函数保维逼近[J].南京理工大学学报（自然科学版）,2026,50(4):472-480,9.基金项目
国家自然科学基金(12071218) （12071218）
基础研究揭榜挂帅专项(30925020210) （30925020210）
