Best approximations of continuous functions by spline functions

An investigation of the approximation on [0, 1] of functionsf (x) by spline functions s(f,; x) of degree 2r-1 and of deficiency r (r>1) depending on the vector function =₁ (x),...,_r-1(x) and interpolatingf (x) at fixed points. For the optimal choice of the vector₀, exact estimates are obtained of the norms f(x)-s (f,₀; x)_C[0,1] and f (x)-s (f,₀; x)_{L[0, 1]} on the function classes HTranslated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 41–46, July, 1970.In conclusion we would like to thank N. P. Korneichuk for suggesting this problem and for his valuable advice.