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m →Choice of inner product: "suppose approximating the vectors are" -> "suppose approximating vectors are" |
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:<math> \langle f,g\rangle = \int f(x) ^* g(x)\,dx </math>
with appropriate [[integral|integration]] boundaries. Here, the asterisk indicates the [[complex conjugate]] of ''f''.
For another perspective on this inner product, suppose approximating vectors <math>\vec{f}</math> and <math>\vec{g}</math> are created whose entries are the values of the functions ''f'' and ''g'', sampled at equally spaced points. Then this inner product between ''f'' and ''g'' can be roughly understood as the dot product between approximating vectors <math>\vec{f}</math> and <math>\vec{g}</math>, in the limit as the number of sampling points goes to infinity. Thus, roughly, two functions are orthogonal if their approximating vectors are perpendicular (under this common inner product).[http://maze5.net/?page_id=369]
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