Completeness of a basis function

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Discussion Overview

The discussion revolves around the concept of completeness of basis functions in various mathematical and physical contexts, particularly in relation to solving ordinary differential equations (ODEs). Participants explore the definitions and implications of completeness, the role of different function spaces, and the challenges associated with boundary conditions in function approximation.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that a basis function is complete if it can represent any function in a given space, while others argue that completeness is context-dependent and relates to specific sets and metrics.
  • There is a discussion about the completeness of polynomials versus Legendre polynomials, with some noting that while polynomials can approximate functions, they may not be complete in certain spaces.
  • One participant questions the interpretation of "complete basis functions" in relation to functionals and the necessity of context regarding the space and norm being used.
  • Another participant emphasizes the importance of specifying the space and metric when discussing completeness, noting that different types of functions form different topological or metric spaces.
  • A participant describes their approach to solving an ODE and the challenges they face in ensuring that trial functions satisfy boundary conditions, raising concerns about the completeness of these functions when recombined.
  • Alternatives to basic polynomials, such as Legendre, Laguerre, and Chebyshev polynomials, are mentioned as potentially more effective for function approximation.

Areas of Agreement / Disagreement

Participants express differing views on the definition and implications of completeness, indicating that multiple competing perspectives remain. There is no consensus on the specific requirements for completeness in the context of the discussion.

Contextual Notes

The discussion highlights the complexity of completeness in relation to different function spaces and boundary conditions. Participants note that the definitions and implications of completeness can vary significantly depending on the context, which remains unresolved.

  • #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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