Heisenberg’s inequality
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I have tried to write a proof for the following inequality, but I couldn't constitute a rigorous one yet.
Suppose $f$ is absolutely continuous on $[−a,a]$ for all $a in mathbb R$ so that $f′$ exists a.e. on $mathbb R$. Then $$int_{- infty}^{infty} |f(x)|^2 dx leq 2 Big ( int_{- infty}^{infty} |xf(x)|^2 dx Big )^{1/2} Big ( int_{- infty}^{infty} |f'(x)|^2 dx Big )^{1/2}$$
Any help will be appreciated.
real-analysis inequality integral-inequality
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add a comment |
$begingroup$
I have tried to write a proof for the following inequality, but I couldn't constitute a rigorous one yet.
Suppose $f$ is absolutely continuous on $[−a,a]$ for all $a in mathbb R$ so that $f′$ exists a.e. on $mathbb R$. Then $$int_{- infty}^{infty} |f(x)|^2 dx leq 2 Big ( int_{- infty}^{infty} |xf(x)|^2 dx Big )^{1/2} Big ( int_{- infty}^{infty} |f'(x)|^2 dx Big )^{1/2}$$
Any help will be appreciated.
real-analysis inequality integral-inequality
$endgroup$
add a comment |
$begingroup$
I have tried to write a proof for the following inequality, but I couldn't constitute a rigorous one yet.
Suppose $f$ is absolutely continuous on $[−a,a]$ for all $a in mathbb R$ so that $f′$ exists a.e. on $mathbb R$. Then $$int_{- infty}^{infty} |f(x)|^2 dx leq 2 Big ( int_{- infty}^{infty} |xf(x)|^2 dx Big )^{1/2} Big ( int_{- infty}^{infty} |f'(x)|^2 dx Big )^{1/2}$$
Any help will be appreciated.
real-analysis inequality integral-inequality
$endgroup$
I have tried to write a proof for the following inequality, but I couldn't constitute a rigorous one yet.
Suppose $f$ is absolutely continuous on $[−a,a]$ for all $a in mathbb R$ so that $f′$ exists a.e. on $mathbb R$. Then $$int_{- infty}^{infty} |f(x)|^2 dx leq 2 Big ( int_{- infty}^{infty} |xf(x)|^2 dx Big )^{1/2} Big ( int_{- infty}^{infty} |f'(x)|^2 dx Big )^{1/2}$$
Any help will be appreciated.
real-analysis inequality integral-inequality
real-analysis inequality integral-inequality
asked Mar 29 at 15:46
user625442
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1 Answer
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To get started, you can apply integration by parts to $displaystyle int_{-a}^a 1 f(x)^2 , dx$ to obtain boundary terms and the integral $$ int_{-a}^a x 2 f(x) f'(x) , dx.$$ Now apply Cauchy-Schwarz, deal with the boundary terms, and send $a to infty$.
answered Mar 29 at 15:57
Umberto P.Umberto P.
40.3k13370
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1 Answer
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1 Answer
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$begingroup$
To get started, you can apply integration by parts to $displaystyle int_{-a}^a 1 f(x)^2 , dx$ to obtain boundary terms and the integral $$ int_{-a}^a x 2 f(x) f'(x) , dx.$$ Now apply Cauchy-Schwarz, deal with the boundary terms, and send $a to infty$.
answered Mar 29 at 15:57
Umberto P.Umberto P.
40.3k13370
$endgroup$
add a comment |
$begingroup$
To get started, you can apply integration by parts to $displaystyle int_{-a}^a 1 f(x)^2 , dx$ to obtain boundary terms and the integral $$ int_{-a}^a x 2 f(x) f'(x) , dx.$$ Now apply Cauchy-Schwarz, deal with the boundary terms, and send $a to infty$.
answered Mar 29 at 15:57
Umberto P.Umberto P.
40.3k13370
$endgroup$
add a comment |
$begingroup$
To get started, you can apply integration by parts to $displaystyle int_{-a}^a 1 f(x)^2 , dx$ to obtain boundary terms and the integral $$ int_{-a}^a x 2 f(x) f'(x) , dx.$$ Now apply Cauchy-Schwarz, deal with the boundary terms, and send $a to infty$.
answered Mar 29 at 15:57
Umberto P.Umberto P.
40.3k13370
$endgroup$
To get started, you can apply integration by parts to $displaystyle int_{-a}^a 1 f(x)^2 , dx$ to obtain boundary terms and the integral $$ int_{-a}^a x 2 f(x) f'(x) , dx.$$ Now apply Cauchy-Schwarz, deal with the boundary terms, and send $a to infty$.
answered Mar 29 at 15:57
Umberto P.Umberto P.
40.3k13370
answered Mar 29 at 15:57
Umberto P.Umberto P.
40.3k13370
answered Mar 29 at 15:57
Umberto P.Umberto P.
40.3k13370
answered Mar 29 at 15:57
Umberto P.Umberto P.
40.3k13370
40.3k13370
add a comment |
add a comment |
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