Scaled series of shifted continuous function












2












$begingroup$


I encountered (a significantly more general version of) the following problem: Consider any sequence $(alpha_k)_kin ell^1$ with $alpha_kneq 0$ for any $k$ and let $psi:[0,infty)rightarrowmathbb{R}$ be a bounded continuous function with $psi(0)=0$ and with the following property: for any $m<0$ we have
$$sumlimits_{j=-infty}^{m}alpha_{|j|}psi(s-j+m)=0$$
for every $sin(0,1)$. I want to conclude that $psi(s)=0$ for $sin[0,epsilon)$ for some $epsilon>0$, but I do not even know where to start. Does anyone have a suggestion?



Note: in my particular case I have that the sequence $(alpha_k)_k$ decays exponentially, but I do not think that is relevant for this part.



I encountered this problem in my own research, where I consider the Hale inner product (see J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, chapter 7) for functional differential equations with infinitely many shifts. In particular, I try to show under which conditions this Hale inner product is non-degenerate, by trying to imitate the proof of Proposition 4.16 of the paper Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations by Mallet-Paret and Verduyn Lunel.






real-analysis sequences-and-series functional-analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    I sincerely doubt that would help in this case but I'll try anyway. I encountered it in my own research, where I consider the Hale inner product (see J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential, chapter 7) for functional differential equations with infinitely many shifts. In particular, I try to show under which conditions this Hale inner product is non-degenerate, by trying to imitate the proof of Proposition 4.16 of the paper Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations by Mallet-Paret and Verduyn Lunel
    $endgroup$
    – WillemMSchouten
    Jan 8 at 14:56






  • 2




    $begingroup$
    It really does help to include that context. It helps answerers to understand the the level that answers should aim for (you tagged this with real-analysis and sequences-and-series, which are two tags used quite a lot by undergraduates in introductory level classes; however, as you are studying function DEs, it seems likely that one needn't provide an answer at an introductory undergraduate level), and it helps answerers to understand what tools might be applicable to the problem.
    $endgroup$
    – Xander Henderson
    Jan 8 at 15:20






  • 2




    $begingroup$
    Moreover, since MSE is meant to be a repository of questions and answers that are useful both to the askers and to future students and/or researchers, the additional context will help other users to find this question in the future. I have taken the liberty of editing your question to add the additional context.
    $endgroup$
    – Xander Henderson
    Jan 8 at 15:20










  • $begingroup$
    Well, hopefully the answer can be understood at an undergraduate level;) But thanks, I will keep these things in mind the next time I ask a question.
    $endgroup$
    – WillemMSchouten
    Jan 8 at 15:27
















2












$begingroup$


I encountered (a significantly more general version of) the following problem: Consider any sequence $(alpha_k)_kin ell^1$ with $alpha_kneq 0$ for any $k$ and let $psi:[0,infty)rightarrowmathbb{R}$ be a bounded continuous function with $psi(0)=0$ and with the following property: for any $m<0$ we have
$$sumlimits_{j=-infty}^{m}alpha_{|j|}psi(s-j+m)=0$$
for every $sin(0,1)$. I want to conclude that $psi(s)=0$ for $sin[0,epsilon)$ for some $epsilon>0$, but I do not even know where to start. Does anyone have a suggestion?



Note: in my particular case I have that the sequence $(alpha_k)_k$ decays exponentially, but I do not think that is relevant for this part.



I encountered this problem in my own research, where I consider the Hale inner product (see J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, chapter 7) for functional differential equations with infinitely many shifts. In particular, I try to show under which conditions this Hale inner product is non-degenerate, by trying to imitate the proof of Proposition 4.16 of the paper Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations by Mallet-Paret and Verduyn Lunel.






real-analysis sequences-and-series functional-analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    I sincerely doubt that would help in this case but I'll try anyway. I encountered it in my own research, where I consider the Hale inner product (see J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential, chapter 7) for functional differential equations with infinitely many shifts. In particular, I try to show under which conditions this Hale inner product is non-degenerate, by trying to imitate the proof of Proposition 4.16 of the paper Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations by Mallet-Paret and Verduyn Lunel
    $endgroup$
    – WillemMSchouten
    Jan 8 at 14:56






  • 2




    $begingroup$
    It really does help to include that context. It helps answerers to understand the the level that answers should aim for (you tagged this with real-analysis and sequences-and-series, which are two tags used quite a lot by undergraduates in introductory level classes; however, as you are studying function DEs, it seems likely that one needn't provide an answer at an introductory undergraduate level), and it helps answerers to understand what tools might be applicable to the problem.
    $endgroup$
    – Xander Henderson
    Jan 8 at 15:20






  • 2




    $begingroup$
    Moreover, since MSE is meant to be a repository of questions and answers that are useful both to the askers and to future students and/or researchers, the additional context will help other users to find this question in the future. I have taken the liberty of editing your question to add the additional context.
    $endgroup$
    – Xander Henderson
    Jan 8 at 15:20










  • $begingroup$
    Well, hopefully the answer can be understood at an undergraduate level;) But thanks, I will keep these things in mind the next time I ask a question.
    $endgroup$
    – WillemMSchouten
    Jan 8 at 15:27














2












2








2





$begingroup$


I encountered (a significantly more general version of) the following problem: Consider any sequence $(alpha_k)_kin ell^1$ with $alpha_kneq 0$ for any $k$ and let $psi:[0,infty)rightarrowmathbb{R}$ be a bounded continuous function with $psi(0)=0$ and with the following property: for any $m<0$ we have
$$sumlimits_{j=-infty}^{m}alpha_{|j|}psi(s-j+m)=0$$
for every $sin(0,1)$. I want to conclude that $psi(s)=0$ for $sin[0,epsilon)$ for some $epsilon>0$, but I do not even know where to start. Does anyone have a suggestion?



Note: in my particular case I have that the sequence $(alpha_k)_k$ decays exponentially, but I do not think that is relevant for this part.



I encountered this problem in my own research, where I consider the Hale inner product (see J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, chapter 7) for functional differential equations with infinitely many shifts. In particular, I try to show under which conditions this Hale inner product is non-degenerate, by trying to imitate the proof of Proposition 4.16 of the paper Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations by Mallet-Paret and Verduyn Lunel.






real-analysis sequences-and-series functional-analysis







share|cite|improve this question











$endgroup$




I encountered (a significantly more general version of) the following problem: Consider any sequence $(alpha_k)_kin ell^1$ with $alpha_kneq 0$ for any $k$ and let $psi:[0,infty)rightarrowmathbb{R}$ be a bounded continuous function with $psi(0)=0$ and with the following property: for any $m<0$ we have
$$sumlimits_{j=-infty}^{m}alpha_{|j|}psi(s-j+m)=0$$
for every $sin(0,1)$. I want to conclude that $psi(s)=0$ for $sin[0,epsilon)$ for some $epsilon>0$, but I do not even know where to start. Does anyone have a suggestion?



Note: in my particular case I have that the sequence $(alpha_k)_k$ decays exponentially, but I do not think that is relevant for this part.



I encountered this problem in my own research, where I consider the Hale inner product (see J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, chapter 7) for functional differential equations with infinitely many shifts. In particular, I try to show under which conditions this Hale inner product is non-degenerate, by trying to imitate the proof of Proposition 4.16 of the paper Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations by Mallet-Paret and Verduyn Lunel.






real-analysis sequences-and-series functional-analysis




real-analysis sequences-and-series functional-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 9 at 13:24







WillemMSchouten

















asked Jan 8 at 14:26









WillemMSchoutenWillemMSchouten

816




816












  • $begingroup$
    I sincerely doubt that would help in this case but I'll try anyway. I encountered it in my own research, where I consider the Hale inner product (see J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential, chapter 7) for functional differential equations with infinitely many shifts. In particular, I try to show under which conditions this Hale inner product is non-degenerate, by trying to imitate the proof of Proposition 4.16 of the paper Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations by Mallet-Paret and Verduyn Lunel
    $endgroup$
    – WillemMSchouten
    Jan 8 at 14:56






  • 2




    $begingroup$
    It really does help to include that context. It helps answerers to understand the the level that answers should aim for (you tagged this with real-analysis and sequences-and-series, which are two tags used quite a lot by undergraduates in introductory level classes; however, as you are studying function DEs, it seems likely that one needn't provide an answer at an introductory undergraduate level), and it helps answerers to understand what tools might be applicable to the problem.
    $endgroup$
    – Xander Henderson
    Jan 8 at 15:20






  • 2




    $begingroup$
    Moreover, since MSE is meant to be a repository of questions and answers that are useful both to the askers and to future students and/or researchers, the additional context will help other users to find this question in the future. I have taken the liberty of editing your question to add the additional context.
    $endgroup$
    – Xander Henderson
    Jan 8 at 15:20










  • $begingroup$
    Well, hopefully the answer can be understood at an undergraduate level;) But thanks, I will keep these things in mind the next time I ask a question.
    $endgroup$
    – WillemMSchouten
    Jan 8 at 15:27


















  • $begingroup$
    I sincerely doubt that would help in this case but I'll try anyway. I encountered it in my own research, where I consider the Hale inner product (see J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential, chapter 7) for functional differential equations with infinitely many shifts. In particular, I try to show under which conditions this Hale inner product is non-degenerate, by trying to imitate the proof of Proposition 4.16 of the paper Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations by Mallet-Paret and Verduyn Lunel
    $endgroup$
    – WillemMSchouten
    Jan 8 at 14:56






  • 2




    $begingroup$
    It really does help to include that context. It helps answerers to understand the the level that answers should aim for (you tagged this with real-analysis and sequences-and-series, which are two tags used quite a lot by undergraduates in introductory level classes; however, as you are studying function DEs, it seems likely that one needn't provide an answer at an introductory undergraduate level), and it helps answerers to understand what tools might be applicable to the problem.
    $endgroup$
    – Xander Henderson
    Jan 8 at 15:20






  • 2




    $begingroup$
    Moreover, since MSE is meant to be a repository of questions and answers that are useful both to the askers and to future students and/or researchers, the additional context will help other users to find this question in the future. I have taken the liberty of editing your question to add the additional context.
    $endgroup$
    – Xander Henderson
    Jan 8 at 15:20










  • $begingroup$
    Well, hopefully the answer can be understood at an undergraduate level;) But thanks, I will keep these things in mind the next time I ask a question.
    $endgroup$
    – WillemMSchouten
    Jan 8 at 15:27
















$begingroup$
I sincerely doubt that would help in this case but I'll try anyway. I encountered it in my own research, where I consider the Hale inner product (see J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential, chapter 7) for functional differential equations with infinitely many shifts. In particular, I try to show under which conditions this Hale inner product is non-degenerate, by trying to imitate the proof of Proposition 4.16 of the paper Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations by Mallet-Paret and Verduyn Lunel
$endgroup$
– WillemMSchouten
Jan 8 at 14:56




$begingroup$
I sincerely doubt that would help in this case but I'll try anyway. I encountered it in my own research, where I consider the Hale inner product (see J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential, chapter 7) for functional differential equations with infinitely many shifts. In particular, I try to show under which conditions this Hale inner product is non-degenerate, by trying to imitate the proof of Proposition 4.16 of the paper Exponential Dichotomies and Wiener-Hopf Factorizations for Mixed-Type Functional Differential Equations by Mallet-Paret and Verduyn Lunel
$endgroup$
– WillemMSchouten
Jan 8 at 14:56




2




2




$begingroup$
It really does help to include that context. It helps answerers to understand the the level that answers should aim for (you tagged this with real-analysis and sequences-and-series, which are two tags used quite a lot by undergraduates in introductory level classes; however, as you are studying function DEs, it seems likely that one needn't provide an answer at an introductory undergraduate level), and it helps answerers to understand what tools might be applicable to the problem.
$endgroup$
– Xander Henderson
Jan 8 at 15:20




$begingroup$
It really does help to include that context. It helps answerers to understand the the level that answers should aim for (you tagged this with real-analysis and sequences-and-series, which are two tags used quite a lot by undergraduates in introductory level classes; however, as you are studying function DEs, it seems likely that one needn't provide an answer at an introductory undergraduate level), and it helps answerers to understand what tools might be applicable to the problem.
$endgroup$
– Xander Henderson
Jan 8 at 15:20




2




2




$begingroup$
Moreover, since MSE is meant to be a repository of questions and answers that are useful both to the askers and to future students and/or researchers, the additional context will help other users to find this question in the future. I have taken the liberty of editing your question to add the additional context.
$endgroup$
– Xander Henderson
Jan 8 at 15:20




$begingroup$
Moreover, since MSE is meant to be a repository of questions and answers that are useful both to the askers and to future students and/or researchers, the additional context will help other users to find this question in the future. I have taken the liberty of editing your question to add the additional context.
$endgroup$
– Xander Henderson
Jan 8 at 15:20












$begingroup$
Well, hopefully the answer can be understood at an undergraduate level;) But thanks, I will keep these things in mind the next time I ask a question.
$endgroup$
– WillemMSchouten
Jan 8 at 15:27




$begingroup$
Well, hopefully the answer can be understood at an undergraduate level;) But thanks, I will keep these things in mind the next time I ask a question.
$endgroup$
– WillemMSchouten
Jan 8 at 15:27










1 Answer
1






active

oldest

votes


















0












$begingroup$

After some more thinking I found a counter example myself:
Consider the sequence $(alpha_k)_k$ with $alpha_k=exp(-k)$ for $kinmathbb{Z}_{geq 0}$ and consider the continuous bounded function $psi:[0,infty)rightarrow mathbb{R}$ which has $psi(k)=0$ for any $kinmathbb{Z}_{geq 0}$, which has $psi(k+frac{1}{2})=1$ for $kinmathbb{Z}_{geq 0}$ even and $psi(k+frac{1}{2})=-e$ for $kinmathbb{Z}_{geq 1}$ odd and which is linear on the intervals I have not defined it yet. Then for any $minmathbb{Z}_{<0}$ and any $sin(0,frac{1}{2}]$, we have
$$sumlimits_{j=-infty}^m alpha_{|j|}psi(s-j+m)=sumlimits_{j=-infty,j-mtext{ even}}^m exp(-|j|)s+sumlimits_{j=-infty,j-mtext{ odd}}^m exp(-|j|)(-ecdot s)=0$$
The same result holds for $sin(frac{1}{2},1)$ by symmetry. However, there is no $epsilon>0$ with $psi(s)=0$ for $sin[0,epsilon)$.






share|cite|improve this answer











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    0












    $begingroup$

    After some more thinking I found a counter example myself:
    Consider the sequence $(alpha_k)_k$ with $alpha_k=exp(-k)$ for $kinmathbb{Z}_{geq 0}$ and consider the continuous bounded function $psi:[0,infty)rightarrow mathbb{R}$ which has $psi(k)=0$ for any $kinmathbb{Z}_{geq 0}$, which has $psi(k+frac{1}{2})=1$ for $kinmathbb{Z}_{geq 0}$ even and $psi(k+frac{1}{2})=-e$ for $kinmathbb{Z}_{geq 1}$ odd and which is linear on the intervals I have not defined it yet. Then for any $minmathbb{Z}_{<0}$ and any $sin(0,frac{1}{2}]$, we have
    $$sumlimits_{j=-infty}^m alpha_{|j|}psi(s-j+m)=sumlimits_{j=-infty,j-mtext{ even}}^m exp(-|j|)s+sumlimits_{j=-infty,j-mtext{ odd}}^m exp(-|j|)(-ecdot s)=0$$
    The same result holds for $sin(frac{1}{2},1)$ by symmetry. However, there is no $epsilon>0$ with $psi(s)=0$ for $sin[0,epsilon)$.






    share|cite|improve this answer











    $endgroup$


















      0












      $begingroup$

      After some more thinking I found a counter example myself:
      Consider the sequence $(alpha_k)_k$ with $alpha_k=exp(-k)$ for $kinmathbb{Z}_{geq 0}$ and consider the continuous bounded function $psi:[0,infty)rightarrow mathbb{R}$ which has $psi(k)=0$ for any $kinmathbb{Z}_{geq 0}$, which has $psi(k+frac{1}{2})=1$ for $kinmathbb{Z}_{geq 0}$ even and $psi(k+frac{1}{2})=-e$ for $kinmathbb{Z}_{geq 1}$ odd and which is linear on the intervals I have not defined it yet. Then for any $minmathbb{Z}_{<0}$ and any $sin(0,frac{1}{2}]$, we have
      $$sumlimits_{j=-infty}^m alpha_{|j|}psi(s-j+m)=sumlimits_{j=-infty,j-mtext{ even}}^m exp(-|j|)s+sumlimits_{j=-infty,j-mtext{ odd}}^m exp(-|j|)(-ecdot s)=0$$
      The same result holds for $sin(frac{1}{2},1)$ by symmetry. However, there is no $epsilon>0$ with $psi(s)=0$ for $sin[0,epsilon)$.






      share|cite|improve this answer











      $endgroup$
















        0












        0








        0





        $begingroup$

        After some more thinking I found a counter example myself:
        Consider the sequence $(alpha_k)_k$ with $alpha_k=exp(-k)$ for $kinmathbb{Z}_{geq 0}$ and consider the continuous bounded function $psi:[0,infty)rightarrow mathbb{R}$ which has $psi(k)=0$ for any $kinmathbb{Z}_{geq 0}$, which has $psi(k+frac{1}{2})=1$ for $kinmathbb{Z}_{geq 0}$ even and $psi(k+frac{1}{2})=-e$ for $kinmathbb{Z}_{geq 1}$ odd and which is linear on the intervals I have not defined it yet. Then for any $minmathbb{Z}_{<0}$ and any $sin(0,frac{1}{2}]$, we have
        $$sumlimits_{j=-infty}^m alpha_{|j|}psi(s-j+m)=sumlimits_{j=-infty,j-mtext{ even}}^m exp(-|j|)s+sumlimits_{j=-infty,j-mtext{ odd}}^m exp(-|j|)(-ecdot s)=0$$
        The same result holds for $sin(frac{1}{2},1)$ by symmetry. However, there is no $epsilon>0$ with $psi(s)=0$ for $sin[0,epsilon)$.






        share|cite|improve this answer











        $endgroup$



        After some more thinking I found a counter example myself:
        Consider the sequence $(alpha_k)_k$ with $alpha_k=exp(-k)$ for $kinmathbb{Z}_{geq 0}$ and consider the continuous bounded function $psi:[0,infty)rightarrow mathbb{R}$ which has $psi(k)=0$ for any $kinmathbb{Z}_{geq 0}$, which has $psi(k+frac{1}{2})=1$ for $kinmathbb{Z}_{geq 0}$ even and $psi(k+frac{1}{2})=-e$ for $kinmathbb{Z}_{geq 1}$ odd and which is linear on the intervals I have not defined it yet. Then for any $minmathbb{Z}_{<0}$ and any $sin(0,frac{1}{2}]$, we have
        $$sumlimits_{j=-infty}^m alpha_{|j|}psi(s-j+m)=sumlimits_{j=-infty,j-mtext{ even}}^m exp(-|j|)s+sumlimits_{j=-infty,j-mtext{ odd}}^m exp(-|j|)(-ecdot s)=0$$
        The same result holds for $sin(frac{1}{2},1)$ by symmetry. However, there is no $epsilon>0$ with $psi(s)=0$ for $sin[0,epsilon)$.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 9 at 13:25

























        answered Jan 9 at 10:50









        WillemMSchoutenWillemMSchouten

        816




        816






























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