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Physgeek64
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Homework Statement
By applying the Gram–Schmidt procedure to the list of monomials 1, x, x2, ..., show that the first three elements of an orthonormal basis for the space L2 (−∞, ∞) with weight function ##w(x) = \frac{1}{\sqrt{\pi}} e^{-x^2} ##
are ##e_0(x)=1## , ##e_1(x)= 2x## ,##e_2(x)= \frac{1}{\sqrt{2}} (2x^2−1)##
Explain why any well-behaved function f : R → C may be expressed as a series of the form
## f(x)= \frac{1}{\sqrt{2 \pi}} \sum {a_n e_n(x) e^{- \frac{1}{2} x^2}} ## where the sum runs from 0 to infinity
Homework Equations
3.The attempted solution[/B]
I have done the first part to do with the Gram–Schmidt procedure, using the usual formula, but cannot understand how to do the second part of the question
My initial thought was to try to isolate one of the co-efficients, ##a_m## and show that this is the same as the 'normal' formula for ##a_m## when using a weight function of one- but this didn't seem to work out.
What I'm confused about it where the ##\frac{1}{\sqrt{2 \pi}}## has come from, since the basis functions are already normalised, and why the square root of the weight function is included in the summation. I get the feeling this is what the question is wanting me to show is okay, so to speak, but I can't quite see it.
Any help would be greatly appreciated :)
(note: i did initially type out all my working, but my computer couldn't cope with it, and kept freezing- sorry)
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