$F(x)=int_{-2}^{2} dy f(x,y)$ 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!










share|cite|improve this question





























    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






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 20 at 2:18

























      asked Nov 20 at 0:11









      bRost03

      34319




      34319






















          1 Answer
          1






          active

          oldest

          votes


















          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





















            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            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

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            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






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.





                    Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


                    Please pay close attention to the following guidance:


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    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





















































                    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







                    Popular posts from this blog

                    How to change which sound is reproduced for terminal bell?

                    Can I use Tabulator js library in my java Spring + Thymeleaf project?

                    Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents