The representation of the trilinear kernel in general orthogonal polynomials and some applications (Q1178510)

If \(p_ n(x)\) is a sequence of orthonormal polynomials on \([-1,1]\) then a study is made of the trilinear kernel \[ D_ n(x,y,z)=\sum_{k=0}^ n p_ k(x)p_ k(y)p_ k(z). \] A representation is obtained from which useful bounds on \(D_ n\) are deduced for orthogonal polynomials in \(M(0,1)\). It is shown how infinite sums of the form \[ K_ n(x,y,z)=\sum_{k=0}^ \infty \lambda_ kp_ k(x)p_ k(y)p_ k(z) \] can be studied. Also the construction of a double-humpbacked majorant for \(D_ n(x,y,z)\) is worked out and used to estimate the Lebesgue quasifunction \[ L_ n(x,y)=\int_{-1}^ 1 | D_ n(x,y,z)| dz. \]