A whole function approximating polynomials

[LWRS]

Given a series of polynomials \(\displaystyle p_{n}\) and a series of open, non-intersecting sets \(\displaystyle V_{n} \subset \mathbb{C}\) show that there exists a function \(\displaystyle g\in \mathcal{O}(\mathbb{C})\) such that \(\displaystyle lim_{n \rightarrow \infty} sup_{z \in V_{n}} |g(z)-p_{n}(z)|=0\).

Normally the approximation goes the other way around so I'm not sure what to do here.

 

[Jhoneine]

Let g(z) denote the function we want to construct. We can use a technique known as the Weierstrass Approximation Theorem to construct g(z). This theorem states that if V_{n} is compact and p_{n} is continuous, then there exists a sequence of polynomials f_{n} such that lim_{n \rightarrow \infty} sup_{z \in V_{n}} |f_{n}(z)-p_{n}(z)|=0.Now, we can define g(z) as the uniform limit of the sequence of polynomials {f_{n}(z)}, i.e., g(z)=lim_{n \rightarrow \infty} f_{n}(z). Since f_{n}(z) converges uniformly to p_{n}(z), it follows that lim_{n \rightarrow \infty} sup_{z \in V_{n}} |g(z)-p_{n}(z)|=0. Hence, we have constructed a function g(z) which satisfies the desired condition.

 

[math771]

Hi there,

It seems like you are trying to prove the Weierstrass Approximation Theorem, which states that for any continuous function f on a closed interval [a,b], there exists a sequence of polynomials p_n that converges uniformly to f on [a,b]. In this case, we are given a sequence of polynomials p_n and a sequence of open sets V_n, and we need to show that there exists a function g\in \mathcal{O}(\mathbb{C}) (the space of holomorphic functions) that approximates the polynomials p_n on the sets V_n.

To prove this, we can use the fact that any continuous function on a compact set can be uniformly approximated by polynomials. Since the sets V_n are open and non-intersecting, they can be covered by a finite number of compact sets. Let's call these compact sets K_n, where K_n \subset V_n for all n. Now, for each n, we can find a polynomial p_n that approximates p_n uniformly on K_n. This means that for all z \in K_n, we have |p_n(z)-p_n(z)|< \epsilon_n, where \epsilon_n is some small positive number.

Next, we can define a function g as follows: g(z)=p_n(z) for z \in K_n and g(z)=0 for z \notin \bigcup_{n}K_n. In other words, g is equal to the polynomial p_n on the compact set K_n and is equal to 0 outside of this set. Since the sets V_n are non-intersecting, this function is well-defined and holomorphic on \mathbb{C}. Also, by construction, we have |g(z)-p_n(z)|< \epsilon_n for all z \in K_n.

Now, since the sets K_n cover the sets V_n, we have that for all z \in V_n, there exists some n_0 such that z \in K_{n_0}. Therefore, we have |g(z)-p_{n_0}(z)|< \epsilon_{n_0} for all z \in V_{n_0}. Taking the supremum over all z \in V_{n_0}, we get lim_{n \rightarrow \infty} sup_{z \in V_{n_0}} |g(z)-