$f in L_1([0,1],m)$ such that $int_0^1 f sin (n^2x) dm= 1$












0












$begingroup$


I have the space of $mathbb{K}$-valued integrable functions with respect to a Lebesgue measure $m$ and I need to find a function $f$ such that $int_0^1 |f| dm=1$ and $int_0^1 f sin(n^2x) dm=1 $, $n geq 2, n in mathbb{N}$. I was thinking to take a continuous function so it's Riemann integrable on $[0,1]$ and then I can "forget" the Lebesgue measure, but I don't know if it's a good idea.






functional-analysis lebesgue-integral lebesgue-measure







share|cite|improve this question











$endgroup$












  • $begingroup$
    What is $mathbb K$?
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:49










  • $begingroup$
    What is the purpose of the cancelling $n$s? It looks like you want a function satisfying $int_0^1 f(x) sin(n^2 x) , dm(x) = 1$ for all $n$.
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:50












  • $begingroup$
    A field, can be $mathbb{R}$ or $mathbb{C}$
    $endgroup$
    – user289143
    Dec 16 '18 at 13:50










  • $begingroup$
    No, that integral should be $n$, not $1$
    $endgroup$
    – user289143
    Dec 16 '18 at 13:53












  • $begingroup$
    What is the $n$ in between $f$ and $sin$ in the integral then?
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:54


















0












$begingroup$


I have the space of $mathbb{K}$-valued integrable functions with respect to a Lebesgue measure $m$ and I need to find a function $f$ such that $int_0^1 |f| dm=1$ and $int_0^1 f sin(n^2x) dm=1 $, $n geq 2, n in mathbb{N}$. I was thinking to take a continuous function so it's Riemann integrable on $[0,1]$ and then I can "forget" the Lebesgue measure, but I don't know if it's a good idea.






functional-analysis lebesgue-integral lebesgue-measure







share|cite|improve this question











$endgroup$












  • $begingroup$
    What is $mathbb K$?
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:49










  • $begingroup$
    What is the purpose of the cancelling $n$s? It looks like you want a function satisfying $int_0^1 f(x) sin(n^2 x) , dm(x) = 1$ for all $n$.
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:50












  • $begingroup$
    A field, can be $mathbb{R}$ or $mathbb{C}$
    $endgroup$
    – user289143
    Dec 16 '18 at 13:50










  • $begingroup$
    No, that integral should be $n$, not $1$
    $endgroup$
    – user289143
    Dec 16 '18 at 13:53












  • $begingroup$
    What is the $n$ in between $f$ and $sin$ in the integral then?
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:54
















0












0








0





$begingroup$


I have the space of $mathbb{K}$-valued integrable functions with respect to a Lebesgue measure $m$ and I need to find a function $f$ such that $int_0^1 |f| dm=1$ and $int_0^1 f sin(n^2x) dm=1 $, $n geq 2, n in mathbb{N}$. I was thinking to take a continuous function so it's Riemann integrable on $[0,1]$ and then I can "forget" the Lebesgue measure, but I don't know if it's a good idea.






functional-analysis lebesgue-integral lebesgue-measure







share|cite|improve this question











$endgroup$




I have the space of $mathbb{K}$-valued integrable functions with respect to a Lebesgue measure $m$ and I need to find a function $f$ such that $int_0^1 |f| dm=1$ and $int_0^1 f sin(n^2x) dm=1 $, $n geq 2, n in mathbb{N}$. I was thinking to take a continuous function so it's Riemann integrable on $[0,1]$ and then I can "forget" the Lebesgue measure, but I don't know if it's a good idea.






functional-analysis lebesgue-integral lebesgue-measure




functional-analysis lebesgue-integral lebesgue-measure






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 16 '18 at 13:58







user289143

















asked Dec 16 '18 at 13:43









user289143user289143

903313




903313












  • $begingroup$
    What is $mathbb K$?
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:49










  • $begingroup$
    What is the purpose of the cancelling $n$s? It looks like you want a function satisfying $int_0^1 f(x) sin(n^2 x) , dm(x) = 1$ for all $n$.
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:50












  • $begingroup$
    A field, can be $mathbb{R}$ or $mathbb{C}$
    $endgroup$
    – user289143
    Dec 16 '18 at 13:50










  • $begingroup$
    No, that integral should be $n$, not $1$
    $endgroup$
    – user289143
    Dec 16 '18 at 13:53












  • $begingroup$
    What is the $n$ in between $f$ and $sin$ in the integral then?
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:54




















  • $begingroup$
    What is $mathbb K$?
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:49










  • $begingroup$
    What is the purpose of the cancelling $n$s? It looks like you want a function satisfying $int_0^1 f(x) sin(n^2 x) , dm(x) = 1$ for all $n$.
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:50












  • $begingroup$
    A field, can be $mathbb{R}$ or $mathbb{C}$
    $endgroup$
    – user289143
    Dec 16 '18 at 13:50










  • $begingroup$
    No, that integral should be $n$, not $1$
    $endgroup$
    – user289143
    Dec 16 '18 at 13:53












  • $begingroup$
    What is the $n$ in between $f$ and $sin$ in the integral then?
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 13:54


















$begingroup$
What is $mathbb K$?
$endgroup$
– Umberto P.
Dec 16 '18 at 13:49




$begingroup$
What is $mathbb K$?
$endgroup$
– Umberto P.
Dec 16 '18 at 13:49












$begingroup$
What is the purpose of the cancelling $n$s? It looks like you want a function satisfying $int_0^1 f(x) sin(n^2 x) , dm(x) = 1$ for all $n$.
$endgroup$
– Umberto P.
Dec 16 '18 at 13:50






$begingroup$
What is the purpose of the cancelling $n$s? It looks like you want a function satisfying $int_0^1 f(x) sin(n^2 x) , dm(x) = 1$ for all $n$.
$endgroup$
– Umberto P.
Dec 16 '18 at 13:50














$begingroup$
A field, can be $mathbb{R}$ or $mathbb{C}$
$endgroup$
– user289143
Dec 16 '18 at 13:50




$begingroup$
A field, can be $mathbb{R}$ or $mathbb{C}$
$endgroup$
– user289143
Dec 16 '18 at 13:50












$begingroup$
No, that integral should be $n$, not $1$
$endgroup$
– user289143
Dec 16 '18 at 13:53






$begingroup$
No, that integral should be $n$, not $1$
$endgroup$
– user289143
Dec 16 '18 at 13:53














$begingroup$
What is the $n$ in between $f$ and $sin$ in the integral then?
$endgroup$
– Umberto P.
Dec 16 '18 at 13:54






$begingroup$
What is the $n$ in between $f$ and $sin$ in the integral then?
$endgroup$
– Umberto P.
Dec 16 '18 at 13:54












1 Answer
1






active

oldest

votes


















1












$begingroup$

You won't find such a function.



If $f in L^1[0,1]$ the Riemann-Lebesgue Lemma tells you that $$lim_{n to infty} int_0^1 f(x) sin(nx) , dx = 0.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    So, how can I show part $a)$ of this question? math.stackexchange.com/questions/3035444/…
    $endgroup$
    – user289143
    Dec 16 '18 at 14:18






  • 1




    $begingroup$
    Comments aren't the place for new questions; perhaps you should post that as a question.
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 16:32











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1 Answer
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1 Answer
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active

oldest

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active

oldest

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active

oldest

votes









1












$begingroup$

You won't find such a function.



If $f in L^1[0,1]$ the Riemann-Lebesgue Lemma tells you that $$lim_{n to infty} int_0^1 f(x) sin(nx) , dx = 0.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    So, how can I show part $a)$ of this question? math.stackexchange.com/questions/3035444/…
    $endgroup$
    – user289143
    Dec 16 '18 at 14:18






  • 1




    $begingroup$
    Comments aren't the place for new questions; perhaps you should post that as a question.
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 16:32
















1












$begingroup$

You won't find such a function.



If $f in L^1[0,1]$ the Riemann-Lebesgue Lemma tells you that $$lim_{n to infty} int_0^1 f(x) sin(nx) , dx = 0.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    So, how can I show part $a)$ of this question? math.stackexchange.com/questions/3035444/…
    $endgroup$
    – user289143
    Dec 16 '18 at 14:18






  • 1




    $begingroup$
    Comments aren't the place for new questions; perhaps you should post that as a question.
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 16:32














1












1








1





$begingroup$

You won't find such a function.



If $f in L^1[0,1]$ the Riemann-Lebesgue Lemma tells you that $$lim_{n to infty} int_0^1 f(x) sin(nx) , dx = 0.$$






share|cite|improve this answer









$endgroup$



You won't find such a function.



If $f in L^1[0,1]$ the Riemann-Lebesgue Lemma tells you that $$lim_{n to infty} int_0^1 f(x) sin(nx) , dx = 0.$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 16 '18 at 14:00









Umberto P.Umberto P.

39.5k13166




39.5k13166












  • $begingroup$
    So, how can I show part $a)$ of this question? math.stackexchange.com/questions/3035444/…
    $endgroup$
    – user289143
    Dec 16 '18 at 14:18






  • 1




    $begingroup$
    Comments aren't the place for new questions; perhaps you should post that as a question.
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 16:32


















  • $begingroup$
    So, how can I show part $a)$ of this question? math.stackexchange.com/questions/3035444/…
    $endgroup$
    – user289143
    Dec 16 '18 at 14:18






  • 1




    $begingroup$
    Comments aren't the place for new questions; perhaps you should post that as a question.
    $endgroup$
    – Umberto P.
    Dec 16 '18 at 16:32
















$begingroup$
So, how can I show part $a)$ of this question? math.stackexchange.com/questions/3035444/…
$endgroup$
– user289143
Dec 16 '18 at 14:18




$begingroup$
So, how can I show part $a)$ of this question? math.stackexchange.com/questions/3035444/…
$endgroup$
– user289143
Dec 16 '18 at 14:18




1




1




$begingroup$
Comments aren't the place for new questions; perhaps you should post that as a question.
$endgroup$
– Umberto P.
Dec 16 '18 at 16:32




$begingroup$
Comments aren't the place for new questions; perhaps you should post that as a question.
$endgroup$
– Umberto P.
Dec 16 '18 at 16:32


















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