Let be an integer and let be a continuous function. Suppose that for all . Show that
We begin by a lemma:
Lemma: For an integer the Hilbert matrix is defined by where
It is known that is invertible and if then .
Since , the Hilbert matrix , is invertible there exist real numbers such that
So the polynomial satisfies the conditions
Clearly is the sum of all the entries of and so . Now let be a real-valued continuous function on such that
Let be the above polynomial.Then since
and the result follows.
The exercise along with its solution was taken from here.