# Smooth lower envelope of a function

I want to determine if this problem has a unique solution. Given a continuous function $f(x)$ in a bounded interval or the real line, say [-1/2,1/2],  the problem is to find a function $L(x)$ that fulfills:

1) Band-limited in the sense of Fourier series, that is: $$L(x)=\sum_{|\nu|<\nu_L}L_\nu*\exp(2 \pi i x \nu)$$ 2) $L(x)\le f(x)$

3) minimizes $||L-f||_2$, that is, the 2-Holder distance.

Solving 2) and 3) is just $f(x)$. Solving 1) and 3) is the truncated Fourier series of $f(x)$. Is the set defined by 1) and 2) closed in the 2-norm? I think this would ensure existence of a minimizer. Is it "convex" in some sense?..something like this would guarantee uniqueness.

Is this a known problem? How can I solve it, that is , anybody knows an algorithm or a reference to construct a solution?

• Is $\nu_L$ fixed or can $L$ have an arbitrarily large (but finite) number of non zero coefficients? – Julián Aguirre Nov 8 '13 at 16:36
• $\nu_L$ is fixed. – Enredanrestos Nov 8 '13 at 16:46
• However, the approach from $L^p$ spaces does not seem to be adequate, since the condition 2) (a.e) defines a closed set with no interior points. – Enredanrestos Nov 10 '13 at 16:14
• I think the method of projection onto convex sets (POCS) might help. – Enredanrestos Nov 18 '13 at 16:53
• Not POCS, but Dijkstra's algorithm on projection onto convex sets. – Enredanrestos Jun 15 '15 at 0:42

## 1 Answer

From the inequality $L(x)\le f(x)$ I gather that $f$ and $L$ are real valued. For simplicity of notation let me cal $\nu_L=N$. Let $$\mathcal{T}_N=\{a_0+\sum_{k=1}^Na_k\cos(2\,k\,\pi\,x)+b_k\sin(2\,k\,\pi\,x):a_k,b_k\in\mathbb{R}\}$$ be the set of trigonometric polynomials of degree $N$. It is a vector space ver $\mathbb{R}$ of dimension $2\,N+1$. If $f\in\mathcal{T}_N$ then the problem is trivial. Assume that $f\not\in\mathcal{T}_N$ and let $V$ be the subspace of $L^2[-1,1]$ generated by $\mathcal{T}_N$ and $f$; $V$ has dimension $2\,N+2$, $\mathcal{T}_N$ is a subspace of $V$ and $f\in V$.

Let $$\mathcal{T}_N(f)=\{T\in\mathcal{T}_N:T(x)\le f(x)\quad \forall x\in[-1,1]\}.$$ Then $\mathcal{T}_N(f)$ is a non empty convex closed subset of $V$. Thus there is a unique $T\in\mathcal{T}_N(f)$ minimizing $\|T-f\|_2$.

As for an algorithm to compute, I am afraid I cannot help you.