Approximation of infinitely differentiable functions on the real line by polynomials in weighted spaces

By a given family of convex functions on the real axis that grow at infinity faster than any linear function and by a certain logarithmically convex sequence of positive numbers, we construct the space of infinitely differentiable functions on the real line. Under the condition of a logarithmic gap between weight functions, we prove the possibility of approximation by polynomials in this space.