Subperiodic trigonometric interpolation and quadrature (Q450290)

The authors study subperiodic trigonometric interpolation and quadrature on \([-\omega, \omega]\) \((0 < \omega \leq \pi)\) at angular nodes \(\theta_j\) \((j=1,\ldots, 2n+1)\) which are the zeros of \(T_{2n+1}(\sin(\theta/2) / \sin(\omega/2))\). Here, \(T_{2n+1}\) denotes the \((2n+1)\)-th Chebyshev polynomial. For \(\omega = \pi\), the nodes \(\theta_j\) are equally spaced in \([-\pi,\pi]\). These angular nodes are unisolvent for trigonometric interpolation of order \(n\) on \([-\omega,\omega]\). The corresponding Lebesgue constant is of order \({\mathcal O}(\log n)\). The associated Fejér-like trigonometric quadrature formula has positive weights. As applications, the equilibrium measure of an arc of the complex unit circle is computed and cubature formulas over circular sectors, which are exact for bivariate algebraic polynomials of degree \(\leq n\), are constructed.