Bases consisting of self-referential functions in Banach spaces

Fractal functions are constructed through iterated function systems on suitable Banach spaces. In this article, first, we introduce a new class of fractal approximants with variable scaling parameters, namely, Bernstein–Kantorovich α-fractal functions. The proposed class of functions is non-differentiable and has good properties of convergence. The construction of bases consisting of self-referential functions in different Banach spaces plays an important role in the studies of non-linearity. In this paper, we show the existence of fractal bases consisting of a new class of self-referential functions. In particular, we establish the existence of frames and fractal bases for the proposed class of functions in the space Lp(I) , where I is a compact interval in R. In the end, we study some special properties with the proposed self-referential functions in the Hilbert space L2(I).