The uniqueness of the element of best mean approximation to a continuous function using splines with fixed nodes

P. V. Galkin

Matematičeskie zametki, Tome 15 (1974) no. 1, pp. 3-14.

Résumé

Suppose that on the interval $[a,b]$ the nodes $$a=x_o\dots{m+1}=b$$ are given and the functions $u_0(t)=\omega_0(t)$, $$u_i(t)=\omega_0(t)=\int_0^t\omega_1(\xi_1)\,d\xi_1\dots\int_a^{\xi_{i-1}}\omega_i(\xi_i)\,d\xi_i,\quad\xi_0=t\quad(i=1,2,\dots,n),$$ where the functions $\omega_i(t)>0$ have continuous $(n-1)$-th derivatives ($i=1,2,\dots,n$). $S_{n,m}$ will designate the subspace of functions that have continuous $(n-1)$-st derivatives on $[a,b]$ and coincide on each of the intervals $[x_j,x_{j+1}]$ ($j=0,1,\dots,m$) with some polynomial from the system $\{u_i(t)\}_{i=0}^n$. THEOREM. {\it For every continuous function on $[a,b]$ there exists in $S_{n,m}$ a unique element of best mean approximation.}