Sobolev orthogonal polynomials on the conic surface

Orthogonal polynomials with respect to the weight function wβ,γ(t) = t^β(1-t)^γ, $γ> -1$, on the conic surface \(x,t): \|x\| = t, \, x ∈ Rᵈ, \, t ≤ 1\ are studied recently, and are shown to be eigenfunctions of a second order differential operator D_γ when $β=-1$. We extend the setting to the Sobolev inner product, defined as the integration of the s-th normal derivative D = d/d t - t⁻¹ x, ∇ₓ of the cone with respect to wβ+s,0 over the conic surface, plus a sum of integrals over the rim of the cone. Our main results provide an explicit construction of an orthogonal basis and a formula for the orthogonal projection operators; the latter is used to exploit the interaction of differential operators and the projection operator, which allows us to study the convergence of the Fourier orthogonal series. The study can be regarded as an extension of the orthogonal structure to the weight function wβ, -s for a positive integer s. It shows, in particular, that the Sobolev orthogonal polynomials are eigenfunctions of D_γ when $γ= -1$.