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Orthonormal basis functions for L^2(R) 
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#1
Feb1813, 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 L^{2}(ℝ)? According to Wikipedia the hermite polynomials form an orthogonal basis (w.r.t. to a certain weight function) for L^{2}(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? 


#2
Feb1813, 08:57 AM

Mentor
P: 18,036

You should do some research on wavelets. For example, the Haar wavelet gives an orthonormal basis apparently: http://en.wikipedia.org/wiki/Haar_wavelet



#3
Feb1813, 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 L^{2}(ℝ) whose support is the whole real line, e.g. rapidly decaying functions. 


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