An error estimate for the trigonometric interpolation of functions of \(m\)-harmonic bounded variation
Let \(V_{m,H}(f;0,2\pi)\) be the \(m\)-harmonic variation of a \(2\pi\)-periodic function \(f\): \[ V_{m,H}(f;0,2\pi)=\sup\sum_{n=1}^{\infty}\frac{|\Delta^mf(I_n)|}{n}, \] where the supremum is taken over all systems \(\{I_n\}\) of disjoint intervals \(I_n=(a_n,b_n)\subset[0,2\pi]\) and \(\Delta^mf(I_n)=\sum_{\nu=0}^m(-1)^{m-\nu}\binom{m}{\nu}f(a_n+\nu h_n),\) \(h_n=(b_n-a_n)/m.\) The author studies the error estimate for the trigonometric interpolations \(L_{n,\xi}(f,x)\) of functions \[ f\in \text{CHBV}_m=\{f\in C_{2\pi}: V_{m,H}(f;0,2\pi)<\infty\} \] at equidistant nodes \(x_{kn}=\xi+2k\pi/(2n+1).\) As a corollary the uniform convergence on \(\mathbb R\) of \(L_{n,\xi}(f,x)\) to the function \(f\in \text{CHBV}_{\infty}=\cup_{m=1}^{\infty}\text{CHBV}_m\) is proved.
scientific article; zbMATH DE number 1752614