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 February 18th, 2013, 07:15 PM #1 Newbie   Joined: Feb 2013 Posts: 16 Thanks: 0 if the function sutisfies conditions For a given real number $c>0$, denote by $(\psi_{n,c}(\cdot))_{n\geq 0}$ are the eigenfunctions of the Sturm-Liouville operators $L_c=(1-x^2)\frac{d^2\psi}{dx^2}-2x \frac{d \psi}{dx}-c^2x^2 \psi$ defined on $C^2[-1,1]$. It happened tha the functions $(\psi_{n,c}(\cdot))_{n\geq 0}$ are also the eigenfunctions of the compact integral operator defined on $L^2[-1,1]$by $F_c(\psi)(x)=\frac{1}{\pi}\int_{-1}^1\frac{\sin c(x-y)}{x-y}\psi(y)dy$. Also, $(\psi_{n,c}(\cdot))_{n\geq 0}$ are normalized as following $\int_{-1}^1|\psi_{n,c}(x)|^2dx=1$ and $\int_R|\psi_{n,c}(x)|^2dx=\frac{1}{\lambda(c)}, n\geq 0$, where $(\lamda_n(c))_n$ are the infinite sequence of the eigenvalues of $F_c$, arranged in the non-increasing order and such that $\left(\frac{c}{25n}\right)^{n-1}\leq \lambda_n(c)\leq c\left(\frac{ec}{4n}\right)^{2n}$. My question is the following: If these functions $(\psi_{n,c}(\cdot))_{n\geq 0}$ sutisfies the following conditions: Let $f(x)=\psi_{n,c}\in L^2[-1,1]$. Fix $\epsilon>0$ . Can one define A so that $\int_{-\infty}^{-A}|f(x)|^2dx< \epsilon^2$ and $\int_A^{\infty}|f(x)|^2dx< \epsilon^2$.
February 18th, 2013, 07:15 PM   #2
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Re: if the function sutisfies conditions

Quote:
 Originally Posted by bvh For a given real number $c>0$, denote by $(\psi_{n,c}(\cdot))_{n\geq 0}$ are the eigenfunctions of the Sturm-Liouville operators $L_c=(1-x^2)\frac{d^2\psi}{dx^2}-2x \frac{d \psi}{dx}-c^2x^2 \psi$ defined on $C^2[-1,1]$. It happened tha the functions $(\psi_{n,c}(\cdot))_{n\geq 0}$ are also the eigenfunctions of the compact integral operator defined on $L^2[-1,1]$by $F_c(\psi)(x)=\frac{1}{\pi}\int_{-1}^1\frac{\sin c(x-y)}{x-y}\psi(y)dy$. Also, $(\psi_{n,c}(\cdot))_{n\geq 0}$ are normalized as following $\int_{-1}^1|\psi_{n,c}(x)|^2dx=1$ and $\int_R|\psi_{n,c}(x)|^2dx=\frac{1}{\lambda(c)}, n\geq 0$, where $(\lamda_n(c))_n$ are the infinite sequence of the eigenvalues of $F_c$, arranged in the non-increasing order and such that $\left(\frac{c}{25n}\right)^{n-1}\leq \lambda_n(c)\leq c\left(\frac{ec}{4n}\right)^{2n}$. My question is the following: If these functions $(\psi_{n,c}(\cdot))_{n\geq 0}$ sutisfies the following conditions: Let $f(x)=\psi_{n,c}\in L^2[-1,1]$. Fix $\epsilon>0$ . Can one define A so that $\int_{-\infty}^{-A}|f(x)|^2dx< \epsilon^2$ and $\int_A^{\infty}|f(x)|^2dx< \epsilon^2$? Thank you.

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