Computing the norm of $hatf_epsilon(xi)=chi_[x-epsilon,x+epsilon](|xi|).$

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Given $fin L^2(mathbbR^n)$. For any $epsilon in(0,x)$, define $$hatf_epsilon(xi)=chi_[x-epsilon,x+epsilon](|xi|).$$ Where $hatf_epsilon(xi)$ is the Fourier Transform $f$.



I am wondering how to compute the following:



  1. $f$

  2. $|f|$

  3. $|hatf|$

  4. $|chi_[x-epsilon,x+epsilon](|xi|)|$.

I know that
begineqnarray*
% nonumber to remove numbering (before each equation)
f(x) &=& 1/(2pi)^nint hatxie^ixxidxi \
&=&1/(2pi)^nint chi_[x-epsilon,x+epsilon](|xi|)e^ixxidxi$\
&=&1/(2pi)^nint_x-epsilon^x+epsilone^ixxidxi\
&=&1/(2pi)^ncdot 1/ixcdot (e^ix(x+epsilon)-e^ix(x-epsilon)).
endeqnarray*



But this is not leading me to anything interesting!!



I also think that after obtaining $f$ I can go on and use $|f|_L^2=1/(2pi)^n/2|hatf|_L^2$ to obtain $hatf$. I need some help please.




functional-analysis fourier-analysis




share|cite|improve this question



















  • Plancherel's Theorem should take care of the hard parts.
    – DisintegratingByParts
    Jul 20 at 18:51










  • How can I use the Placherel's Theorem? I need some ideas
    – Sulayman
    Jul 21 at 21:50














up vote
0
down vote

favorite












Given $fin L^2(mathbbR^n)$. For any $epsilon in(0,x)$, define $$hatf_epsilon(xi)=chi_[x-epsilon,x+epsilon](|xi|).$$ Where $hatf_epsilon(xi)$ is the Fourier Transform $f$.



I am wondering how to compute the following:



  1. $f$

  2. $|f|$

  3. $|hatf|$

  4. $|chi_[x-epsilon,x+epsilon](|xi|)|$.

I know that
begineqnarray*
% nonumber to remove numbering (before each equation)
f(x) &=& 1/(2pi)^nint hatxie^ixxidxi \
&=&1/(2pi)^nint chi_[x-epsilon,x+epsilon](|xi|)e^ixxidxi$\
&=&1/(2pi)^nint_x-epsilon^x+epsilone^ixxidxi\
&=&1/(2pi)^ncdot 1/ixcdot (e^ix(x+epsilon)-e^ix(x-epsilon)).
endeqnarray*



But this is not leading me to anything interesting!!



I also think that after obtaining $f$ I can go on and use $|f|_L^2=1/(2pi)^n/2|hatf|_L^2$ to obtain $hatf$. I need some help please.




functional-analysis fourier-analysis




share|cite|improve this question



















  • Plancherel's Theorem should take care of the hard parts.
    – DisintegratingByParts
    Jul 20 at 18:51










  • How can I use the Placherel's Theorem? I need some ideas
    – Sulayman
    Jul 21 at 21:50












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Given $fin L^2(mathbbR^n)$. For any $epsilon in(0,x)$, define $$hatf_epsilon(xi)=chi_[x-epsilon,x+epsilon](|xi|).$$ Where $hatf_epsilon(xi)$ is the Fourier Transform $f$.



I am wondering how to compute the following:



  1. $f$

  2. $|f|$

  3. $|hatf|$

  4. $|chi_[x-epsilon,x+epsilon](|xi|)|$.

I know that
begineqnarray*
% nonumber to remove numbering (before each equation)
f(x) &=& 1/(2pi)^nint hatxie^ixxidxi \
&=&1/(2pi)^nint chi_[x-epsilon,x+epsilon](|xi|)e^ixxidxi$\
&=&1/(2pi)^nint_x-epsilon^x+epsilone^ixxidxi\
&=&1/(2pi)^ncdot 1/ixcdot (e^ix(x+epsilon)-e^ix(x-epsilon)).
endeqnarray*



But this is not leading me to anything interesting!!



I also think that after obtaining $f$ I can go on and use $|f|_L^2=1/(2pi)^n/2|hatf|_L^2$ to obtain $hatf$. I need some help please.




functional-analysis fourier-analysis




share|cite|improve this question











Given $fin L^2(mathbbR^n)$. For any $epsilon in(0,x)$, define $$hatf_epsilon(xi)=chi_[x-epsilon,x+epsilon](|xi|).$$ Where $hatf_epsilon(xi)$ is the Fourier Transform $f$.



I am wondering how to compute the following:



  1. $f$

  2. $|f|$

  3. $|hatf|$

  4. $|chi_[x-epsilon,x+epsilon](|xi|)|$.

I know that
begineqnarray*
% nonumber to remove numbering (before each equation)
f(x) &=& 1/(2pi)^nint hatxie^ixxidxi \
&=&1/(2pi)^nint chi_[x-epsilon,x+epsilon](|xi|)e^ixxidxi$\
&=&1/(2pi)^nint_x-epsilon^x+epsilone^ixxidxi\
&=&1/(2pi)^ncdot 1/ixcdot (e^ix(x+epsilon)-e^ix(x-epsilon)).
endeqnarray*



But this is not leading me to anything interesting!!



I also think that after obtaining $f$ I can go on and use $|f|_L^2=1/(2pi)^n/2|hatf|_L^2$ to obtain $hatf$. I need some help please.





functional-analysis fourier-analysis





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 20 at 11:57









Sulayman

18717




18717











  • Plancherel's Theorem should take care of the hard parts.
    – DisintegratingByParts
    Jul 20 at 18:51










  • How can I use the Placherel's Theorem? I need some ideas
    – Sulayman
    Jul 21 at 21:50
















  • Plancherel's Theorem should take care of the hard parts.
    – DisintegratingByParts
    Jul 20 at 18:51










  • How can I use the Placherel's Theorem? I need some ideas
    – Sulayman
    Jul 21 at 21:50















Plancherel's Theorem should take care of the hard parts.
– DisintegratingByParts
Jul 20 at 18:51




Plancherel's Theorem should take care of the hard parts.
– DisintegratingByParts
Jul 20 at 18:51












How can I use the Placherel's Theorem? I need some ideas
– Sulayman
Jul 21 at 21:50




How can I use the Placherel's Theorem? I need some ideas
– Sulayman
Jul 21 at 21:50















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