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

In summary, Wiki defines orthogonal functions as functions that are perpendicular to each other in a certain sense. This can be determined by the inner-product chosen, which may include the requirement that it be done over a specific interval. For example, the sine and cosine functions are orthogonal over the interval [0,π], but may not be orthogonal over other intervals. The Chevy Chase polynomials are an example of orthogonal functions.
  • #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
askmathquestions said:
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.

##\ ##
 
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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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1. What are orthogonal functions?

Orthogonal functions are mathematical functions that are perpendicular to each other when plotted on a graph. This means that they intersect at a right angle and have a dot product of zero.

2. What is an example of orthogonal functions?

Sine and cosine are an example of orthogonal functions. When plotted on a graph, they intersect at a right angle and have a dot product of zero.

3. Do sine and cosine qualify as orthogonal functions?

Yes, sine and cosine qualify as orthogonal functions because they are perpendicular to each other and have a dot product of zero.

4. Can two linear functions be orthogonal?

No, two linear functions cannot be orthogonal because they will always intersect at a non-zero angle and have a non-zero dot product.

5. What is the importance of orthogonal functions in mathematics?

Orthogonal functions are important in mathematics because they have many applications in solving mathematical problems, such as in Fourier series and signal processing. They also have properties that make them useful in simplifying calculations and proving theorems.

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