Best approximations of continuous functions by spline functions

An investigation of the approximation on $[0, 1]$ of functions $f(x)$ by spline functions $s(f,\varphi;x)$ of degree $2r-1$ and of deficiency $r$ ($r>1$) depending on the vector function $\varphi=\{\varphi_1(x),\dots,\varphi_{r-1}(x)\}$ and interpolating $f(x)$ at fixed points. For the optimal choice of the vector $\varphi_0$, exact estimates are obtained of the norms $||f(x)-s(f,\varphi_0;x)||_{C[0,1]}$ and $||f(x)-s(f,\varphi_0;x)||_{L[0,1]}$ on the function classes $H_\omega$.