In the book "Index Transforms" by S.B. Yakubovich (World Scientific, 1996), I encountered the notion of a "smooth function of the order two" with compact support on $\mathbb R_+$, denoted $C^{(2)}(\mathbb R_+)$, cf. here (I hope the link works).

What does that mean?

At first I thought it refers to the space of functions with continuous first and second derivative, but then it is the term "smooth" which puzzles me in this context. So maybe it means something different.

Annoyingly, the notation is not explained in the book, and it also is not obvious for me from the context. Yakubovich cites the result that $C^{(2)}(\mathbb R_+)$ is dense in $L_{1,2}(\mathbb R_+):=L^2 (\mathbb R_+, x\, dx)$ and refers to the book of Titchmarsh, "Introduction to the Theory of Fourier Integrals" (1937), but I could not find the result in there. So maybe someone of you knows more about this. Thank you in advance!