The discussion centers around the concept of orthogonality in functions, particularly in the context of Fourier series and Sturm-Liouville problems. Participants explore the implications and applications of proving functions are orthogonal, touching on theoretical and practical aspects.
Participants generally agree on the importance of orthogonality in simplifying calculations and its relevance in various applications, but the discussion remains open regarding specific implications in different contexts.
Some assumptions about the mathematical framework and definitions of orthogonality may not be fully articulated, and the discussion does not resolve the broader implications of orthogonality beyond the mentioned applications.
mfb said:That depends on the topic, there are many applications. Quantum mechanics relies heavily on orthogonal functions, for example, but many fields do that.
Engineerbrah said:I understand that a function is orthogonal if the inner product of any two functions of an infinite series equal to zero.
My question is why do we prove functions are orthogonal? What can we do with this information?
pasmith said:Two vectors are orthogonal if their inner product vanishes.
Why do we care about basis vectors being orthogonal? It makes it trivial to find the components: if [itex]\{e_1, \dots, e_n\}[/itex] are orthogonal then [tex] v = \sum_{k = 1}^n \frac{\langle v, e_k \rangle}{\langle e_k, e_k \rangle}e_k.[/tex] If the basis vectors were not orthogonal it would be necessary to solve a system of [itex]n[/itex] simultaneous equations to find the components.
Functions in this context are vectors, with the complication that the dimension of the space is infinite. Do you want to have to solve a countably infinite system of simultaneous equations to find the components with respect to some basis?