For a partition
ξ
=
(
0
=
ξ
0
>
ξ
1
>
⋯
>
ξ
n
>
ξ
n
+
1
=
1
)
\xi = (0 = {\xi _0} > {\xi _1} > \cdots > {\xi _n} > {\xi _{n + 1}} = 1)
of the unit interval,
S
ξ
k
m
S_\xi ^{km}
,
k
>
m
k > m
, denotes the space of piecewise-polynomials of order k and of smoothness
m
−
1
m - 1
; this space can be represented as the graph of the appropriate linear operator between two finite-dimensional Hilbert spaces. It gives an approach to the C. de Boor problem, 1972, on uniform boundedness (with respect to
ξ
\xi
) in the
L
∞
{L_\infty }
-norm of the orthogonal projections onto
S
ξ
k
m
S_\xi ^{km}
, and we give the detailed analysis of a quadratic pencil (matrix-valued polynomial of the second degree) which appears in the case of a geometric mesh
ξ
\xi
if
2
m
⩽
k
2m \leqslant k
. The explicit calculations and estimates of zeros of the "characteristic" polynomial show that in the case
S
ξ
(
x
)
63
S_{\xi (x)}^{63}
,
ξ
(
x
)
\xi (x)
me geometric mesh with the parameter x,
0
>
x
>
∞
0 > x > \infty
, the orthogonal projectors are uniformly bounded.