# Questions tagged [approximation-theory]

Approximation theory is concerned with how functions can best be approximated with simpler functions, and with quantitatively characterizing the errors introduced thereby.

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### How to prove the best polynomial approximation operator is continuous?

For any function $f\in C[0,1]$, it is well known that there exist an unique polynomial $p^{*}\in P_n[0,1]$ such that $||p^{*}-f||_{\infty}\leq ||p-f||_{\infty}$ for any $p\in P_n[0,1]$. In this ...
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### Closed form of this product or approximate?

What is the closed form of this product: $$\prod_{i=1}^{k-1}\left(1-e^{-a(b- ic)^2}\right)$$ where $a,b,c$ are constants?
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### tan nx is not as popular as Chebyshev polyomials?

I noticed that $\tan(nx)$ can be written as the ratio $P(\tan x) / Q(\tan x)$, where $P$ and $Q$ are polynomials. Indeed, I found a short proof here Expansion of $\tan(nx)$ in powers of $\tan(x)$ ...
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### Approximating Log(Gamma(z)) for small z as Log(Gamma(z + 1)) - Log(z)

I'd like to implement a numerical approximation to the log Gamma function, and I found Gergő Nemes' approximation described here: https://en.wikipedia.org/wiki/Stirling%27s_approximation. This seems ...
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### How far apart can L1 and L2 lines fit to the same data be?

Given $n$ points $(x_i, y_i)$ in the unit square with $x_i = {{i - 1} \over {n - 1}}$ uniformly spaced and $0 \leq y_i \leq 1$, consider the best-fit L1 line and the best-fit L2 line: $\qquad$ L1 ...
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### Stirling's Approximation Proof

I was searching for the reason why Stirling's Approximation holds true. I found the website Stirling's Approximation which apparently shows why this is the case. Does this part of the equation make ...
I have difficulty proving the following claim from a paper (a free version is here, see Lemma 2.4 on page 9): Let in a Banach space $X$ a sequence $\{x_n\}_{n=1}^\infty$ be given. Assume that for ...