Convergence of Fourier serie : if $fin L(0,2pi)$ is $2pi$ periodic and locally $alpha -$Holder continuous,...
$begingroup$
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
asked Jan 2 at 11:43
NewMathNewMath
4059
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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
asked Jan 2 at 11:43
NewMathNewMath
4059
$endgroup$
add a comment |
$begingroup$
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
asked Jan 2 at 11:43
NewMathNewMath
4059
$endgroup$
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 $fin mathcal C^{0,alpha }([a-delta ,a+delta ])$ ?
fourier-series
fourier-series
asked Jan 2 at 11:43
NewMathNewMath
4059
asked Jan 2 at 11:43
NewMathNewMath
4059
edited Jan 2 at 13:07
NewMath
asked Jan 2 at 11:43
NewMathNewMath
4059
asked Jan 2 at 11:43
NewMathNewMath
4059
asked Jan 2 at 11:43
NewMathNewMath
4059
4059
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