Exercise about convolution of functions
$begingroup$
I have found this excercise in theory of convolution (I started it the last week). I have been thinking about it for two days but I don't get solve it:
Let be $1<p<2<q<infty$ and $f:mathbb{R^2}rightarrow{mathbb{R}}$ $fin{L^p(mathbb{R^2})}textrm{ and }{L^qmathbb{(R^2)}}$ prove that :
$$g(y,z)=displaystylefrac{(-z,y)}{2pisqrt{y^2+z^2}}*f; in{L^infty(mathbb{R^2})}$$
Where $*$ denotes covolution of two functions.
Edit
I want to show that the following function is in $L^infty(mathbb{R^2})$
$$int_{mathbb{R^2}}f(x_1-z_1,x_2-z_2)frac{(-z_2,z_1)}{sqrt{z_1^2+z_2^2}}dz_1dz_2$$
I would appreciate if someone help me.
Thanks.
real-analysis functional-analysis convolution
$endgroup$
|
show 7 more comments
$begingroup$
I have found this excercise in theory of convolution (I started it the last week). I have been thinking about it for two days but I don't get solve it:
Let be $1<p<2<q<infty$ and $f:mathbb{R^2}rightarrow{mathbb{R}}$ $fin{L^p(mathbb{R^2})}textrm{ and }{L^qmathbb{(R^2)}}$ prove that :
$$g(y,z)=displaystylefrac{(-z,y)}{2pisqrt{y^2+z^2}}*f; in{L^infty(mathbb{R^2})}$$
Where $*$ denotes covolution of two functions.
Edit
I want to show that the following function is in $L^infty(mathbb{R^2})$
$$int_{mathbb{R^2}}f(x_1-z_1,x_2-z_2)frac{(-z_2,z_1)}{sqrt{z_1^2+z_2^2}}dz_1dz_2$$
I would appreciate if someone help me.
Thanks.
real-analysis functional-analysis convolution
$endgroup$
3
$begingroup$
This question is ill-posed. You use $(y, z)$ as arguments of $g$ but also in the convolution, it is like writing $f(x)=int g(x), dx$, the $x$ is at the same time a true variable and a dummy one. What exactly do you mean by that convolution? Write it as an integral, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:20
$begingroup$
I wan to say $f*g(x)=int_{mathbb{R^2}}f(x-z)g(z)dz$. In this exercise we have to see that the following function is in $L^infty $: $int_{mathbb{R^2}}f((x_1-z_1,x_2-z_2))frac{(-z_2,z_1)}{sqrt{z_1^2+z_2^2}}dz_1dz_2$
$endgroup$
– mathlife
Dec 9 '18 at 11:53
2
$begingroup$
Yeah, please, edit your post. This edit will improve it and raise the odds that it gets a good answer. Make sure that the integral be dimensionally consistent; I see a vector there, is there a scalar product? Or is it a vector-valued integral? Edit the post and explain all these details, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:58
$begingroup$
My book, where I found this excersice doesn't give more details about this. I suppose that it follows the definition of convolution in two dimensions
$endgroup$
– mathlife
Dec 9 '18 at 12:20
$begingroup$
This reminds me of the Biot-Savart law.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 17:11
|
show 7 more comments
$begingroup$
I have found this excercise in theory of convolution (I started it the last week). I have been thinking about it for two days but I don't get solve it:
Let be $1<p<2<q<infty$ and $f:mathbb{R^2}rightarrow{mathbb{R}}$ $fin{L^p(mathbb{R^2})}textrm{ and }{L^qmathbb{(R^2)}}$ prove that :
$$g(y,z)=displaystylefrac{(-z,y)}{2pisqrt{y^2+z^2}}*f; in{L^infty(mathbb{R^2})}$$
Where $*$ denotes covolution of two functions.
Edit
I want to show that the following function is in $L^infty(mathbb{R^2})$
$$int_{mathbb{R^2}}f(x_1-z_1,x_2-z_2)frac{(-z_2,z_1)}{sqrt{z_1^2+z_2^2}}dz_1dz_2$$
I would appreciate if someone help me.
Thanks.
real-analysis functional-analysis convolution
$endgroup$
I have found this excercise in theory of convolution (I started it the last week). I have been thinking about it for two days but I don't get solve it:
Let be $1<p<2<q<infty$ and $f:mathbb{R^2}rightarrow{mathbb{R}}$ $fin{L^p(mathbb{R^2})}textrm{ and }{L^qmathbb{(R^2)}}$ prove that :
$$g(y,z)=displaystylefrac{(-z,y)}{2pisqrt{y^2+z^2}}*f; in{L^infty(mathbb{R^2})}$$
Where $*$ denotes covolution of two functions.
Edit
I want to show that the following function is in $L^infty(mathbb{R^2})$
$$int_{mathbb{R^2}}f(x_1-z_1,x_2-z_2)frac{(-z_2,z_1)}{sqrt{z_1^2+z_2^2}}dz_1dz_2$$
I would appreciate if someone help me.
Thanks.
real-analysis functional-analysis convolution
real-analysis functional-analysis convolution
edited Dec 13 '18 at 18:28
mathlife
asked Dec 6 '18 at 16:29
mathlifemathlife
629
629
3
$begingroup$
This question is ill-posed. You use $(y, z)$ as arguments of $g$ but also in the convolution, it is like writing $f(x)=int g(x), dx$, the $x$ is at the same time a true variable and a dummy one. What exactly do you mean by that convolution? Write it as an integral, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:20
$begingroup$
I wan to say $f*g(x)=int_{mathbb{R^2}}f(x-z)g(z)dz$. In this exercise we have to see that the following function is in $L^infty $: $int_{mathbb{R^2}}f((x_1-z_1,x_2-z_2))frac{(-z_2,z_1)}{sqrt{z_1^2+z_2^2}}dz_1dz_2$
$endgroup$
– mathlife
Dec 9 '18 at 11:53
2
$begingroup$
Yeah, please, edit your post. This edit will improve it and raise the odds that it gets a good answer. Make sure that the integral be dimensionally consistent; I see a vector there, is there a scalar product? Or is it a vector-valued integral? Edit the post and explain all these details, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:58
$begingroup$
My book, where I found this excersice doesn't give more details about this. I suppose that it follows the definition of convolution in two dimensions
$endgroup$
– mathlife
Dec 9 '18 at 12:20
$begingroup$
This reminds me of the Biot-Savart law.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 17:11
|
show 7 more comments
3
$begingroup$
This question is ill-posed. You use $(y, z)$ as arguments of $g$ but also in the convolution, it is like writing $f(x)=int g(x), dx$, the $x$ is at the same time a true variable and a dummy one. What exactly do you mean by that convolution? Write it as an integral, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:20
$begingroup$
I wan to say $f*g(x)=int_{mathbb{R^2}}f(x-z)g(z)dz$. In this exercise we have to see that the following function is in $L^infty $: $int_{mathbb{R^2}}f((x_1-z_1,x_2-z_2))frac{(-z_2,z_1)}{sqrt{z_1^2+z_2^2}}dz_1dz_2$
$endgroup$
– mathlife
Dec 9 '18 at 11:53
2
$begingroup$
Yeah, please, edit your post. This edit will improve it and raise the odds that it gets a good answer. Make sure that the integral be dimensionally consistent; I see a vector there, is there a scalar product? Or is it a vector-valued integral? Edit the post and explain all these details, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:58
$begingroup$
My book, where I found this excersice doesn't give more details about this. I suppose that it follows the definition of convolution in two dimensions
$endgroup$
– mathlife
Dec 9 '18 at 12:20
$begingroup$
This reminds me of the Biot-Savart law.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 17:11
3
3
$begingroup$
This question is ill-posed. You use $(y, z)$ as arguments of $g$ but also in the convolution, it is like writing $f(x)=int g(x), dx$, the $x$ is at the same time a true variable and a dummy one. What exactly do you mean by that convolution? Write it as an integral, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:20
$begingroup$
This question is ill-posed. You use $(y, z)$ as arguments of $g$ but also in the convolution, it is like writing $f(x)=int g(x), dx$, the $x$ is at the same time a true variable and a dummy one. What exactly do you mean by that convolution? Write it as an integral, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:20
$begingroup$
I wan to say $f*g(x)=int_{mathbb{R^2}}f(x-z)g(z)dz$. In this exercise we have to see that the following function is in $L^infty $: $int_{mathbb{R^2}}f((x_1-z_1,x_2-z_2))frac{(-z_2,z_1)}{sqrt{z_1^2+z_2^2}}dz_1dz_2$
$endgroup$
– mathlife
Dec 9 '18 at 11:53
$begingroup$
I wan to say $f*g(x)=int_{mathbb{R^2}}f(x-z)g(z)dz$. In this exercise we have to see that the following function is in $L^infty $: $int_{mathbb{R^2}}f((x_1-z_1,x_2-z_2))frac{(-z_2,z_1)}{sqrt{z_1^2+z_2^2}}dz_1dz_2$
$endgroup$
– mathlife
Dec 9 '18 at 11:53
2
2
$begingroup$
Yeah, please, edit your post. This edit will improve it and raise the odds that it gets a good answer. Make sure that the integral be dimensionally consistent; I see a vector there, is there a scalar product? Or is it a vector-valued integral? Edit the post and explain all these details, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:58
$begingroup$
Yeah, please, edit your post. This edit will improve it and raise the odds that it gets a good answer. Make sure that the integral be dimensionally consistent; I see a vector there, is there a scalar product? Or is it a vector-valued integral? Edit the post and explain all these details, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:58
$begingroup$
My book, where I found this excersice doesn't give more details about this. I suppose that it follows the definition of convolution in two dimensions
$endgroup$
– mathlife
Dec 9 '18 at 12:20
$begingroup$
My book, where I found this excersice doesn't give more details about this. I suppose that it follows the definition of convolution in two dimensions
$endgroup$
– mathlife
Dec 9 '18 at 12:20
$begingroup$
This reminds me of the Biot-Savart law.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 17:11
$begingroup$
This reminds me of the Biot-Savart law.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 17:11
|
show 7 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Take a positive function such that $f(x) sim |x|^{-alpha}$ near $0$ and $f(x) sim |x|^{-beta}$ near $infty$ with $0<alpha<beta<2$.
Then $fin L^pcap L^q(mathbb R^2)$ iff $alpha<frac2q<frac2p<beta$. In particular, $fnotin L^1$, so the integral
$$
int_{mathbb R^2} frac{y_1}{|y|} f(x-y) ,dy
$$
is not even defined in the Lebesgue sense.
$endgroup$
1
$begingroup$
Perhaps simpler: Let $U$ be the exterior of the unit disc, and set $f(y) = |y|^{-2}chi_U(y).$ (This $fin L^p$ for all $p>1.$)
$endgroup$
– zhw.
Dec 15 '18 at 19:29
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Take a positive function such that $f(x) sim |x|^{-alpha}$ near $0$ and $f(x) sim |x|^{-beta}$ near $infty$ with $0<alpha<beta<2$.
Then $fin L^pcap L^q(mathbb R^2)$ iff $alpha<frac2q<frac2p<beta$. In particular, $fnotin L^1$, so the integral
$$
int_{mathbb R^2} frac{y_1}{|y|} f(x-y) ,dy
$$
is not even defined in the Lebesgue sense.
$endgroup$
1
$begingroup$
Perhaps simpler: Let $U$ be the exterior of the unit disc, and set $f(y) = |y|^{-2}chi_U(y).$ (This $fin L^p$ for all $p>1.$)
$endgroup$
– zhw.
Dec 15 '18 at 19:29
add a comment |
$begingroup$
Take a positive function such that $f(x) sim |x|^{-alpha}$ near $0$ and $f(x) sim |x|^{-beta}$ near $infty$ with $0<alpha<beta<2$.
Then $fin L^pcap L^q(mathbb R^2)$ iff $alpha<frac2q<frac2p<beta$. In particular, $fnotin L^1$, so the integral
$$
int_{mathbb R^2} frac{y_1}{|y|} f(x-y) ,dy
$$
is not even defined in the Lebesgue sense.
$endgroup$
1
$begingroup$
Perhaps simpler: Let $U$ be the exterior of the unit disc, and set $f(y) = |y|^{-2}chi_U(y).$ (This $fin L^p$ for all $p>1.$)
$endgroup$
– zhw.
Dec 15 '18 at 19:29
add a comment |
$begingroup$
Take a positive function such that $f(x) sim |x|^{-alpha}$ near $0$ and $f(x) sim |x|^{-beta}$ near $infty$ with $0<alpha<beta<2$.
Then $fin L^pcap L^q(mathbb R^2)$ iff $alpha<frac2q<frac2p<beta$. In particular, $fnotin L^1$, so the integral
$$
int_{mathbb R^2} frac{y_1}{|y|} f(x-y) ,dy
$$
is not even defined in the Lebesgue sense.
$endgroup$
Take a positive function such that $f(x) sim |x|^{-alpha}$ near $0$ and $f(x) sim |x|^{-beta}$ near $infty$ with $0<alpha<beta<2$.
Then $fin L^pcap L^q(mathbb R^2)$ iff $alpha<frac2q<frac2p<beta$. In particular, $fnotin L^1$, so the integral
$$
int_{mathbb R^2} frac{y_1}{|y|} f(x-y) ,dy
$$
is not even defined in the Lebesgue sense.
answered Dec 13 '18 at 19:19
FedericoFederico
4,954514
4,954514
1
$begingroup$
Perhaps simpler: Let $U$ be the exterior of the unit disc, and set $f(y) = |y|^{-2}chi_U(y).$ (This $fin L^p$ for all $p>1.$)
$endgroup$
– zhw.
Dec 15 '18 at 19:29
add a comment |
1
$begingroup$
Perhaps simpler: Let $U$ be the exterior of the unit disc, and set $f(y) = |y|^{-2}chi_U(y).$ (This $fin L^p$ for all $p>1.$)
$endgroup$
– zhw.
Dec 15 '18 at 19:29
1
1
$begingroup$
Perhaps simpler: Let $U$ be the exterior of the unit disc, and set $f(y) = |y|^{-2}chi_U(y).$ (This $fin L^p$ for all $p>1.$)
$endgroup$
– zhw.
Dec 15 '18 at 19:29
$begingroup$
Perhaps simpler: Let $U$ be the exterior of the unit disc, and set $f(y) = |y|^{-2}chi_U(y).$ (This $fin L^p$ for all $p>1.$)
$endgroup$
– zhw.
Dec 15 '18 at 19:29
add a comment |
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3
$begingroup$
This question is ill-posed. You use $(y, z)$ as arguments of $g$ but also in the convolution, it is like writing $f(x)=int g(x), dx$, the $x$ is at the same time a true variable and a dummy one. What exactly do you mean by that convolution? Write it as an integral, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:20
$begingroup$
I wan to say $f*g(x)=int_{mathbb{R^2}}f(x-z)g(z)dz$. In this exercise we have to see that the following function is in $L^infty $: $int_{mathbb{R^2}}f((x_1-z_1,x_2-z_2))frac{(-z_2,z_1)}{sqrt{z_1^2+z_2^2}}dz_1dz_2$
$endgroup$
– mathlife
Dec 9 '18 at 11:53
2
$begingroup$
Yeah, please, edit your post. This edit will improve it and raise the odds that it gets a good answer. Make sure that the integral be dimensionally consistent; I see a vector there, is there a scalar product? Or is it a vector-valued integral? Edit the post and explain all these details, please.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 11:58
$begingroup$
My book, where I found this excersice doesn't give more details about this. I suppose that it follows the definition of convolution in two dimensions
$endgroup$
– mathlife
Dec 9 '18 at 12:20
$begingroup$
This reminds me of the Biot-Savart law.
$endgroup$
– Giuseppe Negro
Dec 9 '18 at 17:11