Uniform approximation of curvature for smooth classes of plane curves (Q1675331)

Given a function \(y=y(x)\), let \(S_{r-1}(y,x,\pi/n)\) denote the interpolation periodic spline of degree \(r-1\) and defect 1 with interpolation nodes \(k\pi/n\) \, \((k\in\mathbb{Z})\) and knots located at the same points if \(r-1\) is odd and at the points \(\pi(2k + 1)/2n\) \, \((k\in\mathbb{Z})\) if \(r-1\) is even. Further, let us denote \[ K(y,x) = \frac{y''(x)}{[1 + (y'(x))^2]^{3/2}}, \qquad \widetilde K(s,x) = \frac{S_{r-2}(y'',x,\pi/n)}{[1 + S^2_{r-1}(y',x,\pi/n)]^{3/2}}. \] It is proved that \[ |K(y,x) - \widetilde K(s,x)| \leq \frac{{\mathcal K}_{r-2}}{n^{r-2}} + \frac{{\mathcal K}_{r-2}{\mathcal K}_{r-1}}{n^{r-1}}, \] where \({\mathcal K}_{r}\) are the Favard constants.