Analysis Problem of the Day 2

Today’s problem is a nice result about smooth non-analytic functions known as Borel’s Theorem.

Problem: Let $a_n \in \mathbb{R}$ be any sequence of real numbers. Show that there exists a smooth function $f \in C^\infty(\mathbb{R})$ such that $f^{(n)}(0)=a_n.$

Solution: The naive approach to this problem (which ignores any convergence issues) would be to set $f(x):= \sum_{n=0}^\infty a_n x^n.$ However, as we are aware from the theory of analytic functions, this power series has a positive radius of convergence only if $\limsup_{n \to \infty} |a_n|^{\frac{1}{n}} < \infty.$ The key point in this problem is that we are working over the category of smooth functions, not real analytic ones. We will modify our naive approach by doing a classic trick – introducing a smooth test function $\chi \in C^\infty(\mathbb{R})$ that equals 1 in a small neighborhood of 0 and vanishes for, say, $|x|>1.$ Then, if we define

$f(x):= \sum_{n=0}^\infty a_n x^n \chi(nx),$

one can easily check the following:

$f(x)$ is clearly defined at $x=0,$ and for $x \not =0,$ the sum is finite since $\chi(nx)=0$ for large enough $n.$ Thus, $f(x)$ is defined pointwise everywhere.
In fact, on any compact interval $K$ avoiding the origin, the sum is finite and therefore (since $\chi$ is smooth) defines a smooth function $f.$ This means that $f^{(k)} \in C^\infty(\mathbb{R} \setminus \{0\})$ for all $k \geq 0,$ and in fact, one may differentiate the power series for $f$ term-by-term.
One may check that
$\lim_{x \to 0} f^{(k)}(x) = \lim_{x \to 0} \sum_{n=0}^\infty a_n \partial_x^k(x^n \chi(nx)) = \sum_{n=0}^\infty a_n \partial_x^k(x^n \chi(nx))|_{x=0} = a_n,$
where we switch the order of the limit and infinite sum by an application of the dominated convergence theorem.
Thus, if one defines $f^{(k)}(0)=a_n,$ this extends $f$ to a smooth function satisfying the required properties.

Remark: Notice that $f$ in general cannot be analytic at 0, since if $\limsup_{n \to \infty} |a_n|^{\frac{1}{n}} = \infty$ the Taylor series would have a zero radius of convergence.

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