Estimates of errors in interpolating trigonometric polynomials with equidistant nodes on classes of continuous periodic functions of several variables
The author discussed the behaviour of the quantity \[ \epsilon_ n^{(N)}(\omega,x)=\sup_{f\in H^{(N)}_{\omega}}| f(x)-S^*_ n(f;x)|, \] where \(S^*_ n(f;x)\) is the interpolating polynomial of order \(n=(n_ 1,n_ 2,...,n_ N) (n_ i=1,2,...,i=1,2,...,N)\) of the function \(f(t)=f(t_ 1,t_ 2,...,t_ N)\) in a system of equidistant nodes \(x^{(k)}=(x_ 1^{(k_ 1)},...,x_ n^{(k_ N)}) (x_ i^{(k_ i)}=2\pi k_ i/2n_ i+1,k_ i=0,\pm 1,\pm 2,...),\) and as usually \[ H^{(N)}_{\omega}=\{f\in C_{2\pi}:| f(t_ 1,...,t_ N)-f(t'\!_ 1,...,t'\!_ N)| \leq \sum^{N}_{i=1}\omega_ i(| t_ i-t'\!_ i|)\}, \] where \(\omega_ i\) are arbitrary fixed moduli of continuity. He finds, for this quantity, an asymptotic equality when the coordinates \(n_ i\) of the vector n increase unboundedly in an arbitrary way. The obtained result yields a full solution of the Kolmogorov-Nikol'skij problem consisting of finding an explicit expression for the function \(\phi (n,\omega\),x) such that \(\epsilon_ n^{(N)}(\omega,x)=\phi(n,\omega,x)+o(\phi(n,\omega,x)).\)
trigonometric interpolation in several variables
scientific article; zbMATH DE number 3861757