
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


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)*(1x)*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 keyphrase "boundary conditions". Suitable SturmLiouville problems > supply many such orthogonal systems. > > For finite sets, take any linearly independent set satisfying your > boundary conditions, and perform GramSchmidt orthogonalization. > > Hope it helps (and hope I understood your question:), > ZVK(Slavek).
Hi,
Thanks a lot for your explanation!

