Let $f\in L^2(0,\infty)$ with $|f(x)| \leq |x|$. Further, define $g(x)=d^x$ for some $d>1$.
Question: Is $f\circ g \in L^2(\mathbb{R})$?
If yes, how do I show this? If no, under which conditions does this hold?
Intuitively, this should hold, but I have no clue if this can be shown based on the above conditions. Any hints?
The question is to determine whether the integral $$ \int_{\mathbb R}\left\lvert f\left(e^{x\ln b}\right)\right\rvert^2\mathrm dx. $$ Since $\ln b\gt 0$, this reduces to determine finiteness of $$ \int_{\mathbb R}\left\lvert f\left(e^{u}\right)\right\rvert^2\mathrm du $$ by letting $u=x\ln b$. Now, letting $t=e^u$, we have to investigate the convergence of $$ \int_{(0,+\infty)}\frac 1t\left\lvert f\left(t\right)\right\rvert^2\mathrm dt. $$ On $(0,1)$, use the condition $\left\lvert f\left(t\right)\right\rvert\leqslant t$ and on $(1,+\infty)$, use square integrability of $f$.