$F(x)=int_{-2}^{2} dy f(x,y)$ is an even function, is $G(x)=int_{-2}^{2} dy [f(x,y)]^2$ even?












3














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!










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    3














    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!










    share|cite|improve this question



























      3












      3








      3


      2





      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!










      share|cite|improve this question















      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






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      edited Nov 20 at 2:18

























      asked Nov 20 at 0:11









      bRost03

      34319




      34319






















          1 Answer
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          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 $f(x,y)$.






          share|cite|improve this answer





















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            1 Answer
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            0














            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 $f(x,y)$.






            share|cite|improve this answer


























              0














              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 $f(x,y)$.






              share|cite|improve this answer
























                0












                0








                0






                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 $f(x,y)$.






                share|cite|improve this answer












                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 $f(x,y)$.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Nov 20 at 3:10









                bRost03

                34319




                34319






























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