A given real continuous function f on [a, b] is approximated by polynomials
P
n
{P_n}
of degree n that are subject to certain restrictions. Let
1
≦
k
1
>
⋯
>
k
p
≦
n
1 \leqq {k_1} > \cdots > {k_p} \leqq n
be given integers,
ε
i
=
±
1
{\varepsilon _i} = \pm 1
, given signs. It is assumed that
P
n
(
k
i
)
(
x
)
P_n^{({k_i})}(x)
has the sign of
ε
i
,
i
=
1
,
…
,
p
,
a
≦
x
≦
b
{\varepsilon _i},i = 1, \ldots ,p,a \leqq x \leqq b
. Theorems are obtained which describe the polynomials of best approximation, and (for
p
=
1
p = 1
) establish their uniqueness. Relations to Birkhoff interpolation problems are of importance. Another tool are the sets A, where
|
f
(
x
)
−
P
n
(
x
)
|
|f(x) - {P_n}(x)|
attains its maximum, and the sets
B
i
{B_i}
with
P
n
(
k
i
)
(
x
)
=
0
P_n^{({k_i})}(x) = 0
. Conditions are discussed which these sets must satisfy for a polynomial
P
n
{P_n}
of best approximation for f. Numbers of the points of sets A,
B
i
{B_i}
are studied, the possibility of certain extreme situations established. For example, if
p
=
1
,
k
1
=
1
,
n
=
2
q
+
1
p = 1,{k_1} = 1,n = 2q + 1
, it is possible that
|
A
|
=
3
,
|
B
|
=
n
|A| = 3,|B| = n
.