Reflection coefficients for the generalized Jacobi weight functions
Let \(w(t)= f(t) \prod^ N_ 1 | e^{it}- e^{it_ j}|^{\gamma_ j}\) be a generalized Jacobi weight on the unit circle, where \(\gamma_ j> -1\) and \(h\) is a smooth differentiable function. Consider the orthonormal polynomials \(\{\Phi_ n\}\) associated with \(w\) and the so-called reflection coefficients \(\Phi_ n(0)\) for them. These coefficients appear in the recurrence formula for the orthogonal polynomials and their behavior is very important in the theory. In the paper the very interesting estimate \[ 0< \limsup_{n\to \infty} n| \Phi_ n(0)|< \infty \] is proven. Then the result is transformed to the real line, where it reads that if \(\{a_ n\}\) and \(\{b_ n\}\) are the recursive coefficients corresponding to the orthogonal polynomials with respect to a generalized Jacobi weight, then \(| a_ n- 1/2|\) and \(| b_ n|\) are bounded by a constant times \(1/n\).
scientific article; zbMATH DE number 636062