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).