On the arclength of trigonometric interpolants

Abstract

As pointed out recently by Strichartz [5], the arclength of the graph \(\Gamma(S_N(f))\) of the partial sums \(S_N(f)\) of the Fourier series of a jump function \(f\) grows with the order of \(\log N\). In this paper we discuss the behaviour of the arclengths of the graphs of trigonometric interpolants to a jump function. Here the boundedness of the arclengths depends essentially on the fact whether the jump discontinuity is at an interpolation point or not. In addition convergence results for the arclengths of interpolants to smoother functions are presented.