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Orthonormal basis functions for L^2(R)

by mnb96
Tags: basis, functions, orthonormal
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mnb96
#1
Feb18-13, 08:48 AM
P: 625
Hello,

are there sets of functions that form an orthonormal basis for the space of square integrable functions over the reals L2(ℝ)?
According to Wikipedia the hermite polynomials form an orthogonal basis (w.r.t. to a certain weight function) for L2(ℝ). So I guess it would be a matter of multiplying the polynomials by suitable scalars in order to make them orthonormal.
Are there other known examples besides the Hermite polynomials?
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micromass
#2
Feb18-13, 08:57 AM
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P: 18,278
You should do some research on wavelets. For example, the Haar wavelet gives an orthonormal basis apparently: http://en.wikipedia.org/wiki/Haar_wavelet
mnb96
#3
Feb18-13, 02:05 PM
P: 625
Hi micromass!
thanks for your reply. Your answer basically answer my question.
Apparently the Haar wavelets "constitute a complete orthogonal system for the functions on the unit interval".

I was now wondering if there are more orthonormal bases for functions in L2(ℝ) whose support is the whole real line, e.g. rapidly decaying functions.


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