Convergence of Fourier serie : if $fin L(0,2pi)$ is $2pi$ periodic and locally $alpha -$Holder continuous,...












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Let $fin L^1(0,2pi)$ a function s.t. $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ for some $delta $, then $$lim_{Nto infty }S_N(f)(a)=frac{f(a+)+f(a-)}{2}.$$
for all $ain[0,2pi]$. I recall that $f(apm)=lim_{xto a^{pm}}f(x)$ and that $S_N(f)$ is the partial Fourier sum.



Q1) So if $a=0$ what is the Fourier series ? Is it $$lim_{Nto infty }S_N(f)(x)frac{f(0^+)+f(2pi^-)}{2} ?$$



Q2) If $anotin [0,2pi]$, do we still have the result, i.e. if $a=5pi$, does $$lim_{Nto infty }S_N(f)(a)=frac{f(5pi^+)+f(5pi^-)}{2}$$ as well or it work on $[0,2pi]$ only ?



Q3) In the condition $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ shouldn't it be $fin mathcal C^{0,alpha }((a-delta ,a))cap mathcal C^{0,alpha }((a,a+delta ))$ ? Because if $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ then $fin mathcal C^{0,alpha }([a-delta ,a+delta ])$ ?






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    $begingroup$


    Let $fin L^1(0,2pi)$ a function s.t. $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ for some $delta $, then $$lim_{Nto infty }S_N(f)(a)=frac{f(a+)+f(a-)}{2}.$$
    for all $ain[0,2pi]$. I recall that $f(apm)=lim_{xto a^{pm}}f(x)$ and that $S_N(f)$ is the partial Fourier sum.



    Q1) So if $a=0$ what is the Fourier series ? Is it $$lim_{Nto infty }S_N(f)(x)frac{f(0^+)+f(2pi^-)}{2} ?$$



    Q2) If $anotin [0,2pi]$, do we still have the result, i.e. if $a=5pi$, does $$lim_{Nto infty }S_N(f)(a)=frac{f(5pi^+)+f(5pi^-)}{2}$$ as well or it work on $[0,2pi]$ only ?



    Q3) In the condition $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ shouldn't it be $fin mathcal C^{0,alpha }((a-delta ,a))cap mathcal C^{0,alpha }((a,a+delta ))$ ? Because if $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ then $fin mathcal C^{0,alpha }([a-delta ,a+delta ])$ ?






    fourier-series







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      $begingroup$


      Let $fin L^1(0,2pi)$ a function s.t. $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ for some $delta $, then $$lim_{Nto infty }S_N(f)(a)=frac{f(a+)+f(a-)}{2}.$$
      for all $ain[0,2pi]$. I recall that $f(apm)=lim_{xto a^{pm}}f(x)$ and that $S_N(f)$ is the partial Fourier sum.



      Q1) So if $a=0$ what is the Fourier series ? Is it $$lim_{Nto infty }S_N(f)(x)frac{f(0^+)+f(2pi^-)}{2} ?$$



      Q2) If $anotin [0,2pi]$, do we still have the result, i.e. if $a=5pi$, does $$lim_{Nto infty }S_N(f)(a)=frac{f(5pi^+)+f(5pi^-)}{2}$$ as well or it work on $[0,2pi]$ only ?



      Q3) In the condition $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ shouldn't it be $fin mathcal C^{0,alpha }((a-delta ,a))cap mathcal C^{0,alpha }((a,a+delta ))$ ? Because if $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ then $fin mathcal C^{0,alpha }([a-delta ,a+delta ])$ ?






      fourier-series







      share|cite|improve this question











      $endgroup$




      Let $fin L^1(0,2pi)$ a function s.t. $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ for some $delta $, then $$lim_{Nto infty }S_N(f)(a)=frac{f(a+)+f(a-)}{2}.$$
      for all $ain[0,2pi]$. I recall that $f(apm)=lim_{xto a^{pm}}f(x)$ and that $S_N(f)$ is the partial Fourier sum.



      Q1) So if $a=0$ what is the Fourier series ? Is it $$lim_{Nto infty }S_N(f)(x)frac{f(0^+)+f(2pi^-)}{2} ?$$



      Q2) If $anotin [0,2pi]$, do we still have the result, i.e. if $a=5pi$, does $$lim_{Nto infty }S_N(f)(a)=frac{f(5pi^+)+f(5pi^-)}{2}$$ as well or it work on $[0,2pi]$ only ?



      Q3) In the condition $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ shouldn't it be $fin mathcal C^{0,alpha }((a-delta ,a))cap mathcal C^{0,alpha }((a,a+delta ))$ ? Because if $fin mathcal C^{0,alpha }([a-delta ,a])cap mathcal C^{0,alpha }([a,a+delta ])$ then $fin mathcal C^{0,alpha }([a-delta ,a+delta ])$ ?






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      edited Jan 2 at 13:07







      NewMath

















      asked Jan 2 at 11:43









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