Mixing
Fourier tranform of the derivative
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I have been recently studing Fourier transform and there is a proposition that says: If $lim limits_{x toinfty}xf(x)=limlimits_{{x}to -infty}xf(x)=0$ then $$hat{f'}(z)=-izhat{f}(z)$$
and this is proven like this:
$(f(x)e^{izx})'=f'(x)e^{izx}+izf(x)e^{izx}$
and intergrating by parts we have
$displaystyleint_{-infty}^{infty}(f(x)e^{izx})'dx=int_{-infty}^{infty}f'(x)e^{izx}dx+int_{-infty}^{infty}izf(x)e^{izx}dx$
or
$f(x)e^{izx}vert_{x=-infty}^{x=+infty}=hat{f'}(z)+izhat{f}(z)$
But
$f(x)e^{izx}vert_{x=-infty}^{x=+infty}=0$ , so we get the desired equality.
My quetion here is why we should suppose that $limlimits_{{x}to infty}xf(x)=limlimits_{{x}to -infty}xf(x)=0$ and not just that $$limlimits_{{x}to infty}f(x)=limlimits_{{x}to -infty}f(x)=0$$ ? Because then we would have $f(-infty)e^{iz(-infty)}=f(infty)e^{iz(infty)}=0$
Thanks in advance for your time and effort.
pde fourier-analysis fourier-transform fast-fourier-transform fourier-restriction
edited Jan 29 at 17:22
Rafa Budría
5,9371825
asked Jan 29 at 16:16
dmtridmtri
1,7802521
$endgroup$
add a comment |
$begingroup$
I have been recently studing Fourier transform and there is a proposition that says: If $lim limits_{x toinfty}xf(x)=limlimits_{{x}to -infty}xf(x)=0$ then $$hat{f'}(z)=-izhat{f}(z)$$
and this is proven like this:
$(f(x)e^{izx})'=f'(x)e^{izx}+izf(x)e^{izx}$
and intergrating by parts we have
$displaystyleint_{-infty}^{infty}(f(x)e^{izx})'dx=int_{-infty}^{infty}f'(x)e^{izx}dx+int_{-infty}^{infty}izf(x)e^{izx}dx$
or
$f(x)e^{izx}vert_{x=-infty}^{x=+infty}=hat{f'}(z)+izhat{f}(z)$
But
$f(x)e^{izx}vert_{x=-infty}^{x=+infty}=0$ , so we get the desired equality.
My quetion here is why we should suppose that $limlimits_{{x}to infty}xf(x)=limlimits_{{x}to -infty}xf(x)=0$ and not just that $$limlimits_{{x}to infty}f(x)=limlimits_{{x}to -infty}f(x)=0$$ ? Because then we would have $f(-infty)e^{iz(-infty)}=f(infty)e^{iz(infty)}=0$
Thanks in advance for your time and effort.
pde fourier-analysis fourier-transform fast-fourier-transform fourier-restriction
edited Jan 29 at 17:22
Rafa Budría
5,9371825
asked Jan 29 at 16:16
dmtridmtri
1,7802521
$endgroup$
1
$begingroup$
See the last half of this answer for a method to avoid integration to prove the theorem math.stackexchange.com/a/3088728/441161
$endgroup$
– Andy Walls
Jan 29 at 17:28
$begingroup$
@AndyWalls, Thanks for the refference (+1) but what I need is to tell me if my statement is correct or wrong not another way to prove it.
$endgroup$
– dmtri
Jan 29 at 19:40
add a comment |
$begingroup$
I have been recently studing Fourier transform and there is a proposition that says: If $lim limits_{x toinfty}xf(x)=limlimits_{{x}to -infty}xf(x)=0$ then $$hat{f'}(z)=-izhat{f}(z)$$
and this is proven like this:
$(f(x)e^{izx})'=f'(x)e^{izx}+izf(x)e^{izx}$
and intergrating by parts we have
$displaystyleint_{-infty}^{infty}(f(x)e^{izx})'dx=int_{-infty}^{infty}f'(x)e^{izx}dx+int_{-infty}^{infty}izf(x)e^{izx}dx$
or
$f(x)e^{izx}vert_{x=-infty}^{x=+infty}=hat{f'}(z)+izhat{f}(z)$
But
$f(x)e^{izx}vert_{x=-infty}^{x=+infty}=0$ , so we get the desired equality.
My quetion here is why we should suppose that $limlimits_{{x}to infty}xf(x)=limlimits_{{x}to -infty}xf(x)=0$ and not just that $$limlimits_{{x}to infty}f(x)=limlimits_{{x}to -infty}f(x)=0$$ ? Because then we would have $f(-infty)e^{iz(-infty)}=f(infty)e^{iz(infty)}=0$
Thanks in advance for your time and effort.
pde fourier-analysis fourier-transform fast-fourier-transform fourier-restriction
edited Jan 29 at 17:22
Rafa Budría
5,9371825
asked Jan 29 at 16:16
dmtridmtri
1,7802521
$endgroup$
I have been recently studing Fourier transform and there is a proposition that says: If $lim limits_{x toinfty}xf(x)=limlimits_{{x}to -infty}xf(x)=0$ then $$hat{f'}(z)=-izhat{f}(z)$$
and this is proven like this:
$(f(x)e^{izx})'=f'(x)e^{izx}+izf(x)e^{izx}$
and intergrating by parts we have
$displaystyleint_{-infty}^{infty}(f(x)e^{izx})'dx=int_{-infty}^{infty}f'(x)e^{izx}dx+int_{-infty}^{infty}izf(x)e^{izx}dx$
or
$f(x)e^{izx}vert_{x=-infty}^{x=+infty}=hat{f'}(z)+izhat{f}(z)$
But
$f(x)e^{izx}vert_{x=-infty}^{x=+infty}=0$ , so we get the desired equality.
My quetion here is why we should suppose that $limlimits_{{x}to infty}xf(x)=limlimits_{{x}to -infty}xf(x)=0$ and not just that $$limlimits_{{x}to infty}f(x)=limlimits_{{x}to -infty}f(x)=0$$ ? Because then we would have $f(-infty)e^{iz(-infty)}=f(infty)e^{iz(infty)}=0$
Thanks in advance for your time and effort.
pde fourier-analysis fourier-transform fast-fourier-transform fourier-restriction
pde fourier-analysis fourier-transform fast-fourier-transform fourier-restriction
edited Jan 29 at 17:22
Rafa Budría
5,9371825
asked Jan 29 at 16:16
dmtridmtri
1,7802521
edited Jan 29 at 17:22
Rafa Budría
5,9371825
asked Jan 29 at 16:16
dmtridmtri
1,7802521
edited Jan 29 at 17:22
Rafa Budría
5,9371825
edited Jan 29 at 17:22
Rafa Budría
5,9371825
edited Jan 29 at 17:22
Rafa Budría
5,9371825
5,9371825
asked Jan 29 at 16:16
dmtridmtri
1,7802521
asked Jan 29 at 16:16
dmtridmtri
1,7802521
asked Jan 29 at 16:16
dmtridmtri
1,7802521
1,7802521
1
$begingroup$
See the last half of this answer for a method to avoid integration to prove the theorem math.stackexchange.com/a/3088728/441161
$endgroup$
– Andy Walls
Jan 29 at 17:28
$begingroup$
@AndyWalls, Thanks for the refference (+1) but what I need is to tell me if my statement is correct or wrong not another way to prove it.
$endgroup$
– dmtri
Jan 29 at 19:40
add a comment |
1
$begingroup$
See the last half of this answer for a method to avoid integration to prove the theorem math.stackexchange.com/a/3088728/441161
$endgroup$
– Andy Walls
Jan 29 at 17:28
$begingroup$
@AndyWalls, Thanks for the refference (+1) but what I need is to tell me if my statement is correct or wrong not another way to prove it.
$endgroup$
– dmtri
Jan 29 at 19:40
1
1
$begingroup$
See the last half of this answer for a method to avoid integration to prove the theorem math.stackexchange.com/a/3088728/441161
$endgroup$
– Andy Walls
Jan 29 at 17:28
$begingroup$
See the last half of this answer for a method to avoid integration to prove the theorem math.stackexchange.com/a/3088728/441161
$endgroup$
– Andy Walls
Jan 29 at 17:28
$begingroup$
@AndyWalls, Thanks for the refference (+1) but what I need is to tell me if my statement is correct or wrong not another way to prove it.
$endgroup$
– dmtri
Jan 29 at 19:40
$begingroup$
@AndyWalls, Thanks for the refference (+1) but what I need is to tell me if my statement is correct or wrong not another way to prove it.
$endgroup$
– dmtri
Jan 29 at 19:40
add a comment |
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$begingroup$
See the last half of this answer for a method to avoid integration to prove the theorem math.stackexchange.com/a/3088728/441161
$endgroup$
– Andy Walls
Jan 29 at 17:28
$begingroup$
@AndyWalls, Thanks for the refference (+1) but what I need is to tell me if my statement is correct or wrong not another way to prove it.
$endgroup$
– dmtri
Jan 29 at 19:40