Is set of all rational linear combination of function sin nx and cos mx $m,nin mathbb N cup {0}$?












1












$begingroup$



Is set of all rational linear combination of function sin nx and cos mx $m,nin mathbb N cup {0}$ is subring of ring of all function from $[0,1]to mathbb R $?




My attempt



$sin x:[0,2 pi]to[-1,1]$: is surjective function.



To show subring we have to show that closed under subtraction and multiplication.



Take $U_1=k_1 sin k_2x+k_3 cos k_4 x , U_2=sin k_6x+k_7 cos k_8x $



Now problem I incurred how to show $U_1-U_2in A$ and $U_1U_2in A$ where A is given subset



Please give me hint










share|cite|improve this question











$endgroup$












  • $begingroup$
    To produce ${$ you can use {, and similarly for $}$ you can use }.
    $endgroup$
    – Asaf Karagila
    Jan 12 at 9:41
















1












$begingroup$



Is set of all rational linear combination of function sin nx and cos mx $m,nin mathbb N cup {0}$ is subring of ring of all function from $[0,1]to mathbb R $?




My attempt



$sin x:[0,2 pi]to[-1,1]$: is surjective function.



To show subring we have to show that closed under subtraction and multiplication.



Take $U_1=k_1 sin k_2x+k_3 cos k_4 x , U_2=sin k_6x+k_7 cos k_8x $



Now problem I incurred how to show $U_1-U_2in A$ and $U_1U_2in A$ where A is given subset



Please give me hint










share|cite|improve this question











$endgroup$












  • $begingroup$
    To produce ${$ you can use {, and similarly for $}$ you can use }.
    $endgroup$
    – Asaf Karagila
    Jan 12 at 9:41














1












1








1





$begingroup$



Is set of all rational linear combination of function sin nx and cos mx $m,nin mathbb N cup {0}$ is subring of ring of all function from $[0,1]to mathbb R $?




My attempt



$sin x:[0,2 pi]to[-1,1]$: is surjective function.



To show subring we have to show that closed under subtraction and multiplication.



Take $U_1=k_1 sin k_2x+k_3 cos k_4 x , U_2=sin k_6x+k_7 cos k_8x $



Now problem I incurred how to show $U_1-U_2in A$ and $U_1U_2in A$ where A is given subset



Please give me hint










share|cite|improve this question











$endgroup$





Is set of all rational linear combination of function sin nx and cos mx $m,nin mathbb N cup {0}$ is subring of ring of all function from $[0,1]to mathbb R $?




My attempt



$sin x:[0,2 pi]to[-1,1]$: is surjective function.



To show subring we have to show that closed under subtraction and multiplication.



Take $U_1=k_1 sin k_2x+k_3 cos k_4 x , U_2=sin k_6x+k_7 cos k_8x $



Now problem I incurred how to show $U_1-U_2in A$ and $U_1U_2in A$ where A is given subset



Please give me hint







abstract-algebra ring-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 12 at 10:44









Fabio Lucchini

8,85811426




8,85811426










asked Jan 12 at 5:41









MathLoverMathLover

53710




53710












  • $begingroup$
    To produce ${$ you can use {, and similarly for $}$ you can use }.
    $endgroup$
    – Asaf Karagila
    Jan 12 at 9:41


















  • $begingroup$
    To produce ${$ you can use {, and similarly for $}$ you can use }.
    $endgroup$
    – Asaf Karagila
    Jan 12 at 9:41
















$begingroup$
To produce ${$ you can use {, and similarly for $}$ you can use }.
$endgroup$
– Asaf Karagila
Jan 12 at 9:41




$begingroup$
To produce ${$ you can use {, and similarly for $}$ you can use }.
$endgroup$
– Asaf Karagila
Jan 12 at 9:41










1 Answer
1






active

oldest

votes


















1












$begingroup$

To get a ring, let $A $ the set of
$$sum_{i=1}^r (a_isin (n_ix)+b_icos (m_ix)) $$
with $a_i,b_iinBbb Q $ and $n_i,m_iinBbb N$.
Then $A $ is closed respect to sum.
Closuresness respect to product follows by Werner formulas.






share|cite|improve this answer











$endgroup$













    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%2f3070632%2fis-set-of-all-rational-linear-combination-of-function-sin-nx-and-cos-mx-m-n-in%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









    1












    $begingroup$

    To get a ring, let $A $ the set of
    $$sum_{i=1}^r (a_isin (n_ix)+b_icos (m_ix)) $$
    with $a_i,b_iinBbb Q $ and $n_i,m_iinBbb N$.
    Then $A $ is closed respect to sum.
    Closuresness respect to product follows by Werner formulas.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      To get a ring, let $A $ the set of
      $$sum_{i=1}^r (a_isin (n_ix)+b_icos (m_ix)) $$
      with $a_i,b_iinBbb Q $ and $n_i,m_iinBbb N$.
      Then $A $ is closed respect to sum.
      Closuresness respect to product follows by Werner formulas.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        To get a ring, let $A $ the set of
        $$sum_{i=1}^r (a_isin (n_ix)+b_icos (m_ix)) $$
        with $a_i,b_iinBbb Q $ and $n_i,m_iinBbb N$.
        Then $A $ is closed respect to sum.
        Closuresness respect to product follows by Werner formulas.






        share|cite|improve this answer











        $endgroup$



        To get a ring, let $A $ the set of
        $$sum_{i=1}^r (a_isin (n_ix)+b_icos (m_ix)) $$
        with $a_i,b_iinBbb Q $ and $n_i,m_iinBbb N$.
        Then $A $ is closed respect to sum.
        Closuresness respect to product follows by Werner formulas.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 12 at 10:45

























        answered Jan 12 at 8:20









        Fabio LucchiniFabio Lucchini

        8,85811426




        8,85811426






























            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.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3070632%2fis-set-of-all-rational-linear-combination-of-function-sin-nx-and-cos-mx-m-n-in%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

            Human spaceflight

            Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

            張江高科駅