# Approximation of Hölder continuous functions "from below"

We assume that we have a $$\alpha$$-Hölder continuous function $$f$$ on an interval $$[0,1]$$ with $$f(0)=0$$.

I am wondering if there exists an explicit construction of a sequence $$f_{n} \in C_c^{\infty}(\mathbb R)$$ such that

$$\lVert f-f_n\rVert_{C^{\alpha}([0,1])} \le \frac{1}{n}$$

and $$\lvert f_n(x)\rvert \le \lvert f(x)\rvert$$ on $$[0,1]$$. The usual convolution idea does not respect the last condition.

• Please revert this edit, accept Alexandre's answer to your original question, and post your modified version as a new question. Commented Apr 11, 2023 at 11:46

It is not possible: your condition $$|f_n|<|f|$$ implies that the zero set of $$f_n$$ is contained in the zero set of $$f$$. So $$f(x)=|x|^{\alpha}$$ cannot be approximated by a smooth function $$f_n$$, since $$f_n(x)=(c+o(1))x$$, and $$\sup|f(x)-f_n(x)|\geq (1+o(x))|x|^\alpha$$ for some small $$|x|$$.
• I think that actually the zero set of $f_n$ must contain, not necessarily be contained in, the zero set of $f$ (and it is a non-strict inequality). Commented Apr 10, 2023 at 13:05
Alexandre has answered correctly, but I wanted to add that the norm closure of $$C^\infty[0,1]$$ in the $$\alpha$$-Hölder space is the "little" $$\alpha$$-Hölder space (a.k.a. little Lipschitz space) consisting of those $$\alpha$$-Hölder functions that are locally flat in the sense that $$\lim_{y \to x}\frac{|f(x) - f(y)|}{|x - y|^\alpha} = 0$$ for all $$x \in [0,1]$$. It's easy enough to see that every $$C^\infty$$ function satisfies this condition and that the set of locally flat $$\alpha$$-Holder functions is norm closed. (Alexandre's function $$|x|^\alpha$$ fails to be locally flat at $$x = 0$$.) To get density we need to use "uniform separation of points"; see Corollary 8.30 of my book Lipschitz Algebras (second edition).
• thank you, does this actually mean the approximation I asked for is possible in every $C^{\beta}$ norm with $\beta< \alpha$? Commented Apr 10, 2023 at 21:46
• Yes, if $\alpha < \beta$ then every $\alpha$-Holder function is not only $\beta$-Holder, but even $\beta$-locally flat. So you are right, anything in $C^\alpha$ can be approximated in $\beta$-Holder norm by $C^\infty$ functions. (That's not to say approximable from below, though.) Commented Apr 10, 2023 at 22:26