Associate fractal functions in LpLp-spaces and in one-sided uniform approximation

• Fractal functions in LpLp-spaces are investigated in detail.
• Fractal versions of the full Müntz theorems in LpLp-spaces are derived.
• One-sided approximation with fractal functions is broached.
• Overall, the article is a step forward in the theory of fractal approximation.

Fractal interpolation function defined with the aid of iterated function system can be employed to show that any continuous real-valued function defined on a compact interval is a special case of a class of fractal functions (self-referential functions). Elements of the iterated function system can be selected appropriately so that the corresponding fractal function enjoys certain properties. In the first part of the paper, we associate a class of self-referential LpLp-functions with a prescribed LpLp-function. Further, we apply our construction of fractal functions in LpLp-spaces in some approximation problems, for instance, to derive fractal versions of the full Müntz theorems in LpLp-spaces. The second part of the paper is devoted to identify parameters so that the fractal functions affiliated to a given continuous function satisfy certain conditions, which in turn facilitate them to find applications in some one-sided uniform approximation problems.