Show that $lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = 0$ $g$ periodic on $mathbb{R}$












0












$begingroup$


Let $g$ be a continuous periodic function on $mathbb{R}$ with $g(x + 1) = g(x)$. Assume that $int_{0}^{1}g(x)dx = 0$.



(a) Let $f$ be continuous on all $mathbb{R}$



Show that $lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = 0$



(b) Let $f$ be Lebesgue integrable on all $mathbb{R}$



Show that $lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = 0$



My attempt:



(a)



For part (a) I dont know how to proceed.
I have a solution that uses the following:



$$lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = int_{0}^{1}f(x)int_{0}^{1}g(x)dx$$ and by hypothesis the results follow. There is another approach ?



(b) (This needs verification)



$g(x)$ is continuous and periodic on $mathbb{R}$, then $exists B > 0$ such that $|g(x)|< B$.



$f$ Lebesgue integrable on $[0,1]$ $rightarrow forall epsilon > 0, exists h$ step function such that $int_{0}^{1}|f - h| < frac{epsilon}{B}, forall x in [0,1]$



then



$$| int_{0}^{1}f(x)g(nx)dx | leq int_{0}^{1}|f(x) - h(x)| |g(nx)|dx + |int_{0}^{1} h(x)g(nx)dx| leq epsilon + |int_{0}^{1} h(x)g(nx)dx|$$



We must show that $lim_{nto infty}int_{0}^{1} h(x)g(nx)dx|$



Since $h$ is a step function,



$$h(x) = sum_{i=1}^{k}lambda_ichi_{O_i}(x), mbox{ where } bigcup_{i=1}^{k}O_i = [0,1]$$



Hence, for $M = max_{i = 1,...,k}|lambda_i|$



$$int_{0}^{1} h(x)g(nx)dx = sum_{i=1}^{k}lambda_iint_{O_i}g(nx)dx$$
$$leq Msum_{i=1}^{k}int_{O_i}g(nx)dx$$
$$= Mint_{0}^{1}g(nx)dx| = frac{M}{n}int_{0}^{n}g(x)dx = Mint_{0}^{n}g(x) = 0$$



Then



$$|int_{0}^{1}f(x)g(nx)dx| < epsilon$$










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  • 2




    $begingroup$
    For $(b)$, you can approximate $f$ by continuous $h$ and use the result of $(a)$.
    $endgroup$
    – Song
    Jan 7 at 17:26
















0












$begingroup$


Let $g$ be a continuous periodic function on $mathbb{R}$ with $g(x + 1) = g(x)$. Assume that $int_{0}^{1}g(x)dx = 0$.



(a) Let $f$ be continuous on all $mathbb{R}$



Show that $lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = 0$



(b) Let $f$ be Lebesgue integrable on all $mathbb{R}$



Show that $lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = 0$



My attempt:



(a)



For part (a) I dont know how to proceed.
I have a solution that uses the following:



$$lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = int_{0}^{1}f(x)int_{0}^{1}g(x)dx$$ and by hypothesis the results follow. There is another approach ?



(b) (This needs verification)



$g(x)$ is continuous and periodic on $mathbb{R}$, then $exists B > 0$ such that $|g(x)|< B$.



$f$ Lebesgue integrable on $[0,1]$ $rightarrow forall epsilon > 0, exists h$ step function such that $int_{0}^{1}|f - h| < frac{epsilon}{B}, forall x in [0,1]$



then



$$| int_{0}^{1}f(x)g(nx)dx | leq int_{0}^{1}|f(x) - h(x)| |g(nx)|dx + |int_{0}^{1} h(x)g(nx)dx| leq epsilon + |int_{0}^{1} h(x)g(nx)dx|$$



We must show that $lim_{nto infty}int_{0}^{1} h(x)g(nx)dx|$



Since $h$ is a step function,



$$h(x) = sum_{i=1}^{k}lambda_ichi_{O_i}(x), mbox{ where } bigcup_{i=1}^{k}O_i = [0,1]$$



Hence, for $M = max_{i = 1,...,k}|lambda_i|$



$$int_{0}^{1} h(x)g(nx)dx = sum_{i=1}^{k}lambda_iint_{O_i}g(nx)dx$$
$$leq Msum_{i=1}^{k}int_{O_i}g(nx)dx$$
$$= Mint_{0}^{1}g(nx)dx| = frac{M}{n}int_{0}^{n}g(x)dx = Mint_{0}^{n}g(x) = 0$$



Then



$$|int_{0}^{1}f(x)g(nx)dx| < epsilon$$










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$endgroup$








  • 2




    $begingroup$
    For $(b)$, you can approximate $f$ by continuous $h$ and use the result of $(a)$.
    $endgroup$
    – Song
    Jan 7 at 17:26














0












0








0





$begingroup$


Let $g$ be a continuous periodic function on $mathbb{R}$ with $g(x + 1) = g(x)$. Assume that $int_{0}^{1}g(x)dx = 0$.



(a) Let $f$ be continuous on all $mathbb{R}$



Show that $lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = 0$



(b) Let $f$ be Lebesgue integrable on all $mathbb{R}$



Show that $lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = 0$



My attempt:



(a)



For part (a) I dont know how to proceed.
I have a solution that uses the following:



$$lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = int_{0}^{1}f(x)int_{0}^{1}g(x)dx$$ and by hypothesis the results follow. There is another approach ?



(b) (This needs verification)



$g(x)$ is continuous and periodic on $mathbb{R}$, then $exists B > 0$ such that $|g(x)|< B$.



$f$ Lebesgue integrable on $[0,1]$ $rightarrow forall epsilon > 0, exists h$ step function such that $int_{0}^{1}|f - h| < frac{epsilon}{B}, forall x in [0,1]$



then



$$| int_{0}^{1}f(x)g(nx)dx | leq int_{0}^{1}|f(x) - h(x)| |g(nx)|dx + |int_{0}^{1} h(x)g(nx)dx| leq epsilon + |int_{0}^{1} h(x)g(nx)dx|$$



We must show that $lim_{nto infty}int_{0}^{1} h(x)g(nx)dx|$



Since $h$ is a step function,



$$h(x) = sum_{i=1}^{k}lambda_ichi_{O_i}(x), mbox{ where } bigcup_{i=1}^{k}O_i = [0,1]$$



Hence, for $M = max_{i = 1,...,k}|lambda_i|$



$$int_{0}^{1} h(x)g(nx)dx = sum_{i=1}^{k}lambda_iint_{O_i}g(nx)dx$$
$$leq Msum_{i=1}^{k}int_{O_i}g(nx)dx$$
$$= Mint_{0}^{1}g(nx)dx| = frac{M}{n}int_{0}^{n}g(x)dx = Mint_{0}^{n}g(x) = 0$$



Then



$$|int_{0}^{1}f(x)g(nx)dx| < epsilon$$










share|cite|improve this question









$endgroup$




Let $g$ be a continuous periodic function on $mathbb{R}$ with $g(x + 1) = g(x)$. Assume that $int_{0}^{1}g(x)dx = 0$.



(a) Let $f$ be continuous on all $mathbb{R}$



Show that $lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = 0$



(b) Let $f$ be Lebesgue integrable on all $mathbb{R}$



Show that $lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = 0$



My attempt:



(a)



For part (a) I dont know how to proceed.
I have a solution that uses the following:



$$lim_{ntoinfty} int_{0}^{1}f(x)g(nx)dx = int_{0}^{1}f(x)int_{0}^{1}g(x)dx$$ and by hypothesis the results follow. There is another approach ?



(b) (This needs verification)



$g(x)$ is continuous and periodic on $mathbb{R}$, then $exists B > 0$ such that $|g(x)|< B$.



$f$ Lebesgue integrable on $[0,1]$ $rightarrow forall epsilon > 0, exists h$ step function such that $int_{0}^{1}|f - h| < frac{epsilon}{B}, forall x in [0,1]$



then



$$| int_{0}^{1}f(x)g(nx)dx | leq int_{0}^{1}|f(x) - h(x)| |g(nx)|dx + |int_{0}^{1} h(x)g(nx)dx| leq epsilon + |int_{0}^{1} h(x)g(nx)dx|$$



We must show that $lim_{nto infty}int_{0}^{1} h(x)g(nx)dx|$



Since $h$ is a step function,



$$h(x) = sum_{i=1}^{k}lambda_ichi_{O_i}(x), mbox{ where } bigcup_{i=1}^{k}O_i = [0,1]$$



Hence, for $M = max_{i = 1,...,k}|lambda_i|$



$$int_{0}^{1} h(x)g(nx)dx = sum_{i=1}^{k}lambda_iint_{O_i}g(nx)dx$$
$$leq Msum_{i=1}^{k}int_{O_i}g(nx)dx$$
$$= Mint_{0}^{1}g(nx)dx| = frac{M}{n}int_{0}^{n}g(x)dx = Mint_{0}^{n}g(x) = 0$$



Then



$$|int_{0}^{1}f(x)g(nx)dx| < epsilon$$







integration measure-theory lebesgue-integral periodic-functions






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asked Jan 7 at 17:18









Richard ClareRichard Clare

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1,054314








  • 2




    $begingroup$
    For $(b)$, you can approximate $f$ by continuous $h$ and use the result of $(a)$.
    $endgroup$
    – Song
    Jan 7 at 17:26














  • 2




    $begingroup$
    For $(b)$, you can approximate $f$ by continuous $h$ and use the result of $(a)$.
    $endgroup$
    – Song
    Jan 7 at 17:26








2




2




$begingroup$
For $(b)$, you can approximate $f$ by continuous $h$ and use the result of $(a)$.
$endgroup$
– Song
Jan 7 at 17:26




$begingroup$
For $(b)$, you can approximate $f$ by continuous $h$ and use the result of $(a)$.
$endgroup$
– Song
Jan 7 at 17:26










2 Answers
2






active

oldest

votes


















1












$begingroup$

For part (a):
Let $I=int_{0}^{1}|g(x)|dx$. Let $epsilon>0$ be given. Note that
$f$ is uniformly continuous on $[0,1]$, so there exists $delta>0$
such that $|f(x)-f(y)|<frac{epsilon}{I+1}$ whenever $x,yin[0,1]$
with $|x-y|<delta$. Choose $Ninmathbb{N}$ such that $frac{1}{N}<delta$.
By considering the substitution $y=nx$, we may rewrite
begin{eqnarray*}
int_{0}^{1}f(x)g(nx)dx & = & frac{1}{n}int_{0}^{n}f(frac{y}{n})g(y)dy\
& = & frac{1}{n}sum_{k=1}^{n}int_{k-1}^{k}f(frac{y}{n})g(y)dy.
end{eqnarray*}

Let $ngeq N$ be arbitrary. Denote $f_{k}=f(frac{k}{n})$, for $k=1,2,ldots n$.
By periodicity of $g$, we have $int_{k-1}^{k}g(y)dy=int_{0}^{1}g(y)dy=0$.
Therefore
begin{eqnarray*}
& & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\
& = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy|\
& leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\
& leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\
& = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\
& = & epsiloncdotfrac{I}{I+1}\
& leq & epsilon.
end{eqnarray*}

Now it is clear that
begin{eqnarray*}
& & |int_{0}^{1}f(x)g(nx)dx|\
& leq & frac{1}{n}sum_{k=1}^{n}|int_{k-1}^{k}f(frac{y}{n})g(y)dy|\
& leq & frac{1}{n}sum_{k=1}^{n}epsilon\
& = & epsilon.
end{eqnarray*}






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$endgroup$













  • $begingroup$
    how you obtain this:begin{eqnarray*} & & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\ & = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy| *****\ & leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\ & leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\ & = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\ & = & epsiloncdotfrac{I}{I+1}\ & leq & epsilon. end{eqnarray*}
    $endgroup$
    – Richard Clare
    Jan 7 at 18:05












  • $begingroup$
    $int_{k-1}^k f_k g(y) dy =0$
    $endgroup$
    – Danny Pak-Keung Chan
    Jan 7 at 18:11










  • $begingroup$
    Sure! Thanks!!!!!!!!
    $endgroup$
    – Richard Clare
    Jan 7 at 18:14



















1












$begingroup$

For (b): Since $g$ is continuous, there exists $M>0$ such that $|g(x)|leq M$
for all $xin[0,1]$. By periodicity of $g$, it follows that $|g(x)|leq M$
for all $xinmathbb{R}$. Recall that Lebesgue integrable function
can be approximated by integrable continuous function in $||cdot||_{1}$-norm.
More precisely, given $epsilon>0$, there exists a continuous function
$h:mathbb{R}rightarrowmathbb{R}$ such that $int_{0}^{1}|f-h|<epsilon$.



Let $epsilon>0$ be given. Choose continuous function $h:mathbb{R}rightarrowmathbb{R}$
such that $int_{0}^{1}|f-h|<frac{epsilon}{2M}.$ By part (a), there
exists $N$ such that $|int_{0}^{1}h(x)g(nx)dx|<frac{epsilon}{2}$
whenever $ngeq N$. Now let $ngeq N$ be aribitrary, then
begin{eqnarray*}
& & |int_{0}^{1}f(x)g(nx)dx|\
& leq & |int_{0}^{1}[f(x)-h(x)]g(nx)dx|+|int_{0}^{1}h(x)g(nx)dx|\
& leq & Mint_{0}^{1}|f(x)-h(x)|dx+frac{epsilon}{2}\
& < & epsilon.
end{eqnarray*}






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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    For part (a):
    Let $I=int_{0}^{1}|g(x)|dx$. Let $epsilon>0$ be given. Note that
    $f$ is uniformly continuous on $[0,1]$, so there exists $delta>0$
    such that $|f(x)-f(y)|<frac{epsilon}{I+1}$ whenever $x,yin[0,1]$
    with $|x-y|<delta$. Choose $Ninmathbb{N}$ such that $frac{1}{N}<delta$.
    By considering the substitution $y=nx$, we may rewrite
    begin{eqnarray*}
    int_{0}^{1}f(x)g(nx)dx & = & frac{1}{n}int_{0}^{n}f(frac{y}{n})g(y)dy\
    & = & frac{1}{n}sum_{k=1}^{n}int_{k-1}^{k}f(frac{y}{n})g(y)dy.
    end{eqnarray*}

    Let $ngeq N$ be arbitrary. Denote $f_{k}=f(frac{k}{n})$, for $k=1,2,ldots n$.
    By periodicity of $g$, we have $int_{k-1}^{k}g(y)dy=int_{0}^{1}g(y)dy=0$.
    Therefore
    begin{eqnarray*}
    & & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\
    & = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy|\
    & leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\
    & leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\
    & = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\
    & = & epsiloncdotfrac{I}{I+1}\
    & leq & epsilon.
    end{eqnarray*}

    Now it is clear that
    begin{eqnarray*}
    & & |int_{0}^{1}f(x)g(nx)dx|\
    & leq & frac{1}{n}sum_{k=1}^{n}|int_{k-1}^{k}f(frac{y}{n})g(y)dy|\
    & leq & frac{1}{n}sum_{k=1}^{n}epsilon\
    & = & epsilon.
    end{eqnarray*}






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      how you obtain this:begin{eqnarray*} & & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\ & = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy| *****\ & leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\ & leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\ & = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\ & = & epsiloncdotfrac{I}{I+1}\ & leq & epsilon. end{eqnarray*}
      $endgroup$
      – Richard Clare
      Jan 7 at 18:05












    • $begingroup$
      $int_{k-1}^k f_k g(y) dy =0$
      $endgroup$
      – Danny Pak-Keung Chan
      Jan 7 at 18:11










    • $begingroup$
      Sure! Thanks!!!!!!!!
      $endgroup$
      – Richard Clare
      Jan 7 at 18:14
















    1












    $begingroup$

    For part (a):
    Let $I=int_{0}^{1}|g(x)|dx$. Let $epsilon>0$ be given. Note that
    $f$ is uniformly continuous on $[0,1]$, so there exists $delta>0$
    such that $|f(x)-f(y)|<frac{epsilon}{I+1}$ whenever $x,yin[0,1]$
    with $|x-y|<delta$. Choose $Ninmathbb{N}$ such that $frac{1}{N}<delta$.
    By considering the substitution $y=nx$, we may rewrite
    begin{eqnarray*}
    int_{0}^{1}f(x)g(nx)dx & = & frac{1}{n}int_{0}^{n}f(frac{y}{n})g(y)dy\
    & = & frac{1}{n}sum_{k=1}^{n}int_{k-1}^{k}f(frac{y}{n})g(y)dy.
    end{eqnarray*}

    Let $ngeq N$ be arbitrary. Denote $f_{k}=f(frac{k}{n})$, for $k=1,2,ldots n$.
    By periodicity of $g$, we have $int_{k-1}^{k}g(y)dy=int_{0}^{1}g(y)dy=0$.
    Therefore
    begin{eqnarray*}
    & & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\
    & = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy|\
    & leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\
    & leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\
    & = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\
    & = & epsiloncdotfrac{I}{I+1}\
    & leq & epsilon.
    end{eqnarray*}

    Now it is clear that
    begin{eqnarray*}
    & & |int_{0}^{1}f(x)g(nx)dx|\
    & leq & frac{1}{n}sum_{k=1}^{n}|int_{k-1}^{k}f(frac{y}{n})g(y)dy|\
    & leq & frac{1}{n}sum_{k=1}^{n}epsilon\
    & = & epsilon.
    end{eqnarray*}






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      how you obtain this:begin{eqnarray*} & & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\ & = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy| *****\ & leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\ & leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\ & = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\ & = & epsiloncdotfrac{I}{I+1}\ & leq & epsilon. end{eqnarray*}
      $endgroup$
      – Richard Clare
      Jan 7 at 18:05












    • $begingroup$
      $int_{k-1}^k f_k g(y) dy =0$
      $endgroup$
      – Danny Pak-Keung Chan
      Jan 7 at 18:11










    • $begingroup$
      Sure! Thanks!!!!!!!!
      $endgroup$
      – Richard Clare
      Jan 7 at 18:14














    1












    1








    1





    $begingroup$

    For part (a):
    Let $I=int_{0}^{1}|g(x)|dx$. Let $epsilon>0$ be given. Note that
    $f$ is uniformly continuous on $[0,1]$, so there exists $delta>0$
    such that $|f(x)-f(y)|<frac{epsilon}{I+1}$ whenever $x,yin[0,1]$
    with $|x-y|<delta$. Choose $Ninmathbb{N}$ such that $frac{1}{N}<delta$.
    By considering the substitution $y=nx$, we may rewrite
    begin{eqnarray*}
    int_{0}^{1}f(x)g(nx)dx & = & frac{1}{n}int_{0}^{n}f(frac{y}{n})g(y)dy\
    & = & frac{1}{n}sum_{k=1}^{n}int_{k-1}^{k}f(frac{y}{n})g(y)dy.
    end{eqnarray*}

    Let $ngeq N$ be arbitrary. Denote $f_{k}=f(frac{k}{n})$, for $k=1,2,ldots n$.
    By periodicity of $g$, we have $int_{k-1}^{k}g(y)dy=int_{0}^{1}g(y)dy=0$.
    Therefore
    begin{eqnarray*}
    & & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\
    & = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy|\
    & leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\
    & leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\
    & = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\
    & = & epsiloncdotfrac{I}{I+1}\
    & leq & epsilon.
    end{eqnarray*}

    Now it is clear that
    begin{eqnarray*}
    & & |int_{0}^{1}f(x)g(nx)dx|\
    & leq & frac{1}{n}sum_{k=1}^{n}|int_{k-1}^{k}f(frac{y}{n})g(y)dy|\
    & leq & frac{1}{n}sum_{k=1}^{n}epsilon\
    & = & epsilon.
    end{eqnarray*}






    share|cite|improve this answer









    $endgroup$



    For part (a):
    Let $I=int_{0}^{1}|g(x)|dx$. Let $epsilon>0$ be given. Note that
    $f$ is uniformly continuous on $[0,1]$, so there exists $delta>0$
    such that $|f(x)-f(y)|<frac{epsilon}{I+1}$ whenever $x,yin[0,1]$
    with $|x-y|<delta$. Choose $Ninmathbb{N}$ such that $frac{1}{N}<delta$.
    By considering the substitution $y=nx$, we may rewrite
    begin{eqnarray*}
    int_{0}^{1}f(x)g(nx)dx & = & frac{1}{n}int_{0}^{n}f(frac{y}{n})g(y)dy\
    & = & frac{1}{n}sum_{k=1}^{n}int_{k-1}^{k}f(frac{y}{n})g(y)dy.
    end{eqnarray*}

    Let $ngeq N$ be arbitrary. Denote $f_{k}=f(frac{k}{n})$, for $k=1,2,ldots n$.
    By periodicity of $g$, we have $int_{k-1}^{k}g(y)dy=int_{0}^{1}g(y)dy=0$.
    Therefore
    begin{eqnarray*}
    & & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\
    & = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy|\
    & leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\
    & leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\
    & = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\
    & = & epsiloncdotfrac{I}{I+1}\
    & leq & epsilon.
    end{eqnarray*}

    Now it is clear that
    begin{eqnarray*}
    & & |int_{0}^{1}f(x)g(nx)dx|\
    & leq & frac{1}{n}sum_{k=1}^{n}|int_{k-1}^{k}f(frac{y}{n})g(y)dy|\
    & leq & frac{1}{n}sum_{k=1}^{n}epsilon\
    & = & epsilon.
    end{eqnarray*}







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Jan 7 at 17:39









    Danny Pak-Keung ChanDanny Pak-Keung Chan

    2,32538




    2,32538












    • $begingroup$
      how you obtain this:begin{eqnarray*} & & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\ & = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy| *****\ & leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\ & leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\ & = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\ & = & epsiloncdotfrac{I}{I+1}\ & leq & epsilon. end{eqnarray*}
      $endgroup$
      – Richard Clare
      Jan 7 at 18:05












    • $begingroup$
      $int_{k-1}^k f_k g(y) dy =0$
      $endgroup$
      – Danny Pak-Keung Chan
      Jan 7 at 18:11










    • $begingroup$
      Sure! Thanks!!!!!!!!
      $endgroup$
      – Richard Clare
      Jan 7 at 18:14


















    • $begingroup$
      how you obtain this:begin{eqnarray*} & & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\ & = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy| *****\ & leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\ & leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\ & = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\ & = & epsiloncdotfrac{I}{I+1}\ & leq & epsilon. end{eqnarray*}
      $endgroup$
      – Richard Clare
      Jan 7 at 18:05












    • $begingroup$
      $int_{k-1}^k f_k g(y) dy =0$
      $endgroup$
      – Danny Pak-Keung Chan
      Jan 7 at 18:11










    • $begingroup$
      Sure! Thanks!!!!!!!!
      $endgroup$
      – Richard Clare
      Jan 7 at 18:14
















    $begingroup$
    how you obtain this:begin{eqnarray*} & & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\ & = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy| *****\ & leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\ & leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\ & = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\ & = & epsiloncdotfrac{I}{I+1}\ & leq & epsilon. end{eqnarray*}
    $endgroup$
    – Richard Clare
    Jan 7 at 18:05






    $begingroup$
    how you obtain this:begin{eqnarray*} & & |int_{k-1}^{k}f(frac{y}{n})g(y)dy|\ & = & |int_{k-1}^{k}f(frac{y}{n})-f_{k}g(y)dy| *****\ & leq & int_{k-1}^{k}|f(frac{y}{n})-f_{k}||g(y)|dy\ & leq & int_{k-1}^{k}frac{epsilon}{I+1}|g(y)|dy\ & = & frac{epsilon}{I+1}int_{0}^{1}|g(y)|dy\ & = & epsiloncdotfrac{I}{I+1}\ & leq & epsilon. end{eqnarray*}
    $endgroup$
    – Richard Clare
    Jan 7 at 18:05














    $begingroup$
    $int_{k-1}^k f_k g(y) dy =0$
    $endgroup$
    – Danny Pak-Keung Chan
    Jan 7 at 18:11




    $begingroup$
    $int_{k-1}^k f_k g(y) dy =0$
    $endgroup$
    – Danny Pak-Keung Chan
    Jan 7 at 18:11












    $begingroup$
    Sure! Thanks!!!!!!!!
    $endgroup$
    – Richard Clare
    Jan 7 at 18:14




    $begingroup$
    Sure! Thanks!!!!!!!!
    $endgroup$
    – Richard Clare
    Jan 7 at 18:14











    1












    $begingroup$

    For (b): Since $g$ is continuous, there exists $M>0$ such that $|g(x)|leq M$
    for all $xin[0,1]$. By periodicity of $g$, it follows that $|g(x)|leq M$
    for all $xinmathbb{R}$. Recall that Lebesgue integrable function
    can be approximated by integrable continuous function in $||cdot||_{1}$-norm.
    More precisely, given $epsilon>0$, there exists a continuous function
    $h:mathbb{R}rightarrowmathbb{R}$ such that $int_{0}^{1}|f-h|<epsilon$.



    Let $epsilon>0$ be given. Choose continuous function $h:mathbb{R}rightarrowmathbb{R}$
    such that $int_{0}^{1}|f-h|<frac{epsilon}{2M}.$ By part (a), there
    exists $N$ such that $|int_{0}^{1}h(x)g(nx)dx|<frac{epsilon}{2}$
    whenever $ngeq N$. Now let $ngeq N$ be aribitrary, then
    begin{eqnarray*}
    & & |int_{0}^{1}f(x)g(nx)dx|\
    & leq & |int_{0}^{1}[f(x)-h(x)]g(nx)dx|+|int_{0}^{1}h(x)g(nx)dx|\
    & leq & Mint_{0}^{1}|f(x)-h(x)|dx+frac{epsilon}{2}\
    & < & epsilon.
    end{eqnarray*}






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      For (b): Since $g$ is continuous, there exists $M>0$ such that $|g(x)|leq M$
      for all $xin[0,1]$. By periodicity of $g$, it follows that $|g(x)|leq M$
      for all $xinmathbb{R}$. Recall that Lebesgue integrable function
      can be approximated by integrable continuous function in $||cdot||_{1}$-norm.
      More precisely, given $epsilon>0$, there exists a continuous function
      $h:mathbb{R}rightarrowmathbb{R}$ such that $int_{0}^{1}|f-h|<epsilon$.



      Let $epsilon>0$ be given. Choose continuous function $h:mathbb{R}rightarrowmathbb{R}$
      such that $int_{0}^{1}|f-h|<frac{epsilon}{2M}.$ By part (a), there
      exists $N$ such that $|int_{0}^{1}h(x)g(nx)dx|<frac{epsilon}{2}$
      whenever $ngeq N$. Now let $ngeq N$ be aribitrary, then
      begin{eqnarray*}
      & & |int_{0}^{1}f(x)g(nx)dx|\
      & leq & |int_{0}^{1}[f(x)-h(x)]g(nx)dx|+|int_{0}^{1}h(x)g(nx)dx|\
      & leq & Mint_{0}^{1}|f(x)-h(x)|dx+frac{epsilon}{2}\
      & < & epsilon.
      end{eqnarray*}






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        For (b): Since $g$ is continuous, there exists $M>0$ such that $|g(x)|leq M$
        for all $xin[0,1]$. By periodicity of $g$, it follows that $|g(x)|leq M$
        for all $xinmathbb{R}$. Recall that Lebesgue integrable function
        can be approximated by integrable continuous function in $||cdot||_{1}$-norm.
        More precisely, given $epsilon>0$, there exists a continuous function
        $h:mathbb{R}rightarrowmathbb{R}$ such that $int_{0}^{1}|f-h|<epsilon$.



        Let $epsilon>0$ be given. Choose continuous function $h:mathbb{R}rightarrowmathbb{R}$
        such that $int_{0}^{1}|f-h|<frac{epsilon}{2M}.$ By part (a), there
        exists $N$ such that $|int_{0}^{1}h(x)g(nx)dx|<frac{epsilon}{2}$
        whenever $ngeq N$. Now let $ngeq N$ be aribitrary, then
        begin{eqnarray*}
        & & |int_{0}^{1}f(x)g(nx)dx|\
        & leq & |int_{0}^{1}[f(x)-h(x)]g(nx)dx|+|int_{0}^{1}h(x)g(nx)dx|\
        & leq & Mint_{0}^{1}|f(x)-h(x)|dx+frac{epsilon}{2}\
        & < & epsilon.
        end{eqnarray*}






        share|cite|improve this answer









        $endgroup$



        For (b): Since $g$ is continuous, there exists $M>0$ such that $|g(x)|leq M$
        for all $xin[0,1]$. By periodicity of $g$, it follows that $|g(x)|leq M$
        for all $xinmathbb{R}$. Recall that Lebesgue integrable function
        can be approximated by integrable continuous function in $||cdot||_{1}$-norm.
        More precisely, given $epsilon>0$, there exists a continuous function
        $h:mathbb{R}rightarrowmathbb{R}$ such that $int_{0}^{1}|f-h|<epsilon$.



        Let $epsilon>0$ be given. Choose continuous function $h:mathbb{R}rightarrowmathbb{R}$
        such that $int_{0}^{1}|f-h|<frac{epsilon}{2M}.$ By part (a), there
        exists $N$ such that $|int_{0}^{1}h(x)g(nx)dx|<frac{epsilon}{2}$
        whenever $ngeq N$. Now let $ngeq N$ be aribitrary, then
        begin{eqnarray*}
        & & |int_{0}^{1}f(x)g(nx)dx|\
        & leq & |int_{0}^{1}[f(x)-h(x)]g(nx)dx|+|int_{0}^{1}h(x)g(nx)dx|\
        & leq & Mint_{0}^{1}|f(x)-h(x)|dx+frac{epsilon}{2}\
        & < & epsilon.
        end{eqnarray*}







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 7 at 17:57









        Danny Pak-Keung ChanDanny Pak-Keung Chan

        2,32538




        2,32538






























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