Analytical and dimensional properties of fractal interpolation functions on the Sierpiński gasket

Abstract

In this article, we construct the fractal interpolation functions (FIFs) on the Sierpiński gasket (SG) by taking data set at the nth level. We discuss the oscillation space, $$\mathcal {L}_q$$ space and some other function spaces on SG, and determine some conditions under which, this FIF belongs to these spaces. We obtain the fractal dimension of the graph of the FIF and the Hausdorff dimension of the invariant measure supported on the graph of the FIF. We also prove that this FIF has finite energy. After that, we discuss the restrictions of the FIF on the bottom line segment of SG and prove that the restriction is again an FIF on the bottom line. We estimate the fractal dimension of the graph of the Riemann-Liouville fractional integral of the restrictions of the FIF on the bottom line segment of SG. Lastly, we prove that the Hausdorff and box-counting dimension of the graph of the restriction of any harmonic function on the bottom line segment of SG are 1 and for the Riemann- Liouville fractional integral of this restriction function, these dimensions are also 1.