Fractal Convolution on the Rectangle

The primary goal of this article is devoted to the study of fractal bases and fractal frames for L2(I× J) , the collection of all square integrable functions on the rectangle I× J. The fractal function when recognized as an internal binary operation paved way for the construction of right and left partial fractal convolution operators on L2(I) , for any real compact interval I. The aforementioned operators defined on one dimensional space have been applied to obtain operators on the space L2(I× J) by the identification of L2(I× J) with the tensor product space L2(I) ⊗ L2(J). In this paper, we establish properties of this bounded linear operator which eventually helps to prove that L2(I× J) admits Bessel sequences, Riesz bases and frames consisting of products of fractal (self-referential) functions in a nice way.