Discussion Overview
The discussion revolves around the existence of a complete set of multi-variable functions that can represent arbitrary functions of two variables, specifically in the context of functional analysis and orthonormality relations.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant proposes the idea of expressing an arbitrary function of two variables, ##f(x,y)=\sum_n c_n \phi_n(x,y)##, using a complete set of functions ##\phi_n(x,y)##.
- Another participant asserts that for arbitrary functions, the answer to finding such a complete set is "no," but under specific conditions in the Hilbert space ##L^2([a,b] \times [a,b]; \mathbb{C})##, it is possible to construct an orthonormal basis from products of functions that form an orthonormal basis of ##L^2([a,b]; \mathbb{C})##.
- A participant questions the meaning of "the space of functions" as used in the initial post, indicating a need for clarification.
- A later reply acknowledges the clarification received from another participant's post, suggesting that the initial question may have been resolved for that individual.
Areas of Agreement / Disagreement
Participants express differing views on the existence of a complete set of functions for arbitrary functions of two variables. While one participant believes it is not possible in general, another provides a specific context where it is achievable, indicating a lack of consensus on the broader question.
Contextual Notes
The discussion highlights the dependence on specific mathematical contexts, such as the properties of Hilbert spaces and the definitions of orthonormal bases, which may not be universally applicable.