Convolved fractal bases and frames

摘要

Based on the theory of fractal functions, in previous papers, the first author introduced fractal versions of functions in GP-spaces, associated fractal operator and some related notions. More recently, it has been realized that the fractalization of a Lebesgue integrable function can be viewed as an internal binary operation, termed fractal convolution, of the germ function and a parameter map. In the current note, we continue to study this fractal convolution with a different viewpoint in mind. In particular, we consider both left and right partial fractal convolution operators on GP-spaces. As an application, we obtain bases and frames consisting of fractal functions by exploring fractal convolutions and their intriguing links with the perturbation theory of Schauder bases and frames. Thus, the theory of fractal functions and theory of bases and frames seem to come together very nicely via fractal convolution. Also established along the way are results involving bases and frames obtained by using partial fractal convolutions with the null function.