Completeness of a basis function

  • #26
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Sorry to keep hammering this, but when you use a Schauder basis this is an ordered basis ( because convergence is not always absolute) , so you may be changing the order of the terms of the sum? Are you doing finite sums ( Hamel bases, equality) or infinite sums ( Schauder bases, convergence).
I'm using finite sums, so Hamel bases. Have you ever encountered anything like what I describe: superimposing basis functions to create new basis functions that are still linearly independent but now also satisfy a new property?
 
  • #27
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I'm using finite sums, so Hamel bases. Have you ever encountered anything like what I describe: superimposing basis functions to create new basis functions that are still linearly independent but now also satisfy a new property?

Not really, sorry. I am a bit confused; your new functions are linear combinations of the previous, so I don't see what you may gain from it. Why not keep the original basis and just use combinations of it?
 
  • #28
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No.
At least your question isn't specifically enough to be answered clearly. Completeness usually relates to a certain set, and there are more than one way to complete such a set. E.g. ##\mathbb{R}## is the topological completion of ##\mathbb{Q}##, whereas ##\mathbb{C}## is the algebraic completion (better: closure) of ##\mathbb{R}##.
Taylor series are series, not polynomials.

.
I think he is referring to the partial sums. But, yes, it does not seem clear.
 
  • #29
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Not really, sorry. I am a bit confused; your new functions are linear combinations of the previous, so I don't see what you may gain from it. Why not keep the original basis and just use combinations of it?
I'm approximately solving a differential eigenvalue problem. The basis functions I use must satisfy the BCs with the EVP. Since they also must be harmonic and satisfy the derivatives I mentioned earlier, I find them generally for those conditions, and then superimpose to satisfy the final BCs. Then I apply a Ritz procedure to them to approximately solve the ODE.

There are two ways to solve: solve the inverse EVP, which introduces a Green's function. In that case, I don't have to recombine basis functions since the Green's function accounts for these. This approach is correct. The second way to solve the EVP is the direct approach. In this approach, I recombine the basis functions, and in this approach I get an incorrect solution for some (not all) parameter values. I just don't know why.

Basis functions work, but when recombined, suddenly the technique can fail. You've never seen anything like this?
 
  • #30
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I'm approximately solving a differential eigenvalue problem. The basis functions I use must satisfy the BCs with the EVP. Since they also must be harmonic and satisfy the derivatives I mentioned earlier, I find them generally for those conditions, and then superimpose to satisfy the final BCs. Then I apply a Ritz procedure to them to approximately solve the ODE.

There are two ways to solve: solve the inverse EVP, which introduces a Green's function. In that case, I don't have to recombine basis functions since the Green's function accounts for these. This approach is correct. The second way to solve the EVP is the direct approach. In this approach, I recombine the basis functions, and in this approach I get an incorrect solution for some (not all) parameter values. I just don't know why. EDIT: Sorry too, for jumping in and sort of getting into a side-discussion. Hope you got something out of it or at least refreshed the material. I hope to get myself back into PDEs and Green's functions some time soon.

Basis functions work, but when recombined, suddenly the technique can fail. You've never seen anything like this?
No, sorry, it has been a while since I have done PDEs or worked with Green's function in general.
 
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  • #31
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No, sorry, it has been a while since I have done PDEs or worked with Green's function in general.
No worries, thanks for taking the time!
 
  • #32
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Not really, sorry. I am a bit confused; your new functions are linear combinations of the previous, so I don't see what you may gain from it. Why not keep the original basis and just use combinations of it?
It is often nice to represent a function in terms of a basis that gives a clean interpretation of the components in the specific current context. So there are reasons to convert from one basis to another, as appropriate.
 
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  • #33
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It is often nice to represent a function in terms of a basis that gives a clean interpretation of the components in the specific current context. So there are reasons to convert from one basis to another, as appropriate.
Yes, the input to the Ritz procedure I employ must satisfy all BCs. As is, the basis functions don't satisfy all the BCs. So we take linearly independent combinations of basis functions to form new basis functions such that the additional BC is satisfied.
 
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