Bump functions of maximal height under smoothness constraints

Suppose I want to find a function that is of maximal height at $0$ and of zero height outside of the unit ball. In other words, a bump function of maximal height.

The constraint I put on this function are bounds on derivatives. For example:

1. "Bounded first derivatives" (but formally $L_0$-Lipschitz continuity).
2. "Bounded second derviatives" (but formally differentiable everywhere with $L_1$-Lipschitz continuous gradients)

etc.

Bounded first derivatives

The zero order constraint is easy, so let us work through it as an example. Without loss of generality (by rescaling) consider $L_0=1$. Then, as the slope can not exceed one, the optimal bump function $p:\mathbb{R}\to\mathbb{R}$ is given by

This is because the derivative is given by$$p'(t) = \begin{cases} 1 & t\in (-1, 0)\\ -1 & t\in (0,1)\\ 0 & |t| > 1\end{cases}$$And we can not move up faster than the maximum allowed slope. This function can now easily be lifted to higher dimensions by$$f(x) = p(\|x\|),$$which is $1$-Lipschitz because$$|f(x) - f(y)| = |p(\|x\|) - p(\|y\|)| \le | \|x\| - \|y\| | \le \|x-y\|$$by the reverse triangle inequality.

Bounded second derivatives

Again, we are going to assume $L_1=1$ without loss of generality.The one dimensional solution $p:\mathbb{R}\to\mathbb{R}$ is given by

$$p(t) = \begin{cases}-\tfrac12 t^2 + \tfrac14 & t\in [-\tfrac12, \tfrac12]\\\tfrac12(|t| -1)^2 & |t| \in (\tfrac12, 1)\\0 & |t| > 1\end{cases}\qquadp''(t) = \begin{cases}-1 & t\in [-\tfrac12, \tfrac12]\\1 & |t| \in (\tfrac12, 1)\\0 & |t| > 1\end{cases}$$

That is, until the halfway point we accelarate maximally quickly, then we decelarate maximally quickly. It now feels intuitive, that the function

$$f(x) = p(\|x\|)$$should be able to lift this bump function to higher dimensions. But now the proof of $1$-Lipschitz continuous gradients is much more difficult. Does $f$ have $1$-Lipschitz continuous gradients or is there some fundamental problem with this approach?

Assuming this approach works...

It is fairly intuitive how to extend this approach to $n$-th order derivative bounds: "accelerate" maximally until the halfway point and then deccelerate maximally.$$p^{(n)}(t) = \begin{cases}-1 & t\in [-\tfrac12, \tfrac12]\\1 & |t| \in (\tfrac12, 1)\\0 & |t| > 1\end{cases}$$

If the requirements are to be combined, accelerate maximally until the bound of the lower order derivative is reached then stop accelerating and coast (like in the zero order case).

Related questions:

• Control over bump function's second derivative
• Construction of bump function with arbitrary choice of bound on second order derivatives