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Does there exist a polynomial $P$ such that $|P(x) − \cos x| \le10^{-6}$ for all (real) x?
I am struggling to understand how to even start with this question, as it is not clear to me what this really means. Obviously $-1\le \cos x\le 1$, so maybe there is a way to use that, or maybe use the small angle approximations?
I am really not too sure, any help would be much appreciated.
The polynomial has to be positive at even multiples of $\pi$ where the cosine is $+1$, negative at odd multiples of $\pi$ where the cosine is $-1$. Therefore infinitely many zeroes. And we're having a bad day.
Yes, the fact that $\cos x$ is bounded implies that it can't be approximated by any polynomial $P(x)$. Consider the largest power of $x$ involved in the polynomial. For $x$ sufficiently large, this term will be much bigger than all others combined, so if this term has a positive coefficient $P(x)$ will be large and positive for $x$ sufficiently large, and if it has a negative coefficient then $P(x)$ will be large and negative.
On any finite interval that would be possible but every non-constant polynomial $P(x)$ diverges to $\pm{\infty}$ as x goes to $\pm{\infty}$ making it impossible.
