Orthogonal basis of polynomials?
Clash Royale CLAN TAG#URR8PPP
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Let us define the basis of polynomials given by:
$$
beginarray
P_0=1, \
P_1=x, \
P_2=x(x-1), \
P_3=x(x-1)(x-2), \
P_4=x(x-1)(x-2)(x-3), ldots\
endarray
$$
I would like to know if this basis is orthogonal with respect to some measure. Thank you very much!
polynomials orthogonal-polynomials
edited 1 hour ago
Mahdi
1,1442723
asked 3 hours ago
fernando
965
add a comment |Â
up vote
3
down vote
favorite
3
Let us define the basis of polynomials given by:
$$
beginarray
P_0=1, \
P_1=x, \
P_2=x(x-1), \
P_3=x(x-1)(x-2), \
P_4=x(x-1)(x-2)(x-3), ldots\
endarray
$$
I would like to know if this basis is orthogonal with respect to some measure. Thank you very much!
polynomials orthogonal-polynomials
edited 1 hour ago
Mahdi
1,1442723
asked 3 hours ago
fernando
965
1
Well, you could define an ad hoc inner product by saying that if $p(x) = sum a_ip_i$ and $q(x)=sum b_ip_i$, then $langle p,qrangle = sum a_ib_i$, which would make it an orthogonal (even orthonormal) basis. But presumably you are looking for more than just "some" measure?
– Arturo Magidin
3 hours ago
Ps: I was thinking about an inner product of the form $int dx P_i(x) P_j(x) mu(x)$ for some measure $mu(x)$
– fernando
2 hours ago
You should explain more clearly what kind of measure you are asking? A real measure on the real line? A complex measure on a subset of the complex plane?
– Alexandre Eremenko
1 hour ago
add a comment |Â
up vote
3
down vote
favorite
3
up vote
3
down vote
favorite
3
3
Let us define the basis of polynomials given by:
$$
beginarray
P_0=1, \
P_1=x, \
P_2=x(x-1), \
P_3=x(x-1)(x-2), \
P_4=x(x-1)(x-2)(x-3), ldots\
endarray
$$
I would like to know if this basis is orthogonal with respect to some measure. Thank you very much!
polynomials orthogonal-polynomials
edited 1 hour ago
Mahdi
1,1442723
asked 3 hours ago
fernando
965
Let us define the basis of polynomials given by:
$$
beginarray
P_0=1, \
P_1=x, \
P_2=x(x-1), \
P_3=x(x-1)(x-2), \
P_4=x(x-1)(x-2)(x-3), ldots\
endarray
$$
I would like to know if this basis is orthogonal with respect to some measure. Thank you very much!
polynomials orthogonal-polynomials
polynomials orthogonal-polynomials
edited 1 hour ago
Mahdi
1,1442723
asked 3 hours ago
fernando
965
edited 1 hour ago
Mahdi
1,1442723
asked 3 hours ago
fernando
965
edited 1 hour ago
Mahdi
1,1442723
edited 1 hour ago
Mahdi
1,1442723
edited 1 hour ago
Mahdi
1,1442723
1,1442723
asked 3 hours ago
fernando
965
asked 3 hours ago
fernando
965
asked 3 hours ago
fernando
965
965
1
Well, you could define an ad hoc inner product by saying that if $p(x) = sum a_ip_i$ and $q(x)=sum b_ip_i$, then $langle p,qrangle = sum a_ib_i$, which would make it an orthogonal (even orthonormal) basis. But presumably you are looking for more than just "some" measure?
– Arturo Magidin
3 hours ago
Ps: I was thinking about an inner product of the form $int dx P_i(x) P_j(x) mu(x)$ for some measure $mu(x)$
– fernando
2 hours ago
You should explain more clearly what kind of measure you are asking? A real measure on the real line? A complex measure on a subset of the complex plane?
– Alexandre Eremenko
1 hour ago
add a comment |Â
1
Well, you could define an ad hoc inner product by saying that if $p(x) = sum a_ip_i$ and $q(x)=sum b_ip_i$, then $langle p,qrangle = sum a_ib_i$, which would make it an orthogonal (even orthonormal) basis. But presumably you are looking for more than just "some" measure?
– Arturo Magidin
3 hours ago
Ps: I was thinking about an inner product of the form $int dx P_i(x) P_j(x) mu(x)$ for some measure $mu(x)$
– fernando
2 hours ago
You should explain more clearly what kind of measure you are asking? A real measure on the real line? A complex measure on a subset of the complex plane?
– Alexandre Eremenko
1 hour ago
1
1
Well, you could define an ad hoc inner product by saying that if $p(x) = sum a_ip_i$ and $q(x)=sum b_ip_i$, then $langle p,qrangle = sum a_ib_i$, which would make it an orthogonal (even orthonormal) basis. But presumably you are looking for more than just "some" measure?
– Arturo Magidin
3 hours ago
Well, you could define an ad hoc inner product by saying that if $p(x) = sum a_ip_i$ and $q(x)=sum b_ip_i$, then $langle p,qrangle = sum a_ib_i$, which would make it an orthogonal (even orthonormal) basis. But presumably you are looking for more than just "some" measure?
– Arturo Magidin
3 hours ago
Ps: I was thinking about an inner product of the form $int dx P_i(x) P_j(x) mu(x)$ for some measure $mu(x)$
– fernando
2 hours ago
Ps: I was thinking about an inner product of the form $int dx P_i(x) P_j(x) mu(x)$ for some measure $mu(x)$
– fernando
2 hours ago
You should explain more clearly what kind of measure you are asking? A real measure on the real line? A complex measure on a subset of the complex plane?
– Alexandre Eremenko
1 hour ago
You should explain more clearly what kind of measure you are asking? A real measure on the real line? A complex measure on a subset of the complex plane?
– Alexandre Eremenko
1 hour ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
If a sequence of monic polynomials is orthogonal with respect a measure, it satisfies a three-term recurrence
[
p_n+1(t) = (t-a_n)p_n(t) - b_n p_n-1(t)
]
where $b_n>0$. From this it follows that consecutive terms in the sequence cannot have a common zero. Your sequence fails badly on this test.
answered 2 hours ago
Chris Godsil
10.8k32756
1
Thank you very much! Indeed you are right. But can we define something similar to a orthogonality condition? e.g. we can also consider the family of polynomials P_n(x) = x^n, which is a complete basis, but in this case we can define, for instance $<P_n P_m> = int fracdzz P_n(x) P_-m(x)$, where the integral is a contour integral around zero.
– fernando
2 hours ago
2
Very nice answer! Do you know of a reference elaborating on this link between orthogonality and recurrences?
– Emanuele Tron
2 hours ago
Any text book on orthogonal polynomials should treat this, it’s quite basic. My “Algebraic Combinatorics†discusses it at lengh too ( if you’ll forgive the plug).
– Chris Godsil
1 hour ago
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
If a sequence of monic polynomials is orthogonal with respect a measure, it satisfies a three-term recurrence
[
p_n+1(t) = (t-a_n)p_n(t) - b_n p_n-1(t)
]
where $b_n>0$. From this it follows that consecutive terms in the sequence cannot have a common zero. Your sequence fails badly on this test.
answered 2 hours ago
Chris Godsil
10.8k32756
1
Thank you very much! Indeed you are right. But can we define something similar to a orthogonality condition? e.g. we can also consider the family of polynomials P_n(x) = x^n, which is a complete basis, but in this case we can define, for instance $<P_n P_m> = int fracdzz P_n(x) P_-m(x)$, where the integral is a contour integral around zero.
– fernando
2 hours ago
2
Very nice answer! Do you know of a reference elaborating on this link between orthogonality and recurrences?
– Emanuele Tron
2 hours ago
Any text book on orthogonal polynomials should treat this, it’s quite basic. My “Algebraic Combinatorics†discusses it at lengh too ( if you’ll forgive the plug).
– Chris Godsil
1 hour ago
add a comment |Â
up vote
4
down vote
accepted
If a sequence of monic polynomials is orthogonal with respect a measure, it satisfies a three-term recurrence
[
p_n+1(t) = (t-a_n)p_n(t) - b_n p_n-1(t)
]
where $b_n>0$. From this it follows that consecutive terms in the sequence cannot have a common zero. Your sequence fails badly on this test.
answered 2 hours ago
Chris Godsil
10.8k32756
1
Thank you very much! Indeed you are right. But can we define something similar to a orthogonality condition? e.g. we can also consider the family of polynomials P_n(x) = x^n, which is a complete basis, but in this case we can define, for instance $<P_n P_m> = int fracdzz P_n(x) P_-m(x)$, where the integral is a contour integral around zero.
– fernando
2 hours ago
2
Very nice answer! Do you know of a reference elaborating on this link between orthogonality and recurrences?
– Emanuele Tron
2 hours ago
Any text book on orthogonal polynomials should treat this, it’s quite basic. My “Algebraic Combinatorics†discusses it at lengh too ( if you’ll forgive the plug).
– Chris Godsil
1 hour ago
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
If a sequence of monic polynomials is orthogonal with respect a measure, it satisfies a three-term recurrence
[
p_n+1(t) = (t-a_n)p_n(t) - b_n p_n-1(t)
]
where $b_n>0$. From this it follows that consecutive terms in the sequence cannot have a common zero. Your sequence fails badly on this test.
answered 2 hours ago
Chris Godsil
10.8k32756
If a sequence of monic polynomials is orthogonal with respect a measure, it satisfies a three-term recurrence
[
p_n+1(t) = (t-a_n)p_n(t) - b_n p_n-1(t)
]
where $b_n>0$. From this it follows that consecutive terms in the sequence cannot have a common zero. Your sequence fails badly on this test.
answered 2 hours ago
Chris Godsil
10.8k32756
answered 2 hours ago
Chris Godsil
10.8k32756
answered 2 hours ago
Chris Godsil
10.8k32756
answered 2 hours ago
Chris Godsil
10.8k32756
10.8k32756
1
Thank you very much! Indeed you are right. But can we define something similar to a orthogonality condition? e.g. we can also consider the family of polynomials P_n(x) = x^n, which is a complete basis, but in this case we can define, for instance $<P_n P_m> = int fracdzz P_n(x) P_-m(x)$, where the integral is a contour integral around zero.
– fernando
2 hours ago
2
Very nice answer! Do you know of a reference elaborating on this link between orthogonality and recurrences?
– Emanuele Tron
2 hours ago
Any text book on orthogonal polynomials should treat this, it’s quite basic. My “Algebraic Combinatorics†discusses it at lengh too ( if you’ll forgive the plug).
– Chris Godsil
1 hour ago
add a comment |Â
1
Thank you very much! Indeed you are right. But can we define something similar to a orthogonality condition? e.g. we can also consider the family of polynomials P_n(x) = x^n, which is a complete basis, but in this case we can define, for instance $<P_n P_m> = int fracdzz P_n(x) P_-m(x)$, where the integral is a contour integral around zero.
– fernando
2 hours ago
2
Very nice answer! Do you know of a reference elaborating on this link between orthogonality and recurrences?
– Emanuele Tron
2 hours ago
Any text book on orthogonal polynomials should treat this, it’s quite basic. My “Algebraic Combinatorics†discusses it at lengh too ( if you’ll forgive the plug).
– Chris Godsil
1 hour ago
1
1
Thank you very much! Indeed you are right. But can we define something similar to a orthogonality condition? e.g. we can also consider the family of polynomials P_n(x) = x^n, which is a complete basis, but in this case we can define, for instance $<P_n P_m> = int fracdzz P_n(x) P_-m(x)$, where the integral is a contour integral around zero.
– fernando
2 hours ago
Thank you very much! Indeed you are right. But can we define something similar to a orthogonality condition? e.g. we can also consider the family of polynomials P_n(x) = x^n, which is a complete basis, but in this case we can define, for instance $<P_n P_m> = int fracdzz P_n(x) P_-m(x)$, where the integral is a contour integral around zero.
– fernando
2 hours ago
2
2
Very nice answer! Do you know of a reference elaborating on this link between orthogonality and recurrences?
– Emanuele Tron
2 hours ago
Very nice answer! Do you know of a reference elaborating on this link between orthogonality and recurrences?
– Emanuele Tron
2 hours ago
Any text book on orthogonal polynomials should treat this, it’s quite basic. My “Algebraic Combinatorics†discusses it at lengh too ( if you’ll forgive the plug).
– Chris Godsil
1 hour ago
Any text book on orthogonal polynomials should treat this, it’s quite basic. My “Algebraic Combinatorics†discusses it at lengh too ( if you’ll forgive the plug).
– Chris Godsil
1 hour ago
add a comment |Â
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1
Well, you could define an ad hoc inner product by saying that if $p(x) = sum a_ip_i$ and $q(x)=sum b_ip_i$, then $langle p,qrangle = sum a_ib_i$, which would make it an orthogonal (even orthonormal) basis. But presumably you are looking for more than just "some" measure?
– Arturo Magidin
3 hours ago
Ps: I was thinking about an inner product of the form $int dx P_i(x) P_j(x) mu(x)$ for some measure $mu(x)$
– fernando
2 hours ago
You should explain more clearly what kind of measure you are asking? A real measure on the real line? A complex measure on a subset of the complex plane?
– Alexandre Eremenko
1 hour ago