Compute the minimum value of the integral: $int_{-1}^{1} |t^3-a-bt-ct^2|^2dt$ [duplicate]












2















This question already has an answer here:




  • Hilbert space for integral

    2 answers



  • Finding the min of an integral

    2 answers




Compute the minimum value of the following integral:



$$int_{-1}^{1} |t^3-a-bt-ct^2|^2dt$$
for $a,b,c in mathbb{C}$.



Any hint? It’s a functional analysis problem so I cannot compute it directly but I don’t understand which functional analysis result can I use to obtain that minimum...






integration functional-analysis optimization complex-numbers maxima-minima







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marked as duplicate by Martin R, Nosrati, DisintegratingByParts functional-analysis
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 2 days ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.















  • Ever heard of the least square method?
    – Jakobian
    2 days ago
















2















This question already has an answer here:




  • Hilbert space for integral

    2 answers



  • Finding the min of an integral

    2 answers




Compute the minimum value of the following integral:



$$int_{-1}^{1} |t^3-a-bt-ct^2|^2dt$$
for $a,b,c in mathbb{C}$.



Any hint? It’s a functional analysis problem so I cannot compute it directly but I don’t understand which functional analysis result can I use to obtain that minimum...






integration functional-analysis optimization complex-numbers maxima-minima







share|cite|improve this question













marked as duplicate by Martin R, Nosrati, DisintegratingByParts functional-analysis
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 2 days ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.















  • Ever heard of the least square method?
    – Jakobian
    2 days ago














2












2








2


1






This question already has an answer here:




  • Hilbert space for integral

    2 answers



  • Finding the min of an integral

    2 answers




Compute the minimum value of the following integral:



$$int_{-1}^{1} |t^3-a-bt-ct^2|^2dt$$
for $a,b,c in mathbb{C}$.



Any hint? It’s a functional analysis problem so I cannot compute it directly but I don’t understand which functional analysis result can I use to obtain that minimum...






integration functional-analysis optimization complex-numbers maxima-minima







share|cite|improve this question














This question already has an answer here:




  • Hilbert space for integral

    2 answers



  • Finding the min of an integral

    2 answers




Compute the minimum value of the following integral:



$$int_{-1}^{1} |t^3-a-bt-ct^2|^2dt$$
for $a,b,c in mathbb{C}$.



Any hint? It’s a functional analysis problem so I cannot compute it directly but I don’t understand which functional analysis result can I use to obtain that minimum...





This question already has an answer here:




  • Hilbert space for integral

    2 answers



  • Finding the min of an integral

    2 answers







integration functional-analysis optimization complex-numbers maxima-minima




integration functional-analysis optimization complex-numbers maxima-minima






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 days ago









Maggie94

1016




1016




marked as duplicate by Martin R, Nosrati, DisintegratingByParts functional-analysis
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marked as duplicate by Martin R, Nosrati, DisintegratingByParts functional-analysis
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 2 days ago


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.














  • Ever heard of the least square method?
    – Jakobian
    2 days ago


















  • Ever heard of the least square method?
    – Jakobian
    2 days ago
















Ever heard of the least square method?
– Jakobian
2 days ago




Ever heard of the least square method?
– Jakobian
2 days ago










1 Answer
1






active

oldest

votes


















3














You are looking for the closest point projection of $t^3$ onto the subspace spanned by ${1,t,t^2}$. This is the same as the orthogonal projection of $t^3$ onto the subspace $[{1,t,t^2}]$, which is the unique element $p(t)=a+bt+ct^2$ such that
$$
langle t^3-p,1rangle =0 \
langle t^3-p,trangle = 0 \
langle t^3-p,t^2rangle = 0.
$$



This results in a linear system of 3 equations in the 3 unknown constants $a,b,c$. The matrix is a covariance matrix, which is invertible because ${ 1,t,t^2 }$ is a linearly independent set of elements.






share|cite|improve this answer





















  • Indeed: math.stackexchange.com/a/1679522/42969.
    – Martin R
    2 days ago










  • Yes it is a duplicate, sorry I didn’t notice it! Thank you all anyway
    – Maggie94
    yesterday


















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









3














You are looking for the closest point projection of $t^3$ onto the subspace spanned by ${1,t,t^2}$. This is the same as the orthogonal projection of $t^3$ onto the subspace $[{1,t,t^2}]$, which is the unique element $p(t)=a+bt+ct^2$ such that
$$
langle t^3-p,1rangle =0 \
langle t^3-p,trangle = 0 \
langle t^3-p,t^2rangle = 0.
$$



This results in a linear system of 3 equations in the 3 unknown constants $a,b,c$. The matrix is a covariance matrix, which is invertible because ${ 1,t,t^2 }$ is a linearly independent set of elements.






share|cite|improve this answer





















  • Indeed: math.stackexchange.com/a/1679522/42969.
    – Martin R
    2 days ago










  • Yes it is a duplicate, sorry I didn’t notice it! Thank you all anyway
    – Maggie94
    yesterday
















3














You are looking for the closest point projection of $t^3$ onto the subspace spanned by ${1,t,t^2}$. This is the same as the orthogonal projection of $t^3$ onto the subspace $[{1,t,t^2}]$, which is the unique element $p(t)=a+bt+ct^2$ such that
$$
langle t^3-p,1rangle =0 \
langle t^3-p,trangle = 0 \
langle t^3-p,t^2rangle = 0.
$$



This results in a linear system of 3 equations in the 3 unknown constants $a,b,c$. The matrix is a covariance matrix, which is invertible because ${ 1,t,t^2 }$ is a linearly independent set of elements.






share|cite|improve this answer





















  • Indeed: math.stackexchange.com/a/1679522/42969.
    – Martin R
    2 days ago










  • Yes it is a duplicate, sorry I didn’t notice it! Thank you all anyway
    – Maggie94
    yesterday














3












3








3






You are looking for the closest point projection of $t^3$ onto the subspace spanned by ${1,t,t^2}$. This is the same as the orthogonal projection of $t^3$ onto the subspace $[{1,t,t^2}]$, which is the unique element $p(t)=a+bt+ct^2$ such that
$$
langle t^3-p,1rangle =0 \
langle t^3-p,trangle = 0 \
langle t^3-p,t^2rangle = 0.
$$



This results in a linear system of 3 equations in the 3 unknown constants $a,b,c$. The matrix is a covariance matrix, which is invertible because ${ 1,t,t^2 }$ is a linearly independent set of elements.






share|cite|improve this answer












You are looking for the closest point projection of $t^3$ onto the subspace spanned by ${1,t,t^2}$. This is the same as the orthogonal projection of $t^3$ onto the subspace $[{1,t,t^2}]$, which is the unique element $p(t)=a+bt+ct^2$ such that
$$
langle t^3-p,1rangle =0 \
langle t^3-p,trangle = 0 \
langle t^3-p,t^2rangle = 0.
$$



This results in a linear system of 3 equations in the 3 unknown constants $a,b,c$. The matrix is a covariance matrix, which is invertible because ${ 1,t,t^2 }$ is a linearly independent set of elements.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 2 days ago









DisintegratingByParts

58.7k42579




58.7k42579












  • Indeed: math.stackexchange.com/a/1679522/42969.
    – Martin R
    2 days ago










  • Yes it is a duplicate, sorry I didn’t notice it! Thank you all anyway
    – Maggie94
    yesterday


















  • Indeed: math.stackexchange.com/a/1679522/42969.
    – Martin R
    2 days ago










  • Yes it is a duplicate, sorry I didn’t notice it! Thank you all anyway
    – Maggie94
    yesterday
















Indeed: math.stackexchange.com/a/1679522/42969.
– Martin R
2 days ago




Indeed: math.stackexchange.com/a/1679522/42969.
– Martin R
2 days ago












Yes it is a duplicate, sorry I didn’t notice it! Thank you all anyway
– Maggie94
yesterday




Yes it is a duplicate, sorry I didn’t notice it! Thank you all anyway
– Maggie94
yesterday



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