Lebesgue Sobolev orthogonality on the unit circle
This is a very nicely written paper extending results for monic orthogonal polynomials \(P_n\) as \(n\rightarrow\infty\) on growth of the norm, pointwise limits and the radius of the disk containing all zeros to the case of the following Sobolev inner product on the unit circle: \[ \langle f(z),g(z)\rangle_s=\int_0^{2\pi} f(e^{i\theta})\overline{g(e^{i\theta})} d\mu(\theta) + \sum_{k=1}^p \lambda_k \int_0^{2\pi} f^{(k)}(e^{i\theta})\overline{g^{(k)}(e^{i\theta})} {d\theta\over 2\pi},\quad z=e^{i\theta}, \] where \(d\mu(\theta)\) is a finite positive Borel measure on \([0,2\pi]\) with infinite support, verifying the Szegő condition and with \(\lambda_1>0\), \(\lambda_k\geq 0\;(2\leq k\leq p), d\theta/2\pi\) the normalized Lebesgue measure on \([0,2\pi]\).
scientific article; zbMATH DE number 1326429