A theorem which in some cases enables one
to give effectively the polynomial and the error
of best integral approximation of a function
.
It was established by
A.A. Markov
in
1898
(see
[1]).
Let
,
,
be a system of linearly independent functions continuous on the interval
,
and let the continuous function
change sign at the points
in
and be such that
If the polynomial
has the property that the difference
changes sign at the points
,
and only at those points, then
is the polynomial of best integral approximation to
and
For the system
on
,
can be taken to be
;
for the system
,
,
can be taken to be
;
and for the system
,
,
one can take
.