For $a,b\in (0,\infty)$ arbitrary is it possible to find a function $f\in L^1(\mathbb R)\cap L^2(\mathbb R)$ with $\|f\|_1=a$ and $\|f\|_2=b$ ? I think this is possible but I haven't such a function.
2026-05-15 14:06:18.1778853978
$f\in L^1\cap L^2$ with $\|f\|_1=a, \|f\|_2=b$
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Sure, take
$$ f(x)=\frac{b^2}{a}\chi_{(0,a^2/b^2)}(x) $$