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I am having trouble with calculating or approximation the following integral: $$ \int_{-\infty}^{\infty}(t^2-1)^{pn}(\mathrm{sgn}(t-1)-\mathrm{sgn}(t+1))^pdt, $$ where $1\leq p<\infty$ and $n \in N$.

Any ideas or references would be very helpful. Thank you.

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1 Answer 1

up vote 2 down vote accepted

$$ \DeclareMathOperator{\sgn}{\operatorname{sgn}} \sgn(t-1)- \sgn(t+1)= \begin{cases} 0, &\text{if } |t| \geq 1, \\ -2, &\text {if } |t| < 1 . \end{cases} $$

$$\begin{align*}\int_{-\infty}^{\infty}(t^2-1)^{pn}(\sgn(t-1)- \sgn(t+1))^p \mathrm dt &=\int_{-1}^{1}(t^2-1)^{pn}(-2)^p \mathrm d t\end{align*}$$

Hint:

Essentially, the problem involves breaking up the integral into relevant pieces and looking the signum function in those pieces. For instance, when $t>1$, note that the integral vanishes and so on.

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