# Projection of a function $f\in L^1(\Omega)$ onto a finite dimensional subspace

Suppose $$\Omega \subset \Bbb R^n$$ be a domain such that $$|\Omega|<\infty$$, $$f\in L^1(\Omega)$$. Let $$Y= \text{span}\{g_1,\dots, g_k\}$$.

Is there a characterization of the set of projections of $$f$$ onto $$Y$$, i.e. the set $$M$$ of all $$g\in Y$$ such that $$\int_\Omega |f(x) - g(x)|dx = \inf_{h\in Y} \int_\Omega |f(x) - h(x)|dx,$$ or, equivalently, $$M = \text{argmin} \{ \int_\Omega |f(x) - g(x)|dx : g \in Y\}$$?

Similar problem in $$L^2$$ (minimizing $$\int_\Omega |f(x) - g(x)|^2 dx$$) is very easy to solve since we have inner product there. However, as $$L^1$$ norm is not strictly convex, our set $$M$$ need to contain a unique element in general.

The solution to the $$1$$-dimensional case $$Y = \text{span}\{1\}$$ is well known, i.e. $$g=c$$ where $$c\in \Bbb R$$ is any median of $$f$$. Is there a complete treatment of this kind of problem anywhere?

• You have the same property: for every $g'\in Y$, the sets where $(f-g)g'> 0$ and $(f-g)g'<0$ have measure at most $\frac12|\Omega|$ but I'm not sure if that can be called a "characterization": it is not easy to verify except in the most trivial cases. What are you really after? – fedja May 14 at 14:49
• @fedja Knowing general properties of $M$ would be a starter, e.g. if it is closed, convex, $G_\delta$ etc. I do agree that the property you mentioned seems hard to verify. Regardless, I still find it quite interesting. Do you perhaps have a reference to that or something similar that I could read? – BigbearZzz May 14 at 14:55
• Closed and convex are (almost) obvious properties. The property I mentioned follows from looking at what happens if you add $g'$ to $g$ (assuming that everything is real valued). As I said, what may be useful to you depends on what you really want. – fedja May 14 at 15:03
• @fedja What I'm looking for is a somewhat effective way to actually compute the projection like how we compute $L^2$ projection. Computing need not mean an explicit formula. A fixed point iteration or a method to find a sequence that converges to a solution would be fine. – BigbearZzz May 14 at 15:10