Error analysis of approximating Fourier transforms












2












$begingroup$


Consider the problem of computing the Fourier transform of a function, $f(x).$ $$ hat{f}(k) = int_{-infty}^{infty} dx~ f(x)~ e^{i k x} .$$



Suppose I want to approximate this transform by a discrete, truncated version,
$$ hat{F}_{Delta, ~L}(k) = sum_{n = -L}^L Delta~ f(Delta n)~ e^{i k Delta n} .$$



I want the approximation to work with error $epsilon$ in some interval $[0, k_{max}]$,



$$sup_{k in [0, k_{max}]} ~| hat{f}(k) - hat{F}_{Delta, ~L}(k) | leq epsilon $$



What values of $Delta$, $L$ should I choose to achieve this error. The answer will obviously depend on the properties of $hat{f}(k)$, like how fast it decays. I have seen this done for specific functions but haven't seen any general rigorous result. Since this seems to be a problem with many practical applications it seems unlikely that no one has worked it out. Are there any rigorous results known for this general case ?






fourier-analysis fourier-transform approximation-theory approximate-integration







share|cite|improve this question









$endgroup$

















    2












    $begingroup$


    Consider the problem of computing the Fourier transform of a function, $f(x).$ $$ hat{f}(k) = int_{-infty}^{infty} dx~ f(x)~ e^{i k x} .$$



    Suppose I want to approximate this transform by a discrete, truncated version,
    $$ hat{F}_{Delta, ~L}(k) = sum_{n = -L}^L Delta~ f(Delta n)~ e^{i k Delta n} .$$



    I want the approximation to work with error $epsilon$ in some interval $[0, k_{max}]$,



    $$sup_{k in [0, k_{max}]} ~| hat{f}(k) - hat{F}_{Delta, ~L}(k) | leq epsilon $$



    What values of $Delta$, $L$ should I choose to achieve this error. The answer will obviously depend on the properties of $hat{f}(k)$, like how fast it decays. I have seen this done for specific functions but haven't seen any general rigorous result. Since this seems to be a problem with many practical applications it seems unlikely that no one has worked it out. Are there any rigorous results known for this general case ?






    fourier-analysis fourier-transform approximation-theory approximate-integration







    share|cite|improve this question









    $endgroup$















      2












      2








      2





      $begingroup$


      Consider the problem of computing the Fourier transform of a function, $f(x).$ $$ hat{f}(k) = int_{-infty}^{infty} dx~ f(x)~ e^{i k x} .$$



      Suppose I want to approximate this transform by a discrete, truncated version,
      $$ hat{F}_{Delta, ~L}(k) = sum_{n = -L}^L Delta~ f(Delta n)~ e^{i k Delta n} .$$



      I want the approximation to work with error $epsilon$ in some interval $[0, k_{max}]$,



      $$sup_{k in [0, k_{max}]} ~| hat{f}(k) - hat{F}_{Delta, ~L}(k) | leq epsilon $$



      What values of $Delta$, $L$ should I choose to achieve this error. The answer will obviously depend on the properties of $hat{f}(k)$, like how fast it decays. I have seen this done for specific functions but haven't seen any general rigorous result. Since this seems to be a problem with many practical applications it seems unlikely that no one has worked it out. Are there any rigorous results known for this general case ?






      fourier-analysis fourier-transform approximation-theory approximate-integration







      share|cite|improve this question









      $endgroup$




      Consider the problem of computing the Fourier transform of a function, $f(x).$ $$ hat{f}(k) = int_{-infty}^{infty} dx~ f(x)~ e^{i k x} .$$



      Suppose I want to approximate this transform by a discrete, truncated version,
      $$ hat{F}_{Delta, ~L}(k) = sum_{n = -L}^L Delta~ f(Delta n)~ e^{i k Delta n} .$$



      I want the approximation to work with error $epsilon$ in some interval $[0, k_{max}]$,



      $$sup_{k in [0, k_{max}]} ~| hat{f}(k) - hat{F}_{Delta, ~L}(k) | leq epsilon $$



      What values of $Delta$, $L$ should I choose to achieve this error. The answer will obviously depend on the properties of $hat{f}(k)$, like how fast it decays. I have seen this done for specific functions but haven't seen any general rigorous result. Since this seems to be a problem with many practical applications it seems unlikely that no one has worked it out. Are there any rigorous results known for this general case ?






      fourier-analysis fourier-transform approximation-theory approximate-integration




      fourier-analysis fourier-transform approximation-theory approximate-integration






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 18 at 8:14









      biryanibiryani

      268413




      268413






















          1 Answer
          1






          active

          oldest

          votes


















          2












          $begingroup$

          If you assume that $f in L^1$, then approximations are not so terrible because $hat{f}$ is uniformly continuous, with $|hat{f}|_{infty}le |f|_{1}$. So, for $epsilon > 0$, there exists $R > 0$ large enough such that
          $$
          left|hat{f}(k)-frac{1}{sqrt{2pi}}int_{-R}^{R}f(x)e^{-ikx}dxright| \
          le frac{1}{sqrt{2pi}}int_{|u|ge R}|f(u)|du < frac{epsilon}{2},;;; kinmathbb{R}.
          $$
          So the Fourier transform is uniformly approximated by the truncated Fourier series integral on $[-R,R]$. Then you can approximate the Fourier integral over $[-R,R]$ by a discrete sum
          $$
          hat{f}(k)approxsum_{n=-N}^{N-1}frac{1}{sqrt{2pi}}int_{Rn/N}^{R(n+1)/N}f(x)dxe^{-ikRn/N}.
          $$
          I'm not sure if that's the type of approximation you want or not.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So you assume $f$ is continuous ($f(x)=e^{-x^2}(1 - 1_{xin mathbb{Q}})$)
            $endgroup$
            – reuns
            Jan 20 at 2:24












          • $begingroup$
            This is the kind of approximation I use. The question is how does $epsilon$ depend on $R$ and $N$.
            $endgroup$
            – biryani
            Jan 21 at 10:39











          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%2f3077962%2ferror-analysis-of-approximating-fourier-transforms%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









          2












          $begingroup$

          If you assume that $f in L^1$, then approximations are not so terrible because $hat{f}$ is uniformly continuous, with $|hat{f}|_{infty}le |f|_{1}$. So, for $epsilon > 0$, there exists $R > 0$ large enough such that
          $$
          left|hat{f}(k)-frac{1}{sqrt{2pi}}int_{-R}^{R}f(x)e^{-ikx}dxright| \
          le frac{1}{sqrt{2pi}}int_{|u|ge R}|f(u)|du < frac{epsilon}{2},;;; kinmathbb{R}.
          $$
          So the Fourier transform is uniformly approximated by the truncated Fourier series integral on $[-R,R]$. Then you can approximate the Fourier integral over $[-R,R]$ by a discrete sum
          $$
          hat{f}(k)approxsum_{n=-N}^{N-1}frac{1}{sqrt{2pi}}int_{Rn/N}^{R(n+1)/N}f(x)dxe^{-ikRn/N}.
          $$
          I'm not sure if that's the type of approximation you want or not.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So you assume $f$ is continuous ($f(x)=e^{-x^2}(1 - 1_{xin mathbb{Q}})$)
            $endgroup$
            – reuns
            Jan 20 at 2:24












          • $begingroup$
            This is the kind of approximation I use. The question is how does $epsilon$ depend on $R$ and $N$.
            $endgroup$
            – biryani
            Jan 21 at 10:39
















          2












          $begingroup$

          If you assume that $f in L^1$, then approximations are not so terrible because $hat{f}$ is uniformly continuous, with $|hat{f}|_{infty}le |f|_{1}$. So, for $epsilon > 0$, there exists $R > 0$ large enough such that
          $$
          left|hat{f}(k)-frac{1}{sqrt{2pi}}int_{-R}^{R}f(x)e^{-ikx}dxright| \
          le frac{1}{sqrt{2pi}}int_{|u|ge R}|f(u)|du < frac{epsilon}{2},;;; kinmathbb{R}.
          $$
          So the Fourier transform is uniformly approximated by the truncated Fourier series integral on $[-R,R]$. Then you can approximate the Fourier integral over $[-R,R]$ by a discrete sum
          $$
          hat{f}(k)approxsum_{n=-N}^{N-1}frac{1}{sqrt{2pi}}int_{Rn/N}^{R(n+1)/N}f(x)dxe^{-ikRn/N}.
          $$
          I'm not sure if that's the type of approximation you want or not.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            So you assume $f$ is continuous ($f(x)=e^{-x^2}(1 - 1_{xin mathbb{Q}})$)
            $endgroup$
            – reuns
            Jan 20 at 2:24












          • $begingroup$
            This is the kind of approximation I use. The question is how does $epsilon$ depend on $R$ and $N$.
            $endgroup$
            – biryani
            Jan 21 at 10:39














          2












          2








          2





          $begingroup$

          If you assume that $f in L^1$, then approximations are not so terrible because $hat{f}$ is uniformly continuous, with $|hat{f}|_{infty}le |f|_{1}$. So, for $epsilon > 0$, there exists $R > 0$ large enough such that
          $$
          left|hat{f}(k)-frac{1}{sqrt{2pi}}int_{-R}^{R}f(x)e^{-ikx}dxright| \
          le frac{1}{sqrt{2pi}}int_{|u|ge R}|f(u)|du < frac{epsilon}{2},;;; kinmathbb{R}.
          $$
          So the Fourier transform is uniformly approximated by the truncated Fourier series integral on $[-R,R]$. Then you can approximate the Fourier integral over $[-R,R]$ by a discrete sum
          $$
          hat{f}(k)approxsum_{n=-N}^{N-1}frac{1}{sqrt{2pi}}int_{Rn/N}^{R(n+1)/N}f(x)dxe^{-ikRn/N}.
          $$
          I'm not sure if that's the type of approximation you want or not.






          share|cite|improve this answer









          $endgroup$



          If you assume that $f in L^1$, then approximations are not so terrible because $hat{f}$ is uniformly continuous, with $|hat{f}|_{infty}le |f|_{1}$. So, for $epsilon > 0$, there exists $R > 0$ large enough such that
          $$
          left|hat{f}(k)-frac{1}{sqrt{2pi}}int_{-R}^{R}f(x)e^{-ikx}dxright| \
          le frac{1}{sqrt{2pi}}int_{|u|ge R}|f(u)|du < frac{epsilon}{2},;;; kinmathbb{R}.
          $$
          So the Fourier transform is uniformly approximated by the truncated Fourier series integral on $[-R,R]$. Then you can approximate the Fourier integral over $[-R,R]$ by a discrete sum
          $$
          hat{f}(k)approxsum_{n=-N}^{N-1}frac{1}{sqrt{2pi}}int_{Rn/N}^{R(n+1)/N}f(x)dxe^{-ikRn/N}.
          $$
          I'm not sure if that's the type of approximation you want or not.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 20 at 0:38









          DisintegratingByPartsDisintegratingByParts

          59.5k42580




          59.5k42580












          • $begingroup$
            So you assume $f$ is continuous ($f(x)=e^{-x^2}(1 - 1_{xin mathbb{Q}})$)
            $endgroup$
            – reuns
            Jan 20 at 2:24












          • $begingroup$
            This is the kind of approximation I use. The question is how does $epsilon$ depend on $R$ and $N$.
            $endgroup$
            – biryani
            Jan 21 at 10:39


















          • $begingroup$
            So you assume $f$ is continuous ($f(x)=e^{-x^2}(1 - 1_{xin mathbb{Q}})$)
            $endgroup$
            – reuns
            Jan 20 at 2:24












          • $begingroup$
            This is the kind of approximation I use. The question is how does $epsilon$ depend on $R$ and $N$.
            $endgroup$
            – biryani
            Jan 21 at 10:39
















          $begingroup$
          So you assume $f$ is continuous ($f(x)=e^{-x^2}(1 - 1_{xin mathbb{Q}})$)
          $endgroup$
          – reuns
          Jan 20 at 2:24






          $begingroup$
          So you assume $f$ is continuous ($f(x)=e^{-x^2}(1 - 1_{xin mathbb{Q}})$)
          $endgroup$
          – reuns
          Jan 20 at 2:24














          $begingroup$
          This is the kind of approximation I use. The question is how does $epsilon$ depend on $R$ and $N$.
          $endgroup$
          – biryani
          Jan 21 at 10:39




          $begingroup$
          This is the kind of approximation I use. The question is how does $epsilon$ depend on $R$ and $N$.
          $endgroup$
          – biryani
          Jan 21 at 10:39


















          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%2f3077962%2ferror-analysis-of-approximating-fourier-transforms%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

          Can a sorcerer learn a 5th-level spell early by creating spell slots using the Font of Magic feature?

          Image
          28 6 $begingroup$ Per the Font of Magic feature, sorcerer can use Flexible Casting to create 5th-level spell slots at level 7, even though they are not typically available until level 9. You can transform unexpended sorcery points into one spell slot as a bonus action on your turn. The Creating Spell Slots table shows the cost of creating a spell slot of [5th level is 7] If such a sorcerer levels up while still having this 5th-level spell slot, can they choose a 5th-level spell as the spell they gain upon levelling up? The Spellcasting feature states: Additionally, when you gain a level in this class, you can choose one of the sorcerer spells you know and replace it with another spell from the sorcerer spell list, which also must be of a level for which you have spell slots.
          Read more

          Does disintegrating a polymorphed enemy still kill it after the 2018 errata?

          Image
          up vote 13 down vote favorite 1 After the 2018 errata, the disintegrate spell description now reads: A creature targeted by this spell must make a Dexterity saving throw. On a failed save, the target takes 10d6 + 40 force damage. The target is disintegrated if this damage leaves it with 0 hit points. If you were to polymorph an enemy into a rat and then disintegrate it, would the enemy be disintegrated or would it just return to its original form? I know that for Druids, it’s not an instant kill anymore, but is this the case for polymorph as well? dnd-5e spells polymorph errata share | improve this question edited 18 hours ago
          Read more

          A Topological Invariant for $pi_3(U(n))$

          Image
          3 3 $begingroup$ Recently, I saw a construction of topological invariant for $pi_3(U(n))$ with $ngeq 2$ : $$ N=frac{1}{24pi^2}int_{S^3} d^3x epsilon^{ijk} Tr[(U^{-1}partial_{x_i}U)(U^{-1}partial_{x_j}U)(U^{-1}partial_{x_k}U)] , $$ where $Uin U(n)$ depends on $boldsymbol{x}=(x_1,x_2,x_3)in S^3$ , $epsilon^{ijk}$ is the Levi-Civita symbol, $i,j,k=1,2,3$ , and the duplicated indexes are summed over. It is claimed that $N$ is an integer, but why? Update 02/02/2019 I think I got an argument for $n=2$ . In this case, $U=e^{i varphi} q$ with $qin SU(2)$ . Due to the trace and the Levi-Civita symbol in $N$ , $varphi$ does not contribute to $N$ . As $Tr[q^{dagger}partial_i q]=0$ and $(q^{dagger}partial_i q)^{dagger}=-q^{dagger}partial_i q$ , $q^{dagger}partial_i q$ in geneeral has the form $q^{dagger}partia
          Read more