2$pi$-peridodic and continuous function with non summable Fourier coeffficients












0












$begingroup$


Can someone tell me an example of 2$pi$-periodic and continuous function, f, with Fourier coefficients $hat{f}(n);forall{nin{mathbb{Z}}}$ such that $sum|hat{f}(n)|>infty$?



Thanks.






real-analysis fourier-analysis fourier-series







share|cite|improve this question









$endgroup$












  • $begingroup$
    try some continuous function that approximates the sign function in $[-pi,pi]$. My idea comes from the fact that the Fourier coefficients of the periodic extension of the sign function in $[-pi,pi]$ are $1/n$ for odd $n$ (an zero for even $n$)
    $endgroup$
    – Masacroso
    Jan 1 at 12:10












  • $begingroup$
    You may also need to prevent your function from being differentiable in order to force the coefficients to be large.
    $endgroup$
    – InequalitiesEverywhere
    Jan 1 at 12:12










  • $begingroup$
    I have tried a Weirstrass type function: $f(t)=sumfrac{1}{n!}cos((n!)^2t)$. But it doesn't work no?
    $endgroup$
    – mathlife
    Jan 1 at 12:20










  • $begingroup$
    Would $sin(csc x)sin x$ work with the value zero at multiples of $pi$ work? The function is continuous but nondifferentiable at these multiples of $pi$.
    $endgroup$
    – Oscar Lanzi
    Jan 1 at 12:29










  • $begingroup$
    I'm quite sure there is no function whose Fourier coefficients sum to a value strictly greater than infinity.
    $endgroup$
    – md2perpe
    Jan 1 at 14:09
















0












$begingroup$


Can someone tell me an example of 2$pi$-periodic and continuous function, f, with Fourier coefficients $hat{f}(n);forall{nin{mathbb{Z}}}$ such that $sum|hat{f}(n)|>infty$?



Thanks.






real-analysis fourier-analysis fourier-series







share|cite|improve this question









$endgroup$












  • $begingroup$
    try some continuous function that approximates the sign function in $[-pi,pi]$. My idea comes from the fact that the Fourier coefficients of the periodic extension of the sign function in $[-pi,pi]$ are $1/n$ for odd $n$ (an zero for even $n$)
    $endgroup$
    – Masacroso
    Jan 1 at 12:10












  • $begingroup$
    You may also need to prevent your function from being differentiable in order to force the coefficients to be large.
    $endgroup$
    – InequalitiesEverywhere
    Jan 1 at 12:12










  • $begingroup$
    I have tried a Weirstrass type function: $f(t)=sumfrac{1}{n!}cos((n!)^2t)$. But it doesn't work no?
    $endgroup$
    – mathlife
    Jan 1 at 12:20










  • $begingroup$
    Would $sin(csc x)sin x$ work with the value zero at multiples of $pi$ work? The function is continuous but nondifferentiable at these multiples of $pi$.
    $endgroup$
    – Oscar Lanzi
    Jan 1 at 12:29










  • $begingroup$
    I'm quite sure there is no function whose Fourier coefficients sum to a value strictly greater than infinity.
    $endgroup$
    – md2perpe
    Jan 1 at 14:09














0












0








0





$begingroup$


Can someone tell me an example of 2$pi$-periodic and continuous function, f, with Fourier coefficients $hat{f}(n);forall{nin{mathbb{Z}}}$ such that $sum|hat{f}(n)|>infty$?



Thanks.






real-analysis fourier-analysis fourier-series







share|cite|improve this question









$endgroup$




Can someone tell me an example of 2$pi$-periodic and continuous function, f, with Fourier coefficients $hat{f}(n);forall{nin{mathbb{Z}}}$ such that $sum|hat{f}(n)|>infty$?



Thanks.






real-analysis fourier-analysis fourier-series




real-analysis fourier-analysis fourier-series






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 1 at 11:47









mathlifemathlife

629




629












  • $begingroup$
    try some continuous function that approximates the sign function in $[-pi,pi]$. My idea comes from the fact that the Fourier coefficients of the periodic extension of the sign function in $[-pi,pi]$ are $1/n$ for odd $n$ (an zero for even $n$)
    $endgroup$
    – Masacroso
    Jan 1 at 12:10












  • $begingroup$
    You may also need to prevent your function from being differentiable in order to force the coefficients to be large.
    $endgroup$
    – InequalitiesEverywhere
    Jan 1 at 12:12










  • $begingroup$
    I have tried a Weirstrass type function: $f(t)=sumfrac{1}{n!}cos((n!)^2t)$. But it doesn't work no?
    $endgroup$
    – mathlife
    Jan 1 at 12:20










  • $begingroup$
    Would $sin(csc x)sin x$ work with the value zero at multiples of $pi$ work? The function is continuous but nondifferentiable at these multiples of $pi$.
    $endgroup$
    – Oscar Lanzi
    Jan 1 at 12:29










  • $begingroup$
    I'm quite sure there is no function whose Fourier coefficients sum to a value strictly greater than infinity.
    $endgroup$
    – md2perpe
    Jan 1 at 14:09


















  • $begingroup$
    try some continuous function that approximates the sign function in $[-pi,pi]$. My idea comes from the fact that the Fourier coefficients of the periodic extension of the sign function in $[-pi,pi]$ are $1/n$ for odd $n$ (an zero for even $n$)
    $endgroup$
    – Masacroso
    Jan 1 at 12:10












  • $begingroup$
    You may also need to prevent your function from being differentiable in order to force the coefficients to be large.
    $endgroup$
    – InequalitiesEverywhere
    Jan 1 at 12:12










  • $begingroup$
    I have tried a Weirstrass type function: $f(t)=sumfrac{1}{n!}cos((n!)^2t)$. But it doesn't work no?
    $endgroup$
    – mathlife
    Jan 1 at 12:20










  • $begingroup$
    Would $sin(csc x)sin x$ work with the value zero at multiples of $pi$ work? The function is continuous but nondifferentiable at these multiples of $pi$.
    $endgroup$
    – Oscar Lanzi
    Jan 1 at 12:29










  • $begingroup$
    I'm quite sure there is no function whose Fourier coefficients sum to a value strictly greater than infinity.
    $endgroup$
    – md2perpe
    Jan 1 at 14:09
















$begingroup$
try some continuous function that approximates the sign function in $[-pi,pi]$. My idea comes from the fact that the Fourier coefficients of the periodic extension of the sign function in $[-pi,pi]$ are $1/n$ for odd $n$ (an zero for even $n$)
$endgroup$
– Masacroso
Jan 1 at 12:10






$begingroup$
try some continuous function that approximates the sign function in $[-pi,pi]$. My idea comes from the fact that the Fourier coefficients of the periodic extension of the sign function in $[-pi,pi]$ are $1/n$ for odd $n$ (an zero for even $n$)
$endgroup$
– Masacroso
Jan 1 at 12:10














$begingroup$
You may also need to prevent your function from being differentiable in order to force the coefficients to be large.
$endgroup$
– InequalitiesEverywhere
Jan 1 at 12:12




$begingroup$
You may also need to prevent your function from being differentiable in order to force the coefficients to be large.
$endgroup$
– InequalitiesEverywhere
Jan 1 at 12:12












$begingroup$
I have tried a Weirstrass type function: $f(t)=sumfrac{1}{n!}cos((n!)^2t)$. But it doesn't work no?
$endgroup$
– mathlife
Jan 1 at 12:20




$begingroup$
I have tried a Weirstrass type function: $f(t)=sumfrac{1}{n!}cos((n!)^2t)$. But it doesn't work no?
$endgroup$
– mathlife
Jan 1 at 12:20












$begingroup$
Would $sin(csc x)sin x$ work with the value zero at multiples of $pi$ work? The function is continuous but nondifferentiable at these multiples of $pi$.
$endgroup$
– Oscar Lanzi
Jan 1 at 12:29




$begingroup$
Would $sin(csc x)sin x$ work with the value zero at multiples of $pi$ work? The function is continuous but nondifferentiable at these multiples of $pi$.
$endgroup$
– Oscar Lanzi
Jan 1 at 12:29












$begingroup$
I'm quite sure there is no function whose Fourier coefficients sum to a value strictly greater than infinity.
$endgroup$
– md2perpe
Jan 1 at 14:09




$begingroup$
I'm quite sure there is no function whose Fourier coefficients sum to a value strictly greater than infinity.
$endgroup$
– md2perpe
Jan 1 at 14:09










1 Answer
1






active

oldest

votes


















3












$begingroup$

Convergence of Fourier Series is a delicate business but everything is known. The theorem you are looking for is the following:




(Du Bois-Reymond 1873) There is a continuous function $f: [-pi, pi] rightarrow mathbb{C}$ such that $limsup_{N} sum_{|n| le N} widehat{f}(n) rightarrow infty$.




A standard proof of this result can be found all over the internet but I think the most elementary construction is given here.



Just as a reference, if you know that your function is in $C^1$ then the Fourier series converges uniformly. Furthermore, if you know that your function is in $L^p$ for $p > 1$ then your Fourier series converges pointwise for everything in your compact interval except possibly on a set of measure $0$. This is the celebrated Carleson-Hunt Theorem.






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    1 Answer
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    1 Answer
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    active

    oldest

    votes






    active

    oldest

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    3












    $begingroup$

    Convergence of Fourier Series is a delicate business but everything is known. The theorem you are looking for is the following:




    (Du Bois-Reymond 1873) There is a continuous function $f: [-pi, pi] rightarrow mathbb{C}$ such that $limsup_{N} sum_{|n| le N} widehat{f}(n) rightarrow infty$.




    A standard proof of this result can be found all over the internet but I think the most elementary construction is given here.



    Just as a reference, if you know that your function is in $C^1$ then the Fourier series converges uniformly. Furthermore, if you know that your function is in $L^p$ for $p > 1$ then your Fourier series converges pointwise for everything in your compact interval except possibly on a set of measure $0$. This is the celebrated Carleson-Hunt Theorem.






    share|cite|improve this answer









    $endgroup$


















      3












      $begingroup$

      Convergence of Fourier Series is a delicate business but everything is known. The theorem you are looking for is the following:




      (Du Bois-Reymond 1873) There is a continuous function $f: [-pi, pi] rightarrow mathbb{C}$ such that $limsup_{N} sum_{|n| le N} widehat{f}(n) rightarrow infty$.




      A standard proof of this result can be found all over the internet but I think the most elementary construction is given here.



      Just as a reference, if you know that your function is in $C^1$ then the Fourier series converges uniformly. Furthermore, if you know that your function is in $L^p$ for $p > 1$ then your Fourier series converges pointwise for everything in your compact interval except possibly on a set of measure $0$. This is the celebrated Carleson-Hunt Theorem.






      share|cite|improve this answer









      $endgroup$
















        3












        3








        3





        $begingroup$

        Convergence of Fourier Series is a delicate business but everything is known. The theorem you are looking for is the following:




        (Du Bois-Reymond 1873) There is a continuous function $f: [-pi, pi] rightarrow mathbb{C}$ such that $limsup_{N} sum_{|n| le N} widehat{f}(n) rightarrow infty$.




        A standard proof of this result can be found all over the internet but I think the most elementary construction is given here.



        Just as a reference, if you know that your function is in $C^1$ then the Fourier series converges uniformly. Furthermore, if you know that your function is in $L^p$ for $p > 1$ then your Fourier series converges pointwise for everything in your compact interval except possibly on a set of measure $0$. This is the celebrated Carleson-Hunt Theorem.






        share|cite|improve this answer









        $endgroup$



        Convergence of Fourier Series is a delicate business but everything is known. The theorem you are looking for is the following:




        (Du Bois-Reymond 1873) There is a continuous function $f: [-pi, pi] rightarrow mathbb{C}$ such that $limsup_{N} sum_{|n| le N} widehat{f}(n) rightarrow infty$.




        A standard proof of this result can be found all over the internet but I think the most elementary construction is given here.



        Just as a reference, if you know that your function is in $C^1$ then the Fourier series converges uniformly. Furthermore, if you know that your function is in $L^p$ for $p > 1$ then your Fourier series converges pointwise for everything in your compact interval except possibly on a set of measure $0$. This is the celebrated Carleson-Hunt Theorem.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 1 at 15:00









        Sandeep SilwalSandeep Silwal

        5,84311236




        5,84311236






























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