Asymptotic expansion of the Lebesgue constant for even-order Hermite-Fejér interpolation on Chebyshev nodes (Q6612198)

Suppose that \(\phi\) is a continuous real-valued function defined on the interval \([-1, 1]\) and \(\mathcal{X} = \big\{x^{(k)}_n : |x^{(k)}_n| \leq 1,\,\, k = 1, 2, \ldots, n,\,\, n = 1, 2, 3,\ldots\big\}\) is a triangular matrix of interpolation nodes. Then for each integer \(t \geq 0\), there is a unique polynomial \(\mathcal{H}_{t,n}(\mathcal{X} ,\phi; x)\) in \(x\) of degree at most \((t + 1)n - 1\) which satisfies \N\[\N\mathcal{H}_{t,n}(\mathcal{X} ,\phi; x^{(i)}_n)=\phi(x^{(i)}_n),\quad 1\leq i\leq n, \N\]\N\[\N\mathcal{H}_{t,n}^{(r)}(\mathcal{X} ,\phi; x^{(i)}_n)=0, \quad 1\leq r\leq t,\,\, \quad 1\leq i\leq n. \N\]\NThis unique polynomial \(\mathcal{H}_{t,n}(\mathcal{X} ,\phi; x)\) is known as the \(t\)-th order Hermite-Fejér interpolation polynomial of \(\phi(x)\) on \(\mathcal{X}\), and it can be expressed as \N\[\N\mathcal{H}_{t,n}(\mathcal{X} ,\phi; x)=\sum_{i=1}^{n}\phi(x^{(i)}_n)\alpha_{t,n,i}(\mathcal{X};x),\N\]\Nwhere \(\alpha_{t,n,i}(\mathcal{X};x)\) is the unique polynomial of degree \((t + 1)n - 1\) which satisfies\N\[\N\alpha_{t,n,i}^{(r)}(\mathcal{X};x^{(j)}_n)= \begin{cases} 1, & i=j,\,\, r=0\\\N0, & i\neq j,\,\, r=0\\\N0, & r\geq 1 \end{cases}\N\]\Nfor \(1 \leq i, j \leq n\) and \(0 \leq r \leq t\). If \(t = 0\), then \(\mathcal{H}_{0,n}(\mathcal{X} ,\phi; x)\) is a classical Lagrange interpolation polynomial of \(\phi(x)\) on \(\mathcal{X}\). For \(t\)-th order Hermite-Fejér interpolation, the Lebesgue constant \(\Lambda_{n}^{(t)}(\mathcal{X})\) is given by \N\[\N\Lambda_{n}^{(t)}(\mathcal{X})=\underset{x\in [-1,1]}\sup\, \sum_{k=1}^{n}|\alpha_{t,n,k}(\mathcal{X};x)|. \N\]\NThe convergence of polynomial interpolation is studied through the Lebesgue constant. The matrix of Chebyshev nodes is defined by \N\[\N\mathcal{T} =\left\{x^{(j)}_n=\cos \frac{(2j - 1)\pi}{2n},\,\, j = 1, 2,\ldots, n,\,\, n = 1, 2, 3 \ldots \right\}.\N\]\NThe author finds an integral representation for \(\Lambda_{n}^{(2s)}(\mathcal{T})\) at \(s\geq 0\) and uses it to obtain the asymptotic expansion of \(\Lambda_{n}^{(2s)}(\mathcal{T})\).