# Functions and their Taylor polynomials

Given a function $f$ from $\Bbb{R}$ to $\Bbb{R}$, we define $P_{f,n,a}$ to be the Taylor polynomial of $f$ of degree $n$ at $a$ (if the function itself is clear from the context, we simply write the more concise $P_{n,a}$). Consider we are given some function $f$ and all its Taylor polynomial at some point $a$ (i.e., the sequence $\{f^{(n)}(a)\}_{n\in \Bbb{N}}$ of derivatives of $f$ at $a$). Then, given a point $b$ different from $a$, we can't say anything about the value that $f$ takes on $b$.

But there are "special" functions for which the sequence $\{f^{(n)}(a)\}_{n\in \Bbb{N}}$ is indeed sufficient for entirely describing the function itself. Such functions satisfy the following property: for each point $p$, the sequence $\{P_{n,a}(p)\}_{n\in\Bbb{N}}$ converges to $f(p)$. Such a function is, for instance, the $\sin$ function; or, another example, every polynomial function. Is there some naming for those functions?

Consider the $\sin$ function and call $S$ the sequence $\{\sin^{(n)}(a)\}_{n\in \Bbb{N}}$. Does the preceding property mean that, among all functions $f$ such that $\{f^{(n)}(a)\}_{n\in \Bbb{N}}=S$, the $\sin$ function is the best fitting (or, in other words, is $\sin$ the natural extension for a function whose sequence at $a$ is $S$)? (Indeed this is not really a question xD$~$)

• You are looking for "analytical functions" or, in the case of sin and exp, "entire (holomorphic) functions". – LutzL Feb 9 '14 at 20:56
• @LutzL, since OP specifies functions on the reals, I think the proper term is "real-analytic" functions. – Gerry Myerson Feb 10 '14 at 8:27

## 1 Answer

Functions that are defined on all of $\Bbb C$ by a (single) power series are called entire functions. The power series (taken around $x=0$, say) must have infinite radius of convergence for this to happen: the absolute value of its coefficients must tend to zero faster than any exponential function of the index$~n$. It is immediate that the coefficients of the series are those of its Taylor series, so in particular they can be read off from the restriction of the function to$~\Bbb R$ (which restriction will be real valued if all coefficients of the series are real), which can therefore be uniqely enxtended to an entire function. I don't think there is a special name for these restrictions to the reals of entire functions; in any case te notion of real analytic function is substantially weaker than this.

Being an entire function with a given Taylor series does not mean in any way being a best fit (to the deriviatives in the point$~a$ where the Taylor series is taken) among all the (smooth) functions that share that Taylor series: every standard estimate by the Taylor series applies to all those functions equally well. It just means being the only entire function in the lot. And again, having rapidly decreasing coefficients is a necessary condition for there to be any entire function at all with a given Taylor series. (But in other cases there still might be a unique real analytic function in the lot.)