Symmetric orthogonal polynomials for Sobolev-type inner products
In this paper the authors investigate monic orthogonal polynomials \(Q_ n (x)\) for the inner product \[ \langle f,g\rangle= \int_{-a}^ a f(x) g(x) d\mu(x)+ \sum_{j=0}^ r M_ j f^{(j)}(0) g^{(j)}(0), \] where \(\mu\) is a symmetric measure. Due to the symmetry, it follows that \(Q_{2n}(x)= U_ n(x^ 2)\) and \(Q_{2n+1}(x)= x V_ n(x^ 2)\), and it is shown that \(U_ n\) and \(V_ n\) are again orthogonal polynomials with a Sobolev-type inner product. The zeros of \(Q_ n(x)\) are investigated and it is shown that in some cases there may be two complex conjugated zeros, a situation which can never occur for orthogonal polynomial in \(L_ 2(\mu)\). Finally, the authors also consider a symmetric Sobolev-type inner product with masses at \(\pm c\).
scientific article; zbMATH DE number 635930