On two-point expansion
A two-point expansion of the form \(f(t)\sim \sum^{\infty}_{j=0}\epsilon_ j[a_{1,j}f^{(j)}(0+)+a_{2,j}f^{(j)}(1-)]P_ j(t)\) is considered. Here \(P_ j\) is a polynomial of degree j; \(\epsilon_ j=-1\) or 1 and \(a_{i,j}=0\) or 1 for \(i=1,2\) and \(j=0,1,...,k,...\). The polynomials \(P_ j\) are defined by \(a_{1,k}P_ j^{(k)}(0)+a_{2,k}P_ j^{(k)}(1)=0\) for \(j=0,1,...,k\) and \(P^ j_ j=\epsilon_ j\). The numbers \(\epsilon_ j\) and \(a_{i,j}\) are defined by certain law from a given 2\(\times n\) matrix \((e_{i,j})\), where \(e_{i,j}=0\) or 1, \(e_{1,j}+e_{2,j}=1\) and \(e_{i,j}+e_{i,2n-j-1}=1\) for \(i=1,2\) and \(j=0,...,2n-1.\) This two-points expansion is connected to the interpolation problem leading to the Abel series. The author associates a convex cone \(C\subset C^{\infty}(0,1)\) to the above problem. The polynomials \(P_ j\) generate extreme rays in C and there is exactly one extreme ray in C generated by a nonpolynomial function \(\phi\) which is certain eigenfunction of the boundary value problem \(y^{(2n)}=\lambda y\) with \(e_{1,j}y^{(j)}(0)+e_{2,j}y^{(j)}(1)=0\) for \(j=0,1,...,2n-1\). The author proves that every f in C can be represented by \[ f(t)=\sum^{\infty}_{j=0}\epsilon_ j[a_{1,j}f^{(j)}(0)+a_{2,j}f^{(j)}(1)]P_ j(t)+a\phi(t) \] where the convergence is uniform and that f coincides in [0,1] with an entire function of growth not exceeding order one and finite type.
scientific article; zbMATH DE number 3857675