Approximation With Fractal Functions by Fractal Dimension

Abstract

On the basis of previous studies, we explore the approximation of continuous functions with fractal structure. We first give the calculation of fractal dimension of the linear combination of continuous functions with different Hausdorff dimension. Fractal dimension estimation of the linear combination of continuous functions with the same Hausdorff dimension has also been discussed elementary. Then, based on Weierstrass Theorem and the related results of Weierstrass function, we give the conclusion that the linear combination of polynomials with the same Hausdorff dimension approximates the objective function. The corresponding results with noninteger and integer Hausdorff dimensions have been investigated. We also give the preliminary applications of the theory in the last section.