For every natural number $n$, we denote by $h_{n}$ the function defined on $[-1,1]$ by $$h_{n}(t) = \frac{\left(1 - t^{2}\right)^{n}}{\lambda_{n}}$$ and zero outside $[-1,1]$, where the real number $\lambda_{n}$ is given by the formula $$\lambda_{n} = \int_{-1}^{1} \left(1 - t^{2}\right)^{n} \mathrm{~d}t$$ a) Show that the sequence of functions $\left(h_{n}\right)_{n \in \mathbb{N}}$ is an approximate identity. b) Show that if $f$ is a continuous function with support included in $\left[-\frac{1}{2}, \frac{1}{2}\right]$, then $f * h_{n}$ is a polynomial function on $\left[-\frac{1}{2}, \frac{1}{2}\right]$ and zero outside the interval $\left[-\frac{3}{2}, \frac{3}{2}\right]$. c) Deduce a proof of Weierstrass's theorem: every complex continuous function on a closed interval of $\mathbb{R}$ is the uniform limit on this interval of a sequence of polynomial functions.
For every natural number $n$, we denote by $h_{n}$ the function defined on $[-1,1]$ by
$$h_{n}(t) = \frac{\left(1 - t^{2}\right)^{n}}{\lambda_{n}}$$
and zero outside $[-1,1]$, where the real number $\lambda_{n}$ is given by the formula
$$\lambda_{n} = \int_{-1}^{1} \left(1 - t^{2}\right)^{n} \mathrm{~d}t$$
a) Show that the sequence of functions $\left(h_{n}\right)_{n \in \mathbb{N}}$ is an approximate identity.\\
b) Show that if $f$ is a continuous function with support included in $\left[-\frac{1}{2}, \frac{1}{2}\right]$, then $f * h_{n}$ is a polynomial function on $\left[-\frac{1}{2}, \frac{1}{2}\right]$ and zero outside the interval $\left[-\frac{3}{2}, \frac{3}{2}\right]$.\\
c) Deduce a proof of Weierstrass's theorem: every complex continuous function on a closed interval of $\mathbb{R}$ is the uniform limit on this interval of a sequence of polynomial functions.