What Is the Effect of the Weyl Fractional Integral on the Hölder Continuous Functions?

Abstract

Let [Formula: see text] be [Formula: see text]-Hölder continuous on [Formula: see text] and well-defined about the Weyl fractional integral. Then, [Formula: see text] where [Formula: see text] and [Formula: see text]. This estimation shows that the Box dimension of [Formula: see text] leads to some similar linear dimension decrease like the Riemann–Liouville fractional integral [Y. S. Liang and W. Y. Su, Fractal dimensions of fractional integral of continuous functions, Acta Math. Sin. 32 (2016) 1494–1508].

A Technique for Estimation of Box Dimension about Fractional Integral

“…Rfs. [26] presented the same estimation of (1.5) for Weyl fractional integral. All these seem to show that Riemann-Liouville integral decreases box dimension linearly.…”

An Investigation on Fractal Characteristics of the Superposition of Fractal Surfaces

“…Of the diverse fractal dimensions, the box dimension mainly considered in the present paper shows its advantage of relatively easy calculation. Up to now, a lot of meaningful work has been done, including fractal interpolation functions [11][12][13][14], α-Hölder continuous functions [15,16], self-similar curves like the Von Koch curve [17,18], and some specific fractal functions like the Weierstrass function [19][20][21][22][23] and the Besicovitch function [24][25][26]. For more details of our latest work, we refer interested readers to [27][28][29][30][31][32].…”

On the Lower and Upper Box Dimensions of the Sum of Two Fractal Functions

“…For Hölder continuous functions, ref. [17,18] estimated the Box dimension of their fractional integral.…”