Does there exists a non-zero continuous function $f$ such that $\displaystyle \int_0^1 x^k (1-x)^{n-k} f(x)dx=0$ for every $k=0,1,2,…,n$ where $n$ is a non-negative integer.
EDIT: The problem is related to Weierstrass approximation using Bernstein approximation of certain function. As you can see the term $x^k (1-x)^{n-k}$ comes from Bernstein polynomials. I want my integral to vanish for non-zero $f$.
Also, the result should hold for every $n$ and for every $k$ running from $0$ to $n$. For example, for $n=2$, we shall have three integrals corresponding to $k=0,1,2$. In this way, it should hold for any $n$.
Best Answer
Here’s the sketch: Since we require this for every $k$ and $n$, it amounts to saying that the integral of f against any $x^k (1-x)^j$ is zero. These are closed under multiplication, so their span is an algebra. By Stone-Weierstrass, find linear combinations $l_n$ of them approaching $f$ uniformly. By the assumption in the problem, $\int l_n f$ is 0 for all $n$, so taking a limit we conclude $f^2$ integrates to 0. It however is positive, so it is identically 0. Therefore $f$ is identically 0.