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

  • #1
askmathquestions
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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?
 

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  • #2
BvU
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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.

##\ ##
 
  • #4
WWGD
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Ultimately, orthogonality is determined by a choice of inner- product , which in this case includes the requirement that it be done over [a,b].
 
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