On an autocorrection phenomenon of the Eckhoff interpolation (Q2895753)

Let \(f\in C^q[-1, 1]\), \(A_k(f):= f^{(k)}(1) - f^{(k)}(-1)\), \(k=0,\dots,q\), and let \(\widehat{B}_n(k)\) be the discrete Fourier coefficients of the Bernoulli polynomials \(B(x;k)\). Suppose that NEWLINE\[NEWLINE \widehat{F}_n=\widehat{f}_n - \sum_{k=0}^{q-1}A_k(f)\widehat{B}_n(k), \quad n=-N, -N+1,\dots, N, \tag{1} NEWLINE\]NEWLINE where \(\widehat{f}_n(k)\) are the discrete Fourier coefficients of \(f\). Then the \textit{Krylov-Lanczos interpolation} \(I_{N,q}(f)\) is defined by NEWLINE\[NEWLINE I_{N,q}(f)(x):= \sum_{n=-N}^N\widehat{F}_ne^{i\pi nx} + \sum_{k=0}^{q-1}A_k(f)B(x;k). \tag{2} NEWLINE\]NEWLINE Now let us consider the following system of linear equation with unknowns \(A_k^i(f,N)\): NEWLINE\[NEWLINE \widehat{f}_n = \sum_{k=0}^{q-1}A_k^i(f,N)\widehat{B}_n(k), \quad n=n_1,n_2,\dots, n_q, NEWLINE\]NEWLINE where the indeces \(n_s\) for even values \(q=2m\) are defined by NEWLINE\[NEWLINE n_s= \begin{cases} \;\;N-s+1,\;\;\quad \;\,s=1,\dots, m\,,\\ \,-(N - s +m +1), \quad s=m+1,\dots,2m\,,\\ \end{cases} NEWLINE\]NEWLINE and for odd values \(q=2m+1\) we have NEWLINE\[NEWLINE n_s= \begin{cases} \;\;N-s+1,\;\;\quad \;\,s=1,\dots, m+1\,,\\ \, -(N - s +m +2), \quad \,s=m+2,\dots, 2m+1\,.\\ \end{cases} NEWLINE\]NEWLINE Then the \textit{Eckhoff interpolation} \(\widetilde I_{N,q}(f)\) can be defined as \(I_{N,q}(f)\), if we replace \(A_k(f)\) by \(A_k^i(f,N)\) in (1) and (2). It is shown that the Eckhoff interpolation converges faster compared with the Krylov-Lanczos interpolation (more precisely, we have an improvement in convergence rate by the factor \(O(N^{2m})\) for \(q=2m\) and \(q=2m+1\)). Numerical experiments confirm theoretical estimates.