Is there a class of real functions strictly between $C^{\infty}$ (smooth) and $C^{\omega}$ (analytic)?

Question

I'm relatively new to the concept of analytic functions, and began wondering: Is it possible to exist a class of functions strictly between $C^{\infty}$ (smooth functions) and $C^{\omega}$ (analytic functions)?

In other words, could there be a class $C^{\alpha}$ of functions from $\mathbb{R}$ to $\mathbb{R}$ such that $f \in C^{\omega} \Rightarrow f \in C^{\alpha} \Rightarrow f \in C^{\infty}$, but $f \in C^{\infty} \nRightarrow f \in C^{\alpha}$ and $f \in C^{\alpha} \nRightarrow f \in C^{\omega}$? If so, what coud be this class?

Thanks.

Answer 1

Yes, for example the Gevrey class.