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Fourier series for complex numbers

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-1

$begingroup$

A certain riemann-integrable function $f:[-pi,pi] rightarrow mathbbC $ and a complex number sequence $c_k$ obey

$ |lvert f(t) -sum_k=-n^n c_ne^ikt|rvert_2 rightarrow 0$ as $n rightarrow infty$

Prove:

For any $ g:[-pi,pi] rightarrow mathbbC$ with $ginmathbbR[-pi,pi]$

$frac12piint_-pi^pi f(t)overlineg(t)=
sum_k=-infty^infty c_koverlinehatg(k)$, where $hatg(k)= int_-pi^pi g(t)e^-iktdt$. Given this, prove $c_k =hatf(k)$ and $sum_k lvert c_k rvert^2 < infty$

I am not sure how to approach this problem. I learnt Fourier analysis for real numbers but the complex numbers are confusing me. Thank you

fourier-series

share|cite|improve this question

edited Mar 13 at 18:11

uniquesolution

9,2911823

asked Mar 13 at 17:44

Kaan YolseverKaan Yolsever

1309

$endgroup$

add a comment | 

-1

$begingroup$

A certain riemann-integrable function $f:[-pi,pi] rightarrow mathbbC $ and a complex number sequence $c_k$ obey

$ |lvert f(t) -sum_k=-n^n c_ne^ikt|rvert_2 rightarrow 0$ as $n rightarrow infty$

Prove:

For any $ g:[-pi,pi] rightarrow mathbbC$ with $ginmathbbR[-pi,pi]$

$frac12piint_-pi^pi f(t)overlineg(t)=
sum_k=-infty^infty c_koverlinehatg(k)$, where $hatg(k)= int_-pi^pi g(t)e^-iktdt$. Given this, prove $c_k =hatf(k)$ and $sum_k lvert c_k rvert^2 < infty$

I am not sure how to approach this problem. I learnt Fourier analysis for real numbers but the complex numbers are confusing me. Thank you

fourier-series

share|cite|improve this question

edited Mar 13 at 18:11

uniquesolution

9,2911823

asked Mar 13 at 17:44

Kaan YolseverKaan Yolsever

1309

$endgroup$

add a comment | 

$begingroup$

A certain riemann-integrable function $f:[-pi,pi] rightarrow mathbbC $ and a complex number sequence $c_k$ obey

$ |lvert f(t) -sum_k=-n^n c_ne^ikt|rvert_2 rightarrow 0$ as $n rightarrow infty$

Prove:

For any $ g:[-pi,pi] rightarrow mathbbC$ with $ginmathbbR[-pi,pi]$

$frac12piint_-pi^pi f(t)overlineg(t)=
sum_k=-infty^infty c_koverlinehatg(k)$, where $hatg(k)= int_-pi^pi g(t)e^-iktdt$. Given this, prove $c_k =hatf(k)$ and $sum_k lvert c_k rvert^2 < infty$

I am not sure how to approach this problem. I learnt Fourier analysis for real numbers but the complex numbers are confusing me. Thank you

fourier-series

share|cite|improve this question

edited Mar 13 at 18:11

uniquesolution

9,2911823

asked Mar 13 at 17:44

Kaan YolseverKaan Yolsever

1309

$endgroup$

share|cite|improve this question

edited Mar 13 at 18:11

uniquesolution

9,2911823

asked Mar 13 at 17:44

Kaan YolseverKaan Yolsever

1309

share|cite|improve this question

edited Mar 13 at 18:11

uniquesolution

9,2911823

asked Mar 13 at 17:44

Kaan YolseverKaan Yolsever

1309

edited Mar 13 at 18:11

uniquesolution

9,2911823

edited Mar 13 at 18:11

uniquesolution

9,2911823

asked Mar 13 at 17:44

Kaan YolseverKaan Yolsever

1309

asked Mar 13 at 17:44

Kaan YolseverKaan Yolsever

1309