Mixing
Every set of orthogonal functions is complete?
Clash Royale CLAN TAG#URR8PPP
up vote
-1
down vote
favorite
Suppose that the functions $ phi_1, phi_2,..., phi_n,... $ are orthogonal on the interval $[a,b]$. That is
$$ int_a^b phi_nphi_m dx = 0 forall nneq m$$
If $$ lim_Nrightarrow infty int_a^b Big(f-sum_n=1^NC_nphi_n Big)^2 dx = 0 $$
for every function $f$ with the property that ( $f$ is square-integrable. )$$ int_a^b f^2 dx < infty$$
we say that the set of functions $phi_1,phi_2,... $ is complete. Where $C_n $ are the Fourier coefficients.
Now, I wonder if there exists are set of orthogonal functions that is not complete, or every set of orthogonal functions is complete?.
pde fourier-series
asked Jul 29 at 22:46
tnt235711
406
add a comment |Â
up vote
-1
down vote
favorite
Suppose that the functions $ phi_1, phi_2,..., phi_n,... $ are orthogonal on the interval $[a,b]$. That is
$$ int_a^b phi_nphi_m dx = 0 forall nneq m$$
If $$ lim_Nrightarrow infty int_a^b Big(f-sum_n=1^NC_nphi_n Big)^2 dx = 0 $$
for every function $f$ with the property that ( $f$ is square-integrable. )$$ int_a^b f^2 dx < infty$$
we say that the set of functions $phi_1,phi_2,... $ is complete. Where $C_n $ are the Fourier coefficients.
Now, I wonder if there exists are set of orthogonal functions that is not complete, or every set of orthogonal functions is complete?.
pde fourier-series
asked Jul 29 at 22:46
tnt235711
406
6
If you remove an element from a complete set or orthogonal, you get a set of orthogonal which is not complete
– Naj Kamp
Jul 29 at 22:51
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Suppose that the functions $ phi_1, phi_2,..., phi_n,... $ are orthogonal on the interval $[a,b]$. That is
$$ int_a^b phi_nphi_m dx = 0 forall nneq m$$
If $$ lim_Nrightarrow infty int_a^b Big(f-sum_n=1^NC_nphi_n Big)^2 dx = 0 $$
for every function $f$ with the property that ( $f$ is square-integrable. )$$ int_a^b f^2 dx < infty$$
we say that the set of functions $phi_1,phi_2,... $ is complete. Where $C_n $ are the Fourier coefficients.
Now, I wonder if there exists are set of orthogonal functions that is not complete, or every set of orthogonal functions is complete?.
pde fourier-series
asked Jul 29 at 22:46
tnt235711
406
Suppose that the functions $ phi_1, phi_2,..., phi_n,... $ are orthogonal on the interval $[a,b]$. That is
$$ int_a^b phi_nphi_m dx = 0 forall nneq m$$
If $$ lim_Nrightarrow infty int_a^b Big(f-sum_n=1^NC_nphi_n Big)^2 dx = 0 $$
for every function $f$ with the property that ( $f$ is square-integrable. )$$ int_a^b f^2 dx < infty$$
we say that the set of functions $phi_1,phi_2,... $ is complete. Where $C_n $ are the Fourier coefficients.
Now, I wonder if there exists are set of orthogonal functions that is not complete, or every set of orthogonal functions is complete?.
pde fourier-series
asked Jul 29 at 22:46
tnt235711
406
asked Jul 29 at 22:46
tnt235711
406
asked Jul 29 at 22:46
tnt235711
406
asked Jul 29 at 22:46
tnt235711
406
406
6
If you remove an element from a complete set or orthogonal, you get a set of orthogonal which is not complete
– Naj Kamp
Jul 29 at 22:51
add a comment |Â
6
If you remove an element from a complete set or orthogonal, you get a set of orthogonal which is not complete
– Naj Kamp
Jul 29 at 22:51
6
6
If you remove an element from a complete set or orthogonal, you get a set of orthogonal which is not complete
– Naj Kamp
Jul 29 at 22:51
If you remove an element from a complete set or orthogonal, you get a set of orthogonal which is not complete
– Naj Kamp
Jul 29 at 22:51
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
2
down vote
accepted
Take any set of complete orthogonal functions, and remove one of the elements. Then the resulting set will no longer be complete.
Edit: in particular, you cannot reach the element you removed (assuming it's nonzero a.e.).
Note:
$$int_a^b(phi_1-sum_n=2^n C_n phi_n)^2 dx=int_a^bphi_1^2+(sum_n=2^NC_n phi_n)^2 dxge int_a^bphi_1^2 dx>0$$
but how do you Know that the resulting set is not complete?
– tnt235711
Jul 29 at 22:56
@tnt235711 You cannot reach the element you removed, see now
– user223391
Jul 29 at 23:01
1
The element you removed is orthogonal to the closure of the span of the other elements.
– copper.hat
Jul 29 at 23:07
@ZacharySelk Let me see if I understand, if we chose $f = phi_1 neq 0 $, then $ lim int_a^b ( f - sum C_nphi_n )^2 > 0 $. Hence the set $ phi_2,phi_3,... $ is not complete.
– tnt235711
Jul 29 at 23:17
@tnt235711 yeah that's right
– user223391
Jul 29 at 23:18
add a comment |Â
up vote
0
down vote
Let our universe be $L^2([0, 1]),$ i.e. $langle f, g rangle = int_0^1 f(x) , g(x) , dx.$
Here are three examples of sets of orthogonal functions which do not form a complete set:
$$
$ x mapsto 1 $
$ x mapsto 1, x mapsto x $
answered Jul 29 at 23:07
md2perpe
5,7821922
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Take any set of complete orthogonal functions, and remove one of the elements. Then the resulting set will no longer be complete.
Edit: in particular, you cannot reach the element you removed (assuming it's nonzero a.e.).
Note:
$$int_a^b(phi_1-sum_n=2^n C_n phi_n)^2 dx=int_a^bphi_1^2+(sum_n=2^NC_n phi_n)^2 dxge int_a^bphi_1^2 dx>0$$
but how do you Know that the resulting set is not complete?
– tnt235711
Jul 29 at 22:56
@tnt235711 You cannot reach the element you removed, see now
– user223391
Jul 29 at 23:01
1
The element you removed is orthogonal to the closure of the span of the other elements.
– copper.hat
Jul 29 at 23:07
@ZacharySelk Let me see if I understand, if we chose $f = phi_1 neq 0 $, then $ lim int_a^b ( f - sum C_nphi_n )^2 > 0 $. Hence the set $ phi_2,phi_3,... $ is not complete.
– tnt235711
Jul 29 at 23:17
@tnt235711 yeah that's right
– user223391
Jul 29 at 23:18
add a comment |Â
up vote
2
down vote
accepted
Take any set of complete orthogonal functions, and remove one of the elements. Then the resulting set will no longer be complete.
Edit: in particular, you cannot reach the element you removed (assuming it's nonzero a.e.).
Note:
$$int_a^b(phi_1-sum_n=2^n C_n phi_n)^2 dx=int_a^bphi_1^2+(sum_n=2^NC_n phi_n)^2 dxge int_a^bphi_1^2 dx>0$$
but how do you Know that the resulting set is not complete?
– tnt235711
Jul 29 at 22:56
@tnt235711 You cannot reach the element you removed, see now
– user223391
Jul 29 at 23:01
1
The element you removed is orthogonal to the closure of the span of the other elements.
– copper.hat
Jul 29 at 23:07
@ZacharySelk Let me see if I understand, if we chose $f = phi_1 neq 0 $, then $ lim int_a^b ( f - sum C_nphi_n )^2 > 0 $. Hence the set $ phi_2,phi_3,... $ is not complete.
– tnt235711
Jul 29 at 23:17
@tnt235711 yeah that's right
– user223391
Jul 29 at 23:18
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Take any set of complete orthogonal functions, and remove one of the elements. Then the resulting set will no longer be complete.
Edit: in particular, you cannot reach the element you removed (assuming it's nonzero a.e.).
Note:
$$int_a^b(phi_1-sum_n=2^n C_n phi_n)^2 dx=int_a^bphi_1^2+(sum_n=2^NC_n phi_n)^2 dxge int_a^bphi_1^2 dx>0$$
Take any set of complete orthogonal functions, and remove one of the elements. Then the resulting set will no longer be complete.
Edit: in particular, you cannot reach the element you removed (assuming it's nonzero a.e.).
Note:
$$int_a^b(phi_1-sum_n=2^n C_n phi_n)^2 dx=int_a^bphi_1^2+(sum_n=2^NC_n phi_n)^2 dxge int_a^bphi_1^2 dx>0$$
edited Jul 29 at 23:00
answered Jul 29 at 22:52
user223391
but how do you Know that the resulting set is not complete?
– tnt235711
Jul 29 at 22:56
@tnt235711 You cannot reach the element you removed, see now
– user223391
Jul 29 at 23:01
1
The element you removed is orthogonal to the closure of the span of the other elements.
– copper.hat
Jul 29 at 23:07
@ZacharySelk Let me see if I understand, if we chose $f = phi_1 neq 0 $, then $ lim int_a^b ( f - sum C_nphi_n )^2 > 0 $. Hence the set $ phi_2,phi_3,... $ is not complete.
– tnt235711
Jul 29 at 23:17
@tnt235711 yeah that's right
– user223391
Jul 29 at 23:18
add a comment |Â
but how do you Know that the resulting set is not complete?
– tnt235711
Jul 29 at 22:56
@tnt235711 You cannot reach the element you removed, see now
– user223391
Jul 29 at 23:01
1
The element you removed is orthogonal to the closure of the span of the other elements.
– copper.hat
Jul 29 at 23:07
@ZacharySelk Let me see if I understand, if we chose $f = phi_1 neq 0 $, then $ lim int_a^b ( f - sum C_nphi_n )^2 > 0 $. Hence the set $ phi_2,phi_3,... $ is not complete.
– tnt235711
Jul 29 at 23:17
@tnt235711 yeah that's right
– user223391
Jul 29 at 23:18
but how do you Know that the resulting set is not complete?
– tnt235711
Jul 29 at 22:56
but how do you Know that the resulting set is not complete?
– tnt235711
Jul 29 at 22:56
@tnt235711 You cannot reach the element you removed, see now
– user223391
Jul 29 at 23:01
@tnt235711 You cannot reach the element you removed, see now
– user223391
Jul 29 at 23:01
1
1
The element you removed is orthogonal to the closure of the span of the other elements.
– copper.hat
Jul 29 at 23:07
The element you removed is orthogonal to the closure of the span of the other elements.
– copper.hat
Jul 29 at 23:07
@ZacharySelk Let me see if I understand, if we chose $f = phi_1 neq 0 $, then $ lim int_a^b ( f - sum C_nphi_n )^2 > 0 $. Hence the set $ phi_2,phi_3,... $ is not complete.
– tnt235711
Jul 29 at 23:17
@ZacharySelk Let me see if I understand, if we chose $f = phi_1 neq 0 $, then $ lim int_a^b ( f - sum C_nphi_n )^2 > 0 $. Hence the set $ phi_2,phi_3,... $ is not complete.
– tnt235711
Jul 29 at 23:17
@tnt235711 yeah that's right
– user223391
Jul 29 at 23:18
@tnt235711 yeah that's right
– user223391
Jul 29 at 23:18
add a comment |Â
up vote
0
down vote
Let our universe be $L^2([0, 1]),$ i.e. $langle f, g rangle = int_0^1 f(x) , g(x) , dx.$
Here are three examples of sets of orthogonal functions which do not form a complete set:
$$
$ x mapsto 1 $
$ x mapsto 1, x mapsto x $
answered Jul 29 at 23:07
md2perpe
5,7821922
add a comment |Â
up vote
0
down vote
Let our universe be $L^2([0, 1]),$ i.e. $langle f, g rangle = int_0^1 f(x) , g(x) , dx.$
Here are three examples of sets of orthogonal functions which do not form a complete set:
$$
$ x mapsto 1 $
$ x mapsto 1, x mapsto x $
answered Jul 29 at 23:07
md2perpe
5,7821922
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Let our universe be $L^2([0, 1]),$ i.e. $langle f, g rangle = int_0^1 f(x) , g(x) , dx.$
Here are three examples of sets of orthogonal functions which do not form a complete set:
$$
$ x mapsto 1 $
$ x mapsto 1, x mapsto x $
answered Jul 29 at 23:07
md2perpe
5,7821922
Let our universe be $L^2([0, 1]),$ i.e. $langle f, g rangle = int_0^1 f(x) , g(x) , dx.$
Here are three examples of sets of orthogonal functions which do not form a complete set:
$$
$ x mapsto 1 $
$ x mapsto 1, x mapsto x $
answered Jul 29 at 23:07
md2perpe
5,7821922
answered Jul 29 at 23:07
md2perpe
5,7821922
answered Jul 29 at 23:07
md2perpe
5,7821922
answered Jul 29 at 23:07
md2perpe
5,7821922
5,7821922
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2866508%2fevery-set-of-orthogonal-functions-is-complete%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
6
If you remove an element from a complete set or orthogonal, you get a set of orthogonal which is not complete
– Naj Kamp
Jul 29 at 22:51