Sturm - Liouville Parseval proof
$begingroup$
Consider the expansion $𝑓(𝑥)=sum_{n=0}^{infty}A_𝑛phi_𝑛(𝑥)$
where $phi_𝑛(𝑥)$ is the set of eigenfunctions of the Sturm-Liouville problem, and $𝑓(𝑥)$ is any piecewise differentiable function on the interval $(𝑎,𝑏)$. Show that $int_{a}^{b}[𝑓(𝑥)]^2𝑤(𝑥)𝑑𝑥=sum_{n=0}^{infty}𝐴_𝑛^2𝑁_𝑛$
where $w(x)$ is weight function and $𝑁_𝑛$ is the normalization integrals of the eigenfunctions $phi_𝑛(𝑥)$. $(𝑁𝑛=〈phi_𝑛(𝑥),phi_𝑛(𝑥)〉)$
sturm-liouville parsevals-identity
asked Jan 8 at 23:15
SquanchSquanch
353
$endgroup$
add a comment |
$begingroup$
Consider the expansion $𝑓(𝑥)=sum_{n=0}^{infty}A_𝑛phi_𝑛(𝑥)$
where $phi_𝑛(𝑥)$ is the set of eigenfunctions of the Sturm-Liouville problem, and $𝑓(𝑥)$ is any piecewise differentiable function on the interval $(𝑎,𝑏)$. Show that $int_{a}^{b}[𝑓(𝑥)]^2𝑤(𝑥)𝑑𝑥=sum_{n=0}^{infty}𝐴_𝑛^2𝑁_𝑛$
where $w(x)$ is weight function and $𝑁_𝑛$ is the normalization integrals of the eigenfunctions $phi_𝑛(𝑥)$. $(𝑁𝑛=〈phi_𝑛(𝑥),phi_𝑛(𝑥)〉)$
sturm-liouville parsevals-identity
asked Jan 8 at 23:15
SquanchSquanch
353
$endgroup$
$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44
add a comment |
$begingroup$
Consider the expansion $𝑓(𝑥)=sum_{n=0}^{infty}A_𝑛phi_𝑛(𝑥)$
where $phi_𝑛(𝑥)$ is the set of eigenfunctions of the Sturm-Liouville problem, and $𝑓(𝑥)$ is any piecewise differentiable function on the interval $(𝑎,𝑏)$. Show that $int_{a}^{b}[𝑓(𝑥)]^2𝑤(𝑥)𝑑𝑥=sum_{n=0}^{infty}𝐴_𝑛^2𝑁_𝑛$
where $w(x)$ is weight function and $𝑁_𝑛$ is the normalization integrals of the eigenfunctions $phi_𝑛(𝑥)$. $(𝑁𝑛=〈phi_𝑛(𝑥),phi_𝑛(𝑥)〉)$
sturm-liouville parsevals-identity
asked Jan 8 at 23:15
SquanchSquanch
353
$endgroup$
Consider the expansion $𝑓(𝑥)=sum_{n=0}^{infty}A_𝑛phi_𝑛(𝑥)$
where $phi_𝑛(𝑥)$ is the set of eigenfunctions of the Sturm-Liouville problem, and $𝑓(𝑥)$ is any piecewise differentiable function on the interval $(𝑎,𝑏)$. Show that $int_{a}^{b}[𝑓(𝑥)]^2𝑤(𝑥)𝑑𝑥=sum_{n=0}^{infty}𝐴_𝑛^2𝑁_𝑛$
where $w(x)$ is weight function and $𝑁_𝑛$ is the normalization integrals of the eigenfunctions $phi_𝑛(𝑥)$. $(𝑁𝑛=〈phi_𝑛(𝑥),phi_𝑛(𝑥)〉)$
sturm-liouville parsevals-identity
sturm-liouville parsevals-identity
asked Jan 8 at 23:15
SquanchSquanch
353
asked Jan 8 at 23:15
SquanchSquanch
353
asked Jan 8 at 23:15
SquanchSquanch
353
asked Jan 8 at 23:15
SquanchSquanch
353
asked Jan 8 at 23:15
SquanchSquanch
353
353
$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44
add a comment |
$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44
$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44
$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44
add a comment |
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$begingroup$
The simplest case would be a regular problem on a finite interval $[a,b]$. In that case, the eigenfunctions form a complete orthogonal basis of $L^2_w[a,b]$, which is equivalent to the Parseval relation you state. However, the proof would be too long for a post in this forum, even for the simplest case of $sin$ and $cos$ functions.
$endgroup$
– DisintegratingByParts
Jan 9 at 6:44