If f and g are orthogonal, are f* and g orthogonal?

  • Thread starter Aziza
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Main Question or Discussion Point

I am curious:

if f and g are (complex) orthogonal functions, are f* and g also orthogonal? (* denotes complex conjugate).

I would think the answer is no, in general, but I just want to confirm
 

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  • #2
Office_Shredder
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What are you integrating over and what functions are you allowed?

For example if you're integrating over the unit circle and all you ask for are integrable functions, then if f = 1/sqrt(z) and g = 1/sqrt(z) (where z has an argument between 0 and 2pi), then
[tex] \oint fg dz = 2\pi i [/tex]
but
[tex] \oint f^* g dz = \oint 1 dz = 0 [/tex]

So f and g are orthogonal, but f* and g are not.
 
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A simpler example for [itex]f,g\in \mathbb C^2,[/itex] (which we could write as [itex]f,g: \{1,2\}\to \mathbb C[/itex] if you like):

Let [itex]f:= (1,i), g:=(1,-i).[/itex] Then [tex]\langle f, g \rangle = (1)(\overline{1})+(i)(\overline{-i}) = (1)(1)+ (i)(i) = 0,[/tex] and [tex]\langle f^*, g \rangle = (\overline{1})(\overline{1})+(\overline{i})(\overline{-i}) = (1)(1)+ (-i)(i) = 2\neq 0,[/tex] so that [itex]f,g[/itex] are orthogonal, but [itex]f^*,g[/itex] aren't.
 

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