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Re: How to find a set of orthogonal functions in a finite time domain
and with both zero start and end?
Posted:
Feb 16, 2010 8:16 AM
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On Feb 10, 10:40 pm, "Zdislav V. Kovarik" <kova...@mcmaster.ca> wrote:
> On Wed, 10 Feb 2010, workaholic wrote:
> > Is there such set of orthogonal function defined in a finite time
> > domain with zero start and zero end? Which kind of book I should look
> > up for such functions?
>
> > Thanks a lot!
>
> There are many such sets. You need only interval (0,1) or (-1,1), because
> you can make a linear change of variable without losing orthogonality. I
> presume the inner product has constant weight 1, and you are looking for
> complete sets.
>
> You have:
>
> sine set, f_n(x) = sin(n*pi*x), n=1, 2, ..., on (0,1),
>
> modified Jacobi polynomials (1+x)*(1-x)*P^(1,1)_n(x) on (-1,1)
> and you can fiddle around with weights to your heart's content.
>
> Literature: look under (you could guess it) "orthogonal sets" with an
> extra key-phrase "boundary conditions". Suitable Sturm-Liouville problems
> supply many such orthogonal systems.
>
> For finite sets, take any linearly independent set satisfying your
> boundary conditions, and perform Gram-Schmidt orthogonalization.
>
> Hope it helps (and hope I understood your question:-),
> ZVK(Slavek).
Hi,
Thanks a lot for your explanation!
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