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

In this article, we construct the fractal interpolation functions (FIFs) on the Sierpinski gasket (SG) by taking data set at the nth level. We discuss the oscillation space, 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.

Automated summary

In this article, fractal interpolation functions (FIFs) are constructed on the Sierpinski gasket (SG) using data set at the nth level. The oscillation space, L-q space, and other function spaces on SG are discussed, and conditions for FIFs belonging to these spaces are determined. The fractal dimension of the FIFs graph and the Hausdorff dimension of the invariant measure on the graph are obtained. It is proved that the FIFs have finite energy and the restrictions of the FIFs on the bottom line segment of SG are again FIFs on the bottom line.