Simultaneous interpolation and approximation of continuous functions
Let us denote: \(P_ n(\pi_ n)\) the space of trigonometric (algebraic) polynomials with degree not greater than n; \(E=C[-1,1]\) or C[0,2\(\pi)\); \(\| x\| = \sup_{t}| x(t)|\), \(x\in E\), \(t\in [-1,1]\) or \(t\in [0,2\pi]\); \(E_ n(x)=\inf_{p}\| x-p\|\), \(p\in \pi_ n\) or \(p\in P_ n\); \(x_ k=\{F_ k\}\), \(F_ k=\{f_ 0,...,f_ k\}\), \(f_ j(x)=x(t_ j)\) \(\forall x\in E\) and \(t_ 0,...,t_ k\) a set of different points from [-1,1] or from [0,2\(\pi)\); \(W_*^ r(W^ r)\) the class of \(x\in C[0,2\pi)\) (x\(\in C[-1,1])\) with (r-1)th absolutely continuous derivative and \(\| x^{(r)}\| <\infty\); \(\omega_ n(x,\delta)\) the modulus of smoothness; \(Z_+\) the set of non-negative integers; M the set of natural numbers. Theorem 1. Let \(k,n\in Z_+\), \(n\geq k/2\), \(x\in W_*^ k\). Then \[ \sup_{F_ k\in x_ k}\inf_{p\in P_ n,f(p)=f(x)\forall f\in F_ k}\| x-p\| \leq C_ k(n+1)^{-k}E_ n(x^{(k)}). \] Theorem 2. Let \(r,k,n\in Z_+\), \(m\in M\), \(n\geq m+r-1\), \(r\geq k\), \(x\in W^ k\). Then for any \(t_ 0,...,t_ k\in [-1,1]\) there exists \(p\in \pi_ n\) such that \(p(t_ j)=x(t_ j)\), \(j=0,...,k\), \[ \| p-x\| \leq C_{r,m}((n+1)^{-1}\sqrt{1-t^ 2}+(n+1)^{-2})^ r\omega_ n(x^{(r)},(n+1)^{-1}\sqrt{1-t^ 2}+(n+1)^{-2}) \] for any \(t\in [- 1,1]\). The algebraic analogue to the first theorem is also proved.
scientific article; zbMATH DE number 3914889