DCT application

Question

If $g_k(x) = k \log\big(1 + \frac{g(x)^2}{k^2}\big) \text{ with} \ g \in L^1([0,1])$, I know the $$\lim\limits_{k}\int_0^1 g_k = 0$$ but my issue is finding the control function.

Using the properties of the log, we can show that $g_k$ is dominated by $\frac{g^2}{k}$. Problem: it is not guaranteed that $g^2 \in L^1([0,1])$?

edit: I think it is bounded by $|g|$.

If $g \in L^1([0,1])$ it does not imply that $g^2 \in L^1([0,1])$. For example let $g(x) = x^{-1/2}$. — LostStatistician18, Feb 10 '23 at 16:19
No worries @C.Master, I just put this here since there is a ? mark in your question, and to make it clear to future readers, if any, the important counter example why $g\in L^1$ does not imply $g\in L^2$. — LostStatistician18, Feb 10 '23 at 17:07

LostStatistician18 · Accepted Answer · 2023-02-10T21:51:43.450

2

Since for $x$ positive, $\log(1+ x^2/2 ) \le x$, which follows from $1+x^2/2 \le e^x$ (look at the Taylor series for $e^x$),

$$ g_k(x) \le k \sqrt{2}\frac{|g(x)|}{k} =\sqrt{2} |g(x)|. $$ Hence when $g$ is integrable, the sequence $g_k$ is bounded by an integrable function.

edited Feb 10 '23 at 21:51

answered Feb 10 '23 at 16:25

LostStatistician18

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DCT application

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