Spline polynomials with a prescribed sequence of extrema

In the present note a theorem about strong suitability of the space of algebraic polynomials of degree $\leqslant n$ in $C_{[a,b]}$ (Theorem A in [1]) is generalized to the space of spline polynomials $\mathcal{S}^{n,k}_{[a,b]}$ ($n\geqslant2$, $k\geqslant0$) in $C_{[a,b]}$. Namely, it is shown that the following theorem is valid: for arbitrary numbers $\eta_0,\eta_1,\dots,\eta_{n+k}$, satisfying the conditions $(\eta_i-\eta_{i-1})(\eta_{i+1}-\eta_i)<0$ ($i=1,\dots,n+k-1$), there is a unique polynomial $s_{n,k}(t)\in \mathcal{S}^{n,k}_{[a,b]}$ and points $a=\xi_0<\xi_1<\dots<\xi_{n+k-1}<\xi_{n+k}=b$ ($\xi_1<z_1<\xi_n,\dots\xi_k<z_k<\xi_{n+k-1}$), such that $s_{n,k}(\xi_i)=\eta_i$ ($i=0,\dots,n+k$), $s'_{n,k}(\xi_i)=0$ ($i=1,\dots,n+k-1$).