We study in this paper one of functional analysis problems, involved with
construction a new class of functions, denoted by . The definition of the new
class depends on definition of Lebesgueclass of functions and on the Holder clas .
We study the
relation between the new class and approximation of the new class
to rational functions.
In this research, we have studied the issue of approximation of complex functions from weighted Lebesgue space ; and (Mukenhoupt weight) to rational functions by using p- Faber polynomials on large group of curves, which called Carlson curves. This
is also considered as a follow-up to the work done by researchers: Israfilov and Testici in 2014 , where they studied approximation of functions from weighted Smirnov space on domains with a Carlson curve boundary.
