Sobolev-type inner product

Polynomials Orthogonal with Respect to Sobolev Type Inner Product Generated by Charlier Polynomials

Authors: Sharapudinov I. I., Guseinov I. G.

The problem of constructing of the Sobol ev orthogonal polynomials sαr,n(x) generated by Char li er polynomialssαn(x) is considered. It is shown that the system of polynomials sαr,n(x) generated by Charlier polynomialsis complete in the space Wrl, consisted of the discrete functions, given on the grid Ω = {0, 1, . . .}. Wrlρisa H ilb ert space with the inner product hf, gi. An explicit formula in the form of sαr,k+r(x) =kPl=0brlx[l+r],where xm]= x(x −1) . . . (x − m + 1), is found.

Recurrence Relations for Polynomials Orthonormal on Sobolev, Generated by Laguerre Polynomials

Authors: Gadzhimirzaev R. M.

In this paper we consider the system of polynomials (l_r,n)^a (r — natural number, n = 0,1,...), orthonormal with respect to the Sobolev inner product (Sobolev orthonormal polynomials) of the following type<f, g> = (sum _(v=0))^(r−1) f^(ν)(0)g ^(ν)(0) + (f _0)^∞ f^(r) (x)g^(r)(x)ρ^(x)dx and generated by the classical orthonormal Laguerre polynomials.Recurrence relations are obtained for the system of Sobolev orthonormal polynomials, which can be used for studying various properties of these polynomials and calculate their values for any x and n.

The Fourier Series of the Meixner Polynomials Orthogonal with Respect to the Sobolev-type Inner Product

Authors: Gadzhimirzaev R. M.

In this paper we consider the system of discrete functions {ϕr,k(x)} ∞ k=0 , which is orthonormal with respect to the Sobolev-type inner product hf, gi = Xr−1 ν=0 ∆ ν f(−r)∆ν g(−r) + X t∈Ωr ∆ r f(t)∆r g(t)µ(t), where µ(t) = q t (1−q), 0 < q < 1. It is shown that the shifted classical Meixner polynomials © M−r k (x + r) ª∞ k=r together with functions n (x+r) [k] k! or−1 k=0 form a complete orthogonal system in the space l2,µ(Ωr) with respect to the Sobolev-type inner product.