Let be an integer and let be a continuous function. Suppose that for all . Show that

**Solution**

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

integrating gives:

and the result follows.

The exercise along with its solution was taken from here.