Generalized polynomials of minimal norm
The authors consider the problem of finding optimal generalized polynomials of minimal \(L_ p\) norm, \(1\leq p<\infty\). Let \(\{u_ i\}^{N+1}_{i=1}\subset C^ N[a,b]\), [0,1]\(\subset (a,b)\), be a given set of functions with the property that \(\{u_ i\}^ K_{i=1}\) forms an ``extended Chebyshev system'' for \(K=N-1,N,N+1\). For \({\bar \tau}=(\tau_ 1,...,\tau_ N)\in R^ N\) the symbol u(\({\bar \tau}\);t) is devoted to denote the unique polynomial of the type \(u=u_{N+1}+\sum^{N}_{i=1}\alpha_ iu_ i\) with real coefficients \(\{\alpha_ i\}\) such that \(u(\tau_ k)=0\), \(k=1,...,N\). Let us consider the multiplicities \(\{\nu_ k\}^ n_{k=1}\), \(\{\mu_ k\}^ m_{k=1}\) such that \(\sum^{n}_{k=1}\nu_ k+\sum^{m}_{k=1}\mu_ k=N\) and the points \({\bar \xi}=(\xi_ 1,...,\xi_ m)\) where \(a<\xi_ 1\leq...\leq \xi_ m<b\). Set \(\Delta_ n=\{\bar x=(x_ 1,...,x_ n)\); \(0\leq x_ 1\leq x_ 2\leq...\leq x_ n\leq 1\}\) and for each \(\bar x=(x_ 1,...,x_ n)\in \Delta_ n\), let \({\bar \tau}={\bar \tau}(\bar x,{\bar \xi})\in R^ N\) be the form \[ {\bar \tau}(\bar x,{\bar \xi})=(x_ 1,...,x_ 1,x_ n,...,x_ n,\xi_ 1,...,\xi_ 1,\xi_ m,...,\xi_ m). \] If \(I(\bar x,{\bar \xi})=\int^{1}_{0}| u({\bar \tau}(\bar x,{\bar \xi});t)|^ p dt,\) \(1\leq p<\infty\), the aim of this important paper is to investigate the existence and the uniqueness of the \(\bar x^*\in \Delta_ n\) with the property \((1)\quad I(\bar x^*,{\bar \xi})=\min_{\bar x\in \Delta_ n}I(\bar x,{\bar \xi}).\) The main results are the following: ``Let \(1\leq p<\infty\). Then there is an \(\bar x^*\in \Delta_ n\) which satisfies (1). Further any such minimizing \(\bar x^*=(x^*_ 1,...,x^*_ n)\) is in interior of \(\Delta_ n\), i.e., \(0<x^*_ 1<...<x^*_ n<1\). For each n,m\(\geq 0\) so that \(n+m\leq N\), there is a unique solution \(\bar x^*\in \Delta_ n\) to the problem cited in (1).'' As an application of the above results, the authors obtain Gaussian quadrature formulas for extended Chebyshev systems.
scientific article; zbMATH DE number 4113208