The paper "Smoothing Noisy Data with Spline Functions" by Grace Wahba addresses the problem of selecting the optimal smoothing parameter for reconstructing a smooth periodic curve from noisy data using a smoothing periodic spline of degree \(2m-1\). The true curve is assumed to be in the Sobolev space \(W_2^{(4m)}\) of periodic functions with specific derivative properties. The goal is to minimize the expected squared error averaged over the data points. The paper derives explicit expressions for the optimum smoothing parameter \(\lambda\) and the optimum choice of the sample size parameter \(S\), which depends on the sample size \(n\), the noise variance \(\sigma^2\), and the smoothness of the true curve. The results are derived for periodic smoothing splines, imposing periodic boundary conditions on the solution. The key findings include the formula for \(\lambda^*\) and the upper bound on the expected error and the optimum choice of \(S\).The paper "Smoothing Noisy Data with Spline Functions" by Grace Wahba addresses the problem of selecting the optimal smoothing parameter for reconstructing a smooth periodic curve from noisy data using a smoothing periodic spline of degree \(2m-1\). The true curve is assumed to be in the Sobolev space \(W_2^{(4m)}\) of periodic functions with specific derivative properties. The goal is to minimize the expected squared error averaged over the data points. The paper derives explicit expressions for the optimum smoothing parameter \(\lambda\) and the optimum choice of the sample size parameter \(S\), which depends on the sample size \(n\), the noise variance \(\sigma^2\), and the smoothness of the true curve. The results are derived for periodic smoothing splines, imposing periodic boundary conditions on the solution. The key findings include the formula for \(\lambda^*\) and the upper bound on the expected error and the optimum choice of \(S\).