Regularity of \((0,1,\dots,r-2,r)\) and \((0,1,\dots,r-2,r)*\) interpolations on some sets of the unit circle (Q1894645)

\((m_0, \ldots, m_q\) interpolation on a set of knots \(Z_n = \{z_1, \ldots, z_n\}\) is called regular, if, for arbitrary complex numbers \(\{c_{m_j,k} |j = 0,1, \ldots, q\), \(k = 1,2, \ldots, n\}\), there exists a unique polynomial \(P\) of degree less or equal \((q + 1) n - 1\) such that \[ P^{(m_j)} (z_k) = c_{m_j,k}, \quad j = 0,1, \ldots, q,\;k = 1,2, \ldots, n. \tag{*} \] The present paper deals with the regularity of \((0,1, \ldots, r - 2,r)\) interpolation on the sets \(Z_n\) obtained by projecting vertically the zeros of \((1 - x^2) P_l^{(\alpha, \beta)} (x)\), \((1 - x) P_l^{(\alpha, \beta)} (x)\), \((1 + x) P_l^{(\alpha, \beta)} (x)\), respectively, onto the unit circle, where \(P_l^{(\alpha, \beta)}\) denotes the \(l\)-th Jacobi polynomial with respect to the real numbers \(\alpha, \beta > - 1\). It is shown that under suitable assumptions on the parameters \(\alpha\) and \(\beta (0,1, \ldots, r - 2,r)\) interpolation is regular on each of these sets \(Z_n\). For \((0,1, \ldots, r - 2,r)^*\) interpolation a similar result is proved.