Completeness of a basis function

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SUMMARY

The discussion centers on the concept of completeness of basis functions in various mathematical spaces. It is established that while the set of polynomials is not complete, Legendre polynomials are complete in specific contexts, particularly in spherical coordinates related to the Laplacian operator. The conversation highlights the importance of specifying the space and metric when discussing completeness, as different mathematical frameworks can yield different interpretations. Additionally, the use of trial functions and boundary conditions in solving ordinary differential equations (ODEs) is emphasized, particularly in relation to the weak formulation and eigenfunction expansions.

PREREQUISITES
  • Understanding of basis functions and their completeness in functional spaces.
  • Familiarity with Legendre polynomials and their applications in differential equations.
  • Knowledge of weak formulations and eigenvalue problems in the context of ODEs.
  • Basic concepts of topological spaces and metrics, particularly in relation to Cauchy sequences.
NEXT STEPS
  • Study the properties and applications of Legendre polynomials in solving differential equations.
  • Learn about the weak formulation of ODEs and how to derive eigenvalue problems from them.
  • Explore the concept of completeness in various function spaces, including L^2 spaces.
  • Investigate the implications of boundary conditions on the choice of trial functions in numerical methods.
USEFUL FOR

Mathematicians, physicists, and engineers involved in solving differential equations, particularly those utilizing eigenfunction expansions and exploring the completeness of function spaces.

  • #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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