Convergence of series to a $C^infty$ function

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Let $S_+ = mathbbR times t>0$ and $,f in C^infty(S^+)$ such that $,f$ and all its derivatives have continuous limits as $t to 0^+$.



In this paper the author defines the following extension of $f$ for $t<0$:
beginequation
Ef(x,t) = sum_k=1^infty a_k ,phi(b_k t), f(x,b_k t)
endequation
where



  1. $phi in C^infty(mathbbR)$ with: $phi(t)=1$ for $0leq tleq1$ and $,phi(t)=0$ for $tgeq$2;



  2. $a_k, b_k$ such that:



    • $b_k < 0$ and $b_k to - infty$,

    • $sum_k=1^infty|a_k||b_k|^n <infty$ for $, n = 0,1, 2, dots$,

    • $sum_k=1^infty (a_k)(b_k)^n = 1$ for $, n = 0,1, 2, dots$.


The existence of sequences $a_k, b_k$ such that 2. holds is proved in the paper.



My observations



  1. Since $b_k to -infty$, for each $t<0$ the sum in $Ef$ is finite and therefore $Ef$ is pointwise well defined.

  2. If $C$ is a compact set in $mathbbRtimes(-infty,0)$ and $bart = max_(x,t) in C t$, let $barN$ be such that for every $k geq barN$ one has $b_k bart > 2$. Then since $b_k t geq b_k bart > 2 $ for all $t in C$, it follows that:
    $$ Ef(x,t) = sum_k=1^barN a_k ,phi(b_k t), f(x,b_k t)qquad forall (x,t) in C $$
    It follows that on every compact set $C subset mathbbRtimes(-infty,0)$ $Ef$ is uniform limit of functions in $C^infty(mathbbRtimes(-infty,0))$. Then it easy to see that the restriction of
    $Ef$ to $t<0$ is in $C^infty(mathbbRtimes(-infty,0))$.

My question:



I have shown that $Ef(t,x)$ is well defined for every $(t,x) in R^2$,
moreover the restriction of $Ef$ to $t<0$ is in $C^infty(mathbbRtimes(-infty,0))$. I'm left with $Ef(t,x)$ has derivative as $tto 0^-$.




Question: How to prove that
$lim_tto 0^- Ef(x,t)$ and
$ lim_tto 0^- partial_t^n partial_x^j Ef(x,t)$
exist finite?




Remarks



a) I believe that to answer the question "dominated convergence" using the second point of 2.2 is what
is needed, but I'm missing the details.



b) If the limit exist then they agree to those for $tto 0^+$, since using the third point in 2.:
$$ lim_tto 0^- Ef(x,t) = sum_k=1^infty lim_tto 0^- a_k ,phi(b_k t), f(x,b_k t) = f(x, 0^+) sum_k=1^infty a_k = f(x, 0^+)$$
$$ lim_tto 0^- partial_t^n partial_x^j Ef(x,t) = sum_k=1^infty lim_tto 0^- a_k sum_j leq n C_j,k partial_t^n-j ,phi(b_k t),partial_t^j partial_x^j f(x,b_k t) = partial_t^m partial_x^n f(x, 0^+) sum_k=1^infty a_k b_k^n = partial_t^m partial_x^n f(x,0^+)$$




real-analysis sequences-and-series derivatives




share|cite|improve this question

























    up vote
    1
    down vote

    favorite












    Let $S_+ = mathbbR times t>0$ and $,f in C^infty(S^+)$ such that $,f$ and all its derivatives have continuous limits as $t to 0^+$.



    In this paper the author defines the following extension of $f$ for $t<0$:
    beginequation
    Ef(x,t) = sum_k=1^infty a_k ,phi(b_k t), f(x,b_k t)
    endequation
    where



    1. $phi in C^infty(mathbbR)$ with: $phi(t)=1$ for $0leq tleq1$ and $,phi(t)=0$ for $tgeq$2;



    2. $a_k, b_k$ such that:



      • $b_k < 0$ and $b_k to - infty$,

      • $sum_k=1^infty|a_k||b_k|^n <infty$ for $, n = 0,1, 2, dots$,

      • $sum_k=1^infty (a_k)(b_k)^n = 1$ for $, n = 0,1, 2, dots$.


    The existence of sequences $a_k, b_k$ such that 2. holds is proved in the paper.



    My observations



    1. Since $b_k to -infty$, for each $t<0$ the sum in $Ef$ is finite and therefore $Ef$ is pointwise well defined.

    2. If $C$ is a compact set in $mathbbRtimes(-infty,0)$ and $bart = max_(x,t) in C t$, let $barN$ be such that for every $k geq barN$ one has $b_k bart > 2$. Then since $b_k t geq b_k bart > 2 $ for all $t in C$, it follows that:
      $$ Ef(x,t) = sum_k=1^barN a_k ,phi(b_k t), f(x,b_k t)qquad forall (x,t) in C $$
      It follows that on every compact set $C subset mathbbRtimes(-infty,0)$ $Ef$ is uniform limit of functions in $C^infty(mathbbRtimes(-infty,0))$. Then it easy to see that the restriction of
      $Ef$ to $t<0$ is in $C^infty(mathbbRtimes(-infty,0))$.

    My question:



    I have shown that $Ef(t,x)$ is well defined for every $(t,x) in R^2$,
    moreover the restriction of $Ef$ to $t<0$ is in $C^infty(mathbbRtimes(-infty,0))$. I'm left with $Ef(t,x)$ has derivative as $tto 0^-$.




    Question: How to prove that
    $lim_tto 0^- Ef(x,t)$ and
    $ lim_tto 0^- partial_t^n partial_x^j Ef(x,t)$
    exist finite?




    Remarks



    a) I believe that to answer the question "dominated convergence" using the second point of 2.2 is what
    is needed, but I'm missing the details.



    b) If the limit exist then they agree to those for $tto 0^+$, since using the third point in 2.:
    $$ lim_tto 0^- Ef(x,t) = sum_k=1^infty lim_tto 0^- a_k ,phi(b_k t), f(x,b_k t) = f(x, 0^+) sum_k=1^infty a_k = f(x, 0^+)$$
    $$ lim_tto 0^- partial_t^n partial_x^j Ef(x,t) = sum_k=1^infty lim_tto 0^- a_k sum_j leq n C_j,k partial_t^n-j ,phi(b_k t),partial_t^j partial_x^j f(x,b_k t) = partial_t^m partial_x^n f(x, 0^+) sum_k=1^infty a_k b_k^n = partial_t^m partial_x^n f(x,0^+)$$




    real-analysis sequences-and-series derivatives




    share|cite|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Let $S_+ = mathbbR times t>0$ and $,f in C^infty(S^+)$ such that $,f$ and all its derivatives have continuous limits as $t to 0^+$.



      In this paper the author defines the following extension of $f$ for $t<0$:
      beginequation
      Ef(x,t) = sum_k=1^infty a_k ,phi(b_k t), f(x,b_k t)
      endequation
      where



      1. $phi in C^infty(mathbbR)$ with: $phi(t)=1$ for $0leq tleq1$ and $,phi(t)=0$ for $tgeq$2;



      2. $a_k, b_k$ such that:



        • $b_k < 0$ and $b_k to - infty$,

        • $sum_k=1^infty|a_k||b_k|^n <infty$ for $, n = 0,1, 2, dots$,

        • $sum_k=1^infty (a_k)(b_k)^n = 1$ for $, n = 0,1, 2, dots$.


      The existence of sequences $a_k, b_k$ such that 2. holds is proved in the paper.



      My observations



      1. Since $b_k to -infty$, for each $t<0$ the sum in $Ef$ is finite and therefore $Ef$ is pointwise well defined.

      2. If $C$ is a compact set in $mathbbRtimes(-infty,0)$ and $bart = max_(x,t) in C t$, let $barN$ be such that for every $k geq barN$ one has $b_k bart > 2$. Then since $b_k t geq b_k bart > 2 $ for all $t in C$, it follows that:
        $$ Ef(x,t) = sum_k=1^barN a_k ,phi(b_k t), f(x,b_k t)qquad forall (x,t) in C $$
        It follows that on every compact set $C subset mathbbRtimes(-infty,0)$ $Ef$ is uniform limit of functions in $C^infty(mathbbRtimes(-infty,0))$. Then it easy to see that the restriction of
        $Ef$ to $t<0$ is in $C^infty(mathbbRtimes(-infty,0))$.

      My question:



      I have shown that $Ef(t,x)$ is well defined for every $(t,x) in R^2$,
      moreover the restriction of $Ef$ to $t<0$ is in $C^infty(mathbbRtimes(-infty,0))$. I'm left with $Ef(t,x)$ has derivative as $tto 0^-$.




      Question: How to prove that
      $lim_tto 0^- Ef(x,t)$ and
      $ lim_tto 0^- partial_t^n partial_x^j Ef(x,t)$
      exist finite?




      Remarks



      a) I believe that to answer the question "dominated convergence" using the second point of 2.2 is what
      is needed, but I'm missing the details.



      b) If the limit exist then they agree to those for $tto 0^+$, since using the third point in 2.:
      $$ lim_tto 0^- Ef(x,t) = sum_k=1^infty lim_tto 0^- a_k ,phi(b_k t), f(x,b_k t) = f(x, 0^+) sum_k=1^infty a_k = f(x, 0^+)$$
      $$ lim_tto 0^- partial_t^n partial_x^j Ef(x,t) = sum_k=1^infty lim_tto 0^- a_k sum_j leq n C_j,k partial_t^n-j ,phi(b_k t),partial_t^j partial_x^j f(x,b_k t) = partial_t^m partial_x^n f(x, 0^+) sum_k=1^infty a_k b_k^n = partial_t^m partial_x^n f(x,0^+)$$




      real-analysis sequences-and-series derivatives




      share|cite|improve this question













      Let $S_+ = mathbbR times t>0$ and $,f in C^infty(S^+)$ such that $,f$ and all its derivatives have continuous limits as $t to 0^+$.



      In this paper the author defines the following extension of $f$ for $t<0$:
      beginequation
      Ef(x,t) = sum_k=1^infty a_k ,phi(b_k t), f(x,b_k t)
      endequation
      where



      1. $phi in C^infty(mathbbR)$ with: $phi(t)=1$ for $0leq tleq1$ and $,phi(t)=0$ for $tgeq$2;



      2. $a_k, b_k$ such that:



        • $b_k < 0$ and $b_k to - infty$,

        • $sum_k=1^infty|a_k||b_k|^n <infty$ for $, n = 0,1, 2, dots$,

        • $sum_k=1^infty (a_k)(b_k)^n = 1$ for $, n = 0,1, 2, dots$.


      The existence of sequences $a_k, b_k$ such that 2. holds is proved in the paper.



      My observations



      1. Since $b_k to -infty$, for each $t<0$ the sum in $Ef$ is finite and therefore $Ef$ is pointwise well defined.

      2. If $C$ is a compact set in $mathbbRtimes(-infty,0)$ and $bart = max_(x,t) in C t$, let $barN$ be such that for every $k geq barN$ one has $b_k bart > 2$. Then since $b_k t geq b_k bart > 2 $ for all $t in C$, it follows that:
        $$ Ef(x,t) = sum_k=1^barN a_k ,phi(b_k t), f(x,b_k t)qquad forall (x,t) in C $$
        It follows that on every compact set $C subset mathbbRtimes(-infty,0)$ $Ef$ is uniform limit of functions in $C^infty(mathbbRtimes(-infty,0))$. Then it easy to see that the restriction of
        $Ef$ to $t<0$ is in $C^infty(mathbbRtimes(-infty,0))$.

      My question:



      I have shown that $Ef(t,x)$ is well defined for every $(t,x) in R^2$,
      moreover the restriction of $Ef$ to $t<0$ is in $C^infty(mathbbRtimes(-infty,0))$. I'm left with $Ef(t,x)$ has derivative as $tto 0^-$.




      Question: How to prove that
      $lim_tto 0^- Ef(x,t)$ and
      $ lim_tto 0^- partial_t^n partial_x^j Ef(x,t)$
      exist finite?




      Remarks



      a) I believe that to answer the question "dominated convergence" using the second point of 2.2 is what
      is needed, but I'm missing the details.



      b) If the limit exist then they agree to those for $tto 0^+$, since using the third point in 2.:
      $$ lim_tto 0^- Ef(x,t) = sum_k=1^infty lim_tto 0^- a_k ,phi(b_k t), f(x,b_k t) = f(x, 0^+) sum_k=1^infty a_k = f(x, 0^+)$$
      $$ lim_tto 0^- partial_t^n partial_x^j Ef(x,t) = sum_k=1^infty lim_tto 0^- a_k sum_j leq n C_j,k partial_t^n-j ,phi(b_k t),partial_t^j partial_x^j f(x,b_k t) = partial_t^m partial_x^n f(x, 0^+) sum_k=1^infty a_k b_k^n = partial_t^m partial_x^n f(x,0^+)$$





      real-analysis sequences-and-series derivatives





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