Convergence rate for trigonometric interpolation of non-smooth functions
For a \(2\pi\)-periodic bounded function \(f\) which is measurable on \([0,2\pi]\), the authors obtain estimates for the order of approximation of \(f\) in \(L_ p\) by trigonometric interpolation polynomials. Let \(n,q\in \mathbb{N}\) and \(0< m_ 1<\cdots< m_ q\) be distinct integers. For given \(n\), \(q\), and \(f\) we consider the interpolation polynomial \(F_ n f\) of order at most \(M= M(n,q)\), which is defined by \[ F_ n(f;\theta_{kn})= f(\theta_{kn}),\;k=0,\dots,n- 1,\quad F^{(j)}_ n(f;\theta_{kn})= 0,\;j= m_ 1,\dots,m_ q,\;k= 0,\dots,n- 1, \] where \(\theta_{kn}= 2k\pi/n\), \(k=0,\dots,n- 1\). One of the main results of the paper is the following inequality \[ \| F_ n f- f\|_{L_ p}\leq C_ p(\widetilde E_ n(f)_ p+ \omega_{m_ 1}(f; 1/n)_ p), \] where \(\widetilde E_ n(f)_ p\), \(1< p<\infty\), is the best one- sided approximation of \(f\) in \(L_ p\) by trigonometric polynomials of order at most \(n\) and \(\omega_{m_ 1}(f;\delta)_ p\) is the \(m_ 1\)th order \(L_ p\) modulus.
scientific article; zbMATH DE number 597640