In this paper, we explore some significant properties associated with a fractal operator on the space of all continuous functions defined on the Sierpiński 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 α-fractal function defined on the SG by using the oscillation of functions.

In this paper, the notion of dimension preserving approximation for real -valued bivariate continuous functions, defined on a rectangular domain ⊏ ⊐, has been introduced and several results, similar to well-known results of bivariate constrained approximation in terms of dimension preserving approximants, have been established. Further, some clue for the construction of bivariate dimension preserving approximants, using the concept of fractal interpolation functions, has been added. In the last part, some multi-valued fractal operators associated with bivariate α-fractal functions are defined and studied..