  In this paper, we explore some significant properties associated with a
fractal operator on the space of all continuous functions defined on the
Sierpi\'nski Gasket (SG). We also provide some results related to constrained
approximation with fractal polynomials and study the best approximation
properties of fractal polynomials defined on the SG. Further we discuss some
remarks on the class of polynomials defined on the SG and try to estimate the
fractal dimensions of the graph of $\alpha$- fractal function defined on the SG
by using the oscillation of functions.
