$F(x)=int_{-2}^{2} dy f(x,y)$ is an even function, is $G(x)=int_{-2}^{2} dy [f(x,y)]^2$ even?
up vote
3
down vote
favorite
I have a real valued function in two real variables $f(x,y)$ which is essentially a black box. The only thing I really know is that
$$
F(x)=int_{-2}^{2} f(x,y) dy
$$
is even and that
$$
int_{-infty}^{infty} F(x) dx=1
$$
Can I conclude that
$$
G(x)=int_{-2}^{2} left[f(x,y)right]^2 dy
$$
is also even? I looked for a counterexample, but couldn't find one. I intuitively feel like this should be true, but am struggling to make that more rigorous. Any ideas? Thanks!
calculus real-analysis multivariable-calculus even-and-odd-functions
asked Nov 20 at 0:11
bRost03
33819
add a comment |
up vote
3
down vote
favorite
I have a real valued function in two real variables $f(x,y)$ which is essentially a black box. The only thing I really know is that
$$
F(x)=int_{-2}^{2} f(x,y) dy
$$
is even and that
$$
int_{-infty}^{infty} F(x) dx=1
$$
Can I conclude that
$$
G(x)=int_{-2}^{2} left[f(x,y)right]^2 dy
$$
is also even? I looked for a counterexample, but couldn't find one. I intuitively feel like this should be true, but am struggling to make that more rigorous. Any ideas? Thanks!
calculus real-analysis multivariable-calculus even-and-odd-functions
asked Nov 20 at 0:11
bRost03
33819
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
I have a real valued function in two real variables $f(x,y)$ which is essentially a black box. The only thing I really know is that
$$
F(x)=int_{-2}^{2} f(x,y) dy
$$
is even and that
$$
int_{-infty}^{infty} F(x) dx=1
$$
Can I conclude that
$$
G(x)=int_{-2}^{2} left[f(x,y)right]^2 dy
$$
is also even? I looked for a counterexample, but couldn't find one. I intuitively feel like this should be true, but am struggling to make that more rigorous. Any ideas? Thanks!
calculus real-analysis multivariable-calculus even-and-odd-functions
asked Nov 20 at 0:11
bRost03
33819
I have a real valued function in two real variables $f(x,y)$ which is essentially a black box. The only thing I really know is that
$$
F(x)=int_{-2}^{2} f(x,y) dy
$$
is even and that
$$
int_{-infty}^{infty} F(x) dx=1
$$
Can I conclude that
$$
G(x)=int_{-2}^{2} left[f(x,y)right]^2 dy
$$
is also even? I looked for a counterexample, but couldn't find one. I intuitively feel like this should be true, but am struggling to make that more rigorous. Any ideas? Thanks!
calculus real-analysis multivariable-calculus even-and-odd-functions
calculus real-analysis multivariable-calculus even-and-odd-functions
asked Nov 20 at 0:11
bRost03
33819
asked Nov 20 at 0:11
bRost03
33819
edited Nov 20 at 2:18
asked Nov 20 at 0:11
bRost03
33819
asked Nov 20 at 0:11
bRost03
33819
asked Nov 20 at 0:11
bRost03
33819
33819
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
I asked the question but after doing some more work I have found a counterexample.
$$f(x,y)=Theta (x) left(x y^2-frac{4 x}{3}right)$$
Where $Theta(x)$ is the step function, then
$$
F(x)=int_{-2}^2 Theta (x) left(x y^2-frac{4 x}{3}right) , dy=0
$$
Is an even function, but
$$
G(x)=int_{-2}^2 left(Theta (x) left(x y^2-frac{4 x}{3}right)right)^2 , dy= frac{256 x^2 Theta (x)}{45}
$$
Is not. However I believe that if $f(x,y)=f(-x,pm y)$ then $G(x)=G(-x)$. I'd be glad to hear if anyone can give more general conditions on $f(x,y)$.
answered Nov 20 at 3:10
bRost03
33819
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
I asked the question but after doing some more work I have found a counterexample.
$$f(x,y)=Theta (x) left(x y^2-frac{4 x}{3}right)$$
Where $Theta(x)$ is the step function, then
$$
F(x)=int_{-2}^2 Theta (x) left(x y^2-frac{4 x}{3}right) , dy=0
$$
Is an even function, but
$$
G(x)=int_{-2}^2 left(Theta (x) left(x y^2-frac{4 x}{3}right)right)^2 , dy= frac{256 x^2 Theta (x)}{45}
$$
Is not. However I believe that if $f(x,y)=f(-x,pm y)$ then $G(x)=G(-x)$. I'd be glad to hear if anyone can give more general conditions on $f(x,y)$.
answered Nov 20 at 3:10
bRost03
33819
add a comment |
up vote
0
down vote
I asked the question but after doing some more work I have found a counterexample.
$$f(x,y)=Theta (x) left(x y^2-frac{4 x}{3}right)$$
Where $Theta(x)$ is the step function, then
$$
F(x)=int_{-2}^2 Theta (x) left(x y^2-frac{4 x}{3}right) , dy=0
$$
Is an even function, but
$$
G(x)=int_{-2}^2 left(Theta (x) left(x y^2-frac{4 x}{3}right)right)^2 , dy= frac{256 x^2 Theta (x)}{45}
$$
Is not. However I believe that if $f(x,y)=f(-x,pm y)$ then $G(x)=G(-x)$. I'd be glad to hear if anyone can give more general conditions on $f(x,y)$.
answered Nov 20 at 3:10
bRost03
33819
add a comment |
up vote
0
down vote
up vote
0
down vote
I asked the question but after doing some more work I have found a counterexample.
$$f(x,y)=Theta (x) left(x y^2-frac{4 x}{3}right)$$
Where $Theta(x)$ is the step function, then
$$
F(x)=int_{-2}^2 Theta (x) left(x y^2-frac{4 x}{3}right) , dy=0
$$
Is an even function, but
$$
G(x)=int_{-2}^2 left(Theta (x) left(x y^2-frac{4 x}{3}right)right)^2 , dy= frac{256 x^2 Theta (x)}{45}
$$
Is not. However I believe that if $f(x,y)=f(-x,pm y)$ then $G(x)=G(-x)$. I'd be glad to hear if anyone can give more general conditions on $f(x,y)$.
answered Nov 20 at 3:10
bRost03
33819
I asked the question but after doing some more work I have found a counterexample.
$$f(x,y)=Theta (x) left(x y^2-frac{4 x}{3}right)$$
Where $Theta(x)$ is the step function, then
$$
F(x)=int_{-2}^2 Theta (x) left(x y^2-frac{4 x}{3}right) , dy=0
$$
Is an even function, but
$$
G(x)=int_{-2}^2 left(Theta (x) left(x y^2-frac{4 x}{3}right)right)^2 , dy= frac{256 x^2 Theta (x)}{45}
$$
Is not. However I believe that if $f(x,y)=f(-x,pm y)$ then $G(x)=G(-x)$. I'd be glad to hear if anyone can give more general conditions on $f(x,y)$.
answered Nov 20 at 3:10
bRost03
33819
answered Nov 20 at 3:10
bRost03
33819
answered Nov 20 at 3:10
bRost03
33819
answered Nov 20 at 3:10
bRost03
33819
33819
add a comment |
add a comment |
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3005731%2ffx-int-22-dy-fx-y-is-an-even-function-is-gx-int-22-dy-f%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
