Extremality of monosplines of minimal deficiency

Let $M_{wN}^r(A,B)$ be the set of monosplines
$$ M(x)=\int_0^1w(t)(x-t)_+^{r-1}\,dt-\sum_{i=1}^n\sum_{j\in\Gamma_i}a_{ij}(x-x_i)_+^{r-1-j}-\sum_{k=0}^{r-1}b_kx^k $$
that satisfy
$$ M^{(i)}(0)=0\quad(i\in A),\qquad M^{(j)}(1)= 0\quad(j\in B),\qquad\sum_{i=1}^n|\Gamma_i|\leqslant N, $$
where $A,B$ and $\Gamma_i$ are subsets of $Z_r=\{0,1,\dots,r-1\}$, $|\Gamma_i|$ is the number of elements in $\Gamma_i$, $M_{wN}^{r0}(A,B)$ is the subset of elements of $M_{wN}^r(A,B)$ for which $n=N$, $\Gamma_i=\{0\}$ ($i=1,\dots,N$), and $\widetilde M_{wN}^r(A,B)$ and $\widetilde M_{wN}^{r0}(A,B)$ are the corresponding sets of periodic monosplines. It was shown that the monosplines that have the smallest $L_p$-norms in $M_{wN}^r(A, B)$ and $\widetilde M_{wN}^r(A,B)$ belong to $M_{wN}^{r0}(A,B)$ and $\widetilde M_{wN}^{r0}(A,B)$, respectively. Some theorems are also obtained on snakes for monosplines.
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