Assume that functions $f_n(t), f(t)\in C_b(R_+)$. For every $\lambda >0$, we have $$ \bigg|\int_0^\infty e^{-\lambda t}f_n(t)d t-\int_0^\infty e^{-\lambda t}f(t)d t\bigg|\leq C_\lambda n^{-1}, $$ where $C_\lambda>0$ is a constant. Can we get that for every $t>0$, $f_n(t)$ converges to $f(t)$? Is it possible to get the order of convergence?
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6$\begingroup$ If $f_n(t) = \sin(n t)$ and $f(t) = 0$, then the left-hand side is $n/(\lambda^2 + n^2)$, which is $O(1/n)$ despite the fact that $f_n(t)$ does not converge to $f(t)$ pointwise. Or am I missing something? $\endgroup$– Mateusz KwaśnickiCommented Feb 25, 2020 at 21:38
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