On orthogonal decomposition of space in spline-fitting problem

A special orthogonal decomposition of a basic space for an abstract quasi-spline-fitting problem is proposed. Using this decomposition, a theorem on representation of smoothing quasi-spline $\sigma_{\alpha}$ is proved. Exact in order convergence estimates of $\sigma_{\alpha}$ to the limit quasi-splines $\sigma_0$ and $\sigma_{\infty}$ are obtained. The monotony and the upper convexity of the function $\psi^{-1}(\beta)$, used in the algorithm of selection of the smoothing parameter $\alpha$ by the residual criterion, are proved.