广义分数阶微积分下的连续函数保维逼近OA

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.