Fractal Convolution on the Rectangle

Abstract

The primary goal of this article is devoted to the study of fractal bases and fractal frames for $L^2(I\times J)$, the collection of all square integrable functions on the rectangle $I\times J$. The fractal function when recognized as an internal binary operation paved way for the construction of right and left partial fractal convolution operators on $L^2(I)$, for any real compact interval I. The aforementioned operators defined on one dimensional space have been applied to obtain operators on the space $L^2(I \times J)$ by the identification of $L^2(I \times J)$ with the tensor product space $L^2(I)\otimes L^2(J)$. In this paper, we establish properties of this bounded linear operator which eventually helps to prove that $L^2(I \times J)$ admits Bessel sequences, Riesz bases and frames consisting of products of fractal (self-referential) functions in a nice way.