Why can the eigenfunctions of a regular SLP be used to form an orthonormal basis?
Conditions
- The Sturm-Liouville problem is regular.
- The eigenfunctions are mutually orthogonal and total.
Reasoning, step by step
- Start from the theorem statement that the eigenfunctions of a regular SLP are orthogonal and total.
- Interpret "total" as meaning the eigenfunctions form a complete basis.
- Normalize each eigenfunction to unit length.
- Conclude that because they are already mutually orthogonal, the normalized family becomes an orthonormal basis.
Example
The board's parenthetical reads "(so can form an ONB)". The lecturer explains: "if you normalize them accordingly, you can get an orthonormal basis of eigenfunctions of your differential operator."
Common misconceptions
- Thinking that totality alone is the same as already being an orthonormal basis; normalization is the extra step.
- Assuming orthogonality alone is sufficient to form a basis; completeness (totality) is also required.
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Related questions
Regular Sturm-Liouville problems have an infinite spectrum , and their eigenfunctions are orthogonal and total. Because they are orthogonal and total, they can form an orthonormal basis (ONB) after normalization.
Conditions: The Sturm-Liouville problem is regular ( and for all ).
Sturm-Liouville problems are the general class of boundary value problems that arise when the separation of variables technique is applied to a PDE or an initial-boundary value problem. They are also situated as part of the extension of the spectral theorem to infinite dimensions.
Conditions: The context is solving PDEs or initial-boundary value problems using separation of variables.
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