[seqfan] Re: person needing help with sequences

[Andrew N W Hone]

Hi - 

There is quite a lot of literature about orthogonal polynomials with non-smooth weights, and even 
discontinuous weights. See for example the following paper: 

http://iopscience.iop.org/0305-4470/38/12/L01/fulltext

This particular weight is still continuous everywhere, even if not differentiable at two points.

Any piecewise continuous weight function w>=0 on an interval [a,b] defines a set of orthogonal polynomials,
by the condition that the integral of P_n times P_m times w over [a,b] is equal to 0 if n and m are different, 
and equal to a non-zero constant if n=m, where the degree of each P_n is n; if the polynomials are fixed to be 
monic, 

P_n = x^n + ..., 

then they are uniquely determined by this condition (using a three-term recurrence relation). One can 
generalize from piecewise continuous weights to more general measures.    

I think the Gibbs phenomenon should depend only on discontinuities in the function being expanded with respect to 
an orthonormal basis, and not on the weight function.

Andy  
________________________________________
From: seqfan-bounces at list.seqfan.eu [seqfan-bounces at list.seqfan.eu] On Behalf Of Richard Mathar [mathar at strw.leidenuniv.nl]
Sent: 14 August 2010 20:57
To: seqfan at seqfan.eu
Cc: jalel.atia at gmail.com
Subject: [seqfan] Re: person needing help with sequences

I doubt that there is any standard literature on the weight function
|x-1/2|/sqrt(1-x)+|x+1/2|/sqrt(1+x). It has two "edges" at
x=+-1/2 where it is not differentiable, so usually one would not try to
implement a set of basis functions with polynomials on
such a thing; this suffers from all the standard Gibbs phenomena.
This is the major difference to the standard case of Jacobi weights over the
same [-1,1] interval.
One can probably derive the expansion coefficients by splitting all
the integrations at x=+-0.5. So a formal definition of the associated
orthogonal polynomials is not a problem.

RJM


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