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Problem: Let $\varphi_{1}, \varphi_{2}, \cdots, \varphi_{n},\cdots$ be non-negative continuous functions on $[0,1]$ such that the limit $$\lim_{n \to \infty} \int\limits_{0}^{1} x^{k} \varphi_{n}(x) \ \text{dx}$$ exists for every $k \in \mathbb{Z}_{+}$. Does this imply the limit $$\lim_{n \to \infty} \int\limits_{0}^{1} f(x) \varphi_{n}(x) \ \text{dx}$$ exists for every continuous function $f(x)$ on $[0,1]$.

No idea on how to proceed. Any help would be useful.

Added. As everybody suggested the idea is to use the Stone-Weierstrass theorem in the space $\mathcal{C}[0,1]$ equipped with the $\text{Sup-norm}$. So let $\{p_{n}\}$ be a sequence of polynomials which converge to $f$ in the $\text{sup-norm}$. With our hypothesis, we can conclude that there is some positive $C$ such that $$\Biggl|\int\limits_{0}^{1} \varphi_{n}(x) \ \text{dx}\Biggr| \leq C$$ for all $n$. Since $p_{n} \to f$, we have for given $\epsilon >0$, there exists an integer $N$ such that $||f-p_{N}||< \epsilon$ for all $k \geq N$. I could only reach till here by applying the definitions.

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Stone-Weierstrass suggests itself... –  Arturo Magidin May 4 '11 at 4:34
Maybe you can try approximating $f(x)$ by polynomials using the Stone-Weierstrass Theorem? –  Adrián Barquero May 4 '11 at 4:36
Enlighten by this Stone-Weierstrass Theorem based problem, I wonder if the following is true: if $\displaystyle \lim_{n\to\infty} \int^1_0 \sin(kx)\phi_n(x) \,dx = 0$ for any $k$, does this implies $\displaystyle \lim_{n \to \infty} \int^1_0 f(x) \phi_{n}(x) = 0$ for any $f\in C([0,1])$? –  Shuhao Cao May 4 '11 at 5:46
Yes, i tried using Ston-Weierstrass and ended up somewhere. –  user9413 May 4 '11 at 6:01
You shouldn't just apply the definitions blindly but apply what the definitions imply! –  Jonas Teuwen May 4 '11 at 12:20
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1 Answer

up vote 5 down vote accepted

We can quickly see that for all polynomials $p$ there holds that

$$\lim_{n \to \infty} \int_0^1 p(x) \phi_n(x) \, dx \text{ exists}$$

So now, approximate your $C[0,1]$ function $f$ uniformly by polynomials $p_k$.

We would now like to show that

$$\lim_{n \to \infty} \int_0^1 \lim_{k \to \infty} p_k(x) \phi_n(x) \, dx \text{ exists}.$$

By uniform continuity we know that $\int f_n \to \int f$ if $f_n \to f$ uniformly. Now note that $p_k \phi_n \to f \phi_n$ uniformly as well as $k \to \infty$.

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