I What's an example of orthogonal functions? Do these qualify?

[askmathquestions]

Wiki defines orthogonal functions here

https://en.wikipedia.org/wiki/Orthogonal_functions

Here's one example, but it's an example that is only true for a specific interval

https://www.wolframalpha.com/input?i=integral+sin(x)cos(x)+from+0+to+pi

So are these functions orthogonal because there simply exists *some* interval where their integral product is $0?$ Or, must the entire integral be identically $0$ over the entire domain? I'm confused. Are $\sin$ and $\cos$ always orthogonal or only sometimes orthogonal?

 

[BvU]

are these functions orthogonal because there simply exists *some* interval where their integral product is 0? Or, must the entire integral be identically 0 over the entire domain?

The latter. However, if you designate the *some* interval as *the* interval alias the domain, the two statements become identical.

I find the wiki lemma pretty clear -- but then, hey, I'm a physicist.

$\$

 

[fresh_42]

https://en.wikipedia.org/wiki/Chebyshev_polynomials#Orthogonality

 

[WWGD]

Ultimately, orthogonality is determined by a choice of inner- product , which in this case includes the requirement that it be done over [a,b].

 

[WWGD]

https://en.wikipedia.org/wiki/Chebyshev_polynomials#Orthogonality

I like the Chevy Chase polynomials.