On some general lacunary interpolation problems
Consider a system \(x_1< \cdots< x_n\) of nodes in the interval \([-1,1]\) and a system \(0=m_0< m_1< \cdots< m_{q-1}\) of natural numbers. The \((0,m_1,\dots,m_n)\) interpolation problem consists in finding a polynomial \(L(x)\), of degree \(\leq nq-1\), such that \(L^{ (m_i)} (x_k)=y_{k,i}\), where \(y_{k,i}\) are arbitrary given numbers, \(k=1,\dots,n\); \(i=1,\dots,q -1\). The problem is called regular if it is uniquely solvable for any set \(\{y_{k,i}\}\) of data. In this case the problem reduces to the determination of the fundamental polynomials \(L_{\nu, m_j} (x)\) of degree at most \(nq-1\), verifying the conditions \(L^{(m_i)}_{\nu,m_j} (x_k)= \delta_{ij} \delta_{\nu k}\), \(\nu,k=1,\dots,n\); \(i,j=1,\dots,q-1\). Assuming \(L_{\nu,m_i} (x)\) known, for \(\nu=1,\dots,n\); \(i= p+1,\dots,q-1\), where \(m_p< q\leq m_{p+1}\), the remaining fundamental polynomials can be given (by an explicit formula) in terms of these known fundamental polynomials and the fundamental Hermite interpolation polynomials for the \((0,1,2,\dots,q-1)\) interpolation problem. The problem of finding the polynomials \(L_{\nu, m_i}\), \(\nu =1,\dots,n\); \(i=p +1,\dots,q-1\), is more complicated. Supposing that the interpolation nodes are the roots of a primitive of the Legendre polynomial \(P_{n-1} (x)\), the authors consider three cases, namely \((0,1,\dots,r-2,r)\), \((0,1,\dots,r-3,r)\), and \((0,1,\dots,r-3, r-1,r)\), show that these problems are regular and give explicit formulae for the last fundamental polynomials \(L_{\nu, m_{q-1}}\). They give also estimates for the Hermite fundamental polynomials as well as for \(L_{\nu,m_i}\). As application, they prove that, in the case of (0,2,3) interpolation, \(|f(x) - R_n(f,x) |=O(\omega_f(1/n) \log n)\), where \(R_n(f,x)=\sum^n_{\nu=1} f(x_k) L_{\nu,0} (x)\).
scientific article; zbMATH DE number 954855