Weierstrass Approximation Theorem/Lemma 2
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Theorem
Let $\map {p_{n, k} } t : \N^2 \times \closedint 0 1 \to \R$ be a real valued function defined as:
- $\map {p_{n, k} } t := \dbinom n k t^k \paren {1 - t}^{n - k}$
where:
- $n, k \in \N$
- $t \in \closedint 0 1$
- $\dbinom n k$ denotes a binomial coefficient.
Then:
- $\ds \sum_{k \mathop = 0}^n \paren {k - n t}^2 \map {p_{n, k} } t = n t \paren {1 - t}$
Proof
From the binomial theorem:
- $\ds 1 = \sum_{k \mathop = 0}^n \binom n k t^k \paren {1 - t}^{n - k}$
From Lemma 1:
\(\ds n t\) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \binom n k k t^k \paren {1 - t}^{n - k}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds n\) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \binom n k k \paren {k t^{k - 1} \paren {1 - t}^{n - k} - t^k \paren {n - k} \paren {1 - t}^{n - k - 1} }\) | Definition of Derivative of Real Function with respect to $t$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 t \sum_{k \mathop = 0}^n k^2 \map {p_{n, k} } t - \frac n {1 - t} \sum_{k \mathop = 0}^n k \map {p_{n, k} } t + \frac 1 {1 - t} \sum_{k \mathop = 0}^n k^2 \map {p_{n, k} } t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {t \paren {1 - t} } \sum_{k \mathop = 0}^n k^2 \map {p_{n, k} } t - \frac n {1 - t} n t\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 0}^n k^2 \map {p_{n, k} } t\) | \(=\) | \(\ds n t \paren {1 - t} + n^2 t^2\) |
Then:
\(\ds \sum_{k \mathop = 0}^n \paren {k - n t}^2 \map {p_{n, k} } t\) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \paren {k^2 - 2 n t k + n^2 t^2} \map {p_{n, k} } t\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n t \paren {1 - t} + n^2 t^2 - 2 n t n t + n^2 t^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n t \paren {1 - t}\) |
$\blacksquare$
Sources
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis: Chapter $1$: Normed and Banach spaces