Undergrad Completeness of a basis function

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Completeness of a basis function refers to its ability to represent any function within a specific space. The set of polynomials is not complete because they cannot approximate all continuous functions, as demonstrated by the convergence of truncated Taylor series to non-polynomial functions. In contrast, Legendre polynomials are considered complete within certain contexts, particularly in solving differential equations. The discussion emphasizes the importance of specifying the metric space and context when discussing completeness, as it varies across different applications, such as quantum physics or engineering. Understanding the completeness of trial functions is crucial for accurately solving boundary conditions in differential equations.
  • #31
WWGD said:
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!
 
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  • #32
WWGD said:
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
FactChecker said:
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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