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. © 2021, Springer Science+Business Media, LLC, part of Springer Nature.