A REVISIT TO STABILITY OF SCHAUDER BASES: FRACTALIZING MULTIVARIATE FABER–SCHAUDER SYSTEM

Abstract

Let [Formula: see text] be a Banach space with a Schauder basis [Formula: see text], and [Formula: see text] be the identity operator on [Formula: see text]. It is known, at least in essence, that if [Formula: see text] is a sequence of bounded linear operators on [Formula: see text] such that [Formula: see text] then [Formula: see text] is also a basis. The first part of this work acts as an expository note to formally record the aforementioned stability result. In the second part, we apply this stability result to construct a Schauder basis consisting of bivariate fractal functions for the space of continuous functions defined on a rectangle. To this end, fractal perturbations of the elements in the classical bivariate Faber–Schauder system are formulated using a sequence of bounded linear fractal operators close to the identity operator in accordance with the stability result mentioned above. This illustration, although emphasized only for the bivariate case, can easily be extended to higher dimensions. Further, the perturbation technique used here acts as a companion for a few researches on fractal bases in the univariate setting.