On a generalized Euler spline and its applications to the study of convergence in cardinal interpolation and solutions of certain extremal problems (Q1314835)

We expand any function \(f\) in \(\Phi(\xi)\) to generalized spline series, in the meantime, the remainder is also given. The remainder can be represented by an integral, the sign pattern of the integral kernel is given in Theorem 2. An extremal problem is solved. The generalized Euler spline function \(E_{n+1}\) and the characteristic function \(A_ n\) play a key role in the proof of these theorems. The non-constant coefficients differential operator \(\mathcal L\) which is relevant to the generalized Euler spline \(E_{n+1}\) in this paper can be described as follows: \({\mathcal L}= {\mathcal L}_{n+1}\), where \({\mathcal L}_ j= D_ j\cdots D_ 1\), \(j=1,\dots,n+ 1\) and \((D_ j g)(t)= \alpha_{j-1}(t)\;D(\beta_{j=1}(t)g(t))\) with \(\alpha_ j\in C^{n-j}(R')\) and \(\beta_ j\in C^{n+ 1-j}(R')\), \(\alpha_ j\) and \(\beta_ j\) are positive functions such that \(\alpha_ j(t+ h)= C_ j \alpha_ j(t)\), \(\beta_ j(t+ h)= C^{-1}_ j \beta_ j(t)\) for all \(t\in R'\) and \(j=0,\dots,n+1\). Here, \(h>0\) is some fixed constant. It is worthwhile to mention that the linear differential operators \({\mathcal L}_ j\) defined above extend those with constant coefficients and real roots. Indeed, by setting \(\alpha_ j(t)= e^{\gamma_ j t}\) and \(\beta_ j(t)= e^{- \gamma_ j t}\), \(\gamma_ j\) real, we have \({\mathcal L}= {\mathcal L}_{n+1}= \prod^ n_{j=0} (D-\gamma_ j)\). Another example is as follows: By setting \(\alpha_ k(t)\equiv 1\) and \(\beta_ j\) the \(h\)-periodic functions, we have \(({\mathcal L}_ j g)\;(t)= D(\beta_{j-1}(t)g(t))\). To simplify our presentation, the additional assumption \(\alpha_{-1}(t)=\beta_{n+1}(t)= 1\) is made through the paper.