A remarkable fact for the box dimensions of fractal interpolation curves of R 3

In this paper, we give the fractal interpolation curves generated on the Koch curve by Hölder continuous functions and function vertical scaling factors, and at the same time, we present the estimation method of box dimensions of curves of $R^3$ through the estimation of lower and upper box dimensions of obtained curves, where a remarkable fact is that the box dimensions are dependent not only on function vertical scaling factors but also on Hölder exponent of harmonic functions on the Koch curve. Especially, by using an interval instead of the Koch curve and by using complicated Hölder continuous functions-affine fractal interpolation functions on an interval instead of harmonic functions on the Koch curve, we give new fractal interpolation functions on an interval that provide strong insights in the mathematical theory and particular application of fractal interpolation.