Projection of functions in $L^1(\Omega)$ onto a $1$-dimensional subspace.

Question

Suppose $f\in L^1(\Omega)$, where $\Omega\subset \Bbb R^n$ with $|\Omega|<\infty$. Let's consider a (probably not unique) constant function $c$ such that $$ \int_\Omega |f-c|\,d \mu = \inf_{t\in\Bbb R} \int_\Omega |f-t|\,d \mu, $$ what can be said about $c$?

More generally, let $Y = \text{span}\{g\}$ for some $g\in L^1(\Omega)$. How do we determine at least one projection of $f$ onto $Y$, i.e. determining a $c \in \Bbb R$ such that $$ \int_\Omega |f-cg|\,d \mu = \inf_{t\in\Bbb R} \int_\Omega |f-tg|\,d \mu, $$ and what can be said about the set of all such $c$?

The similar problem of minimizing similar expression in $L^2$, i.e. finding $c$ such that $ \int_\Omega |f-c|^2\,d \mu = \inf_{t\in\Bbb R} \int_\Omega |f-t|^2\,d \mu, $ since $c$ is just the average of $f$ on $\Omega$. However, the $L^1$ norm is not strictly convex and such a $c$ need not even be unique.

Proving that at least one such $c$ exists shouldn't be difficult, but is there a characterization of the set of all the minimizers?

It seems quite likely that $\int|f-c| \ d \mu$ won't be finite unless $c=0$... — kccu, May 14 '19 at 11:28
For the existence: The map $c\mapsto \int|f-cg|d\mu$ is continuous by Lebesgue's theorem on dominated convergence and it tends to $\infty$ for $c\to \pm \infty$ (if $g\neq 0$) by Fatou's lemma. — Jochen, May 14 '19 at 12:58

Adrián González Pérez · Answer 1 · 2019-05-15T13:34:56.997

0

Edit I misread the question. I post a correction. This is not a solution.

In the case of $\Omega$ being a probability space and $g = 1_\Omega$ the constant functions. I think the characterization is the minimizing $c$ is given by the set of medians. I.e. $c$ such that $$ \mu\{ x \in \Omega : f(x) \geq c ) \geq \frac12 \mbox{ and } \mu\{ x \in \Omega : f(x) \leq c \} \geq \frac12. $$ See: https://en.wikipedia.org/wiki/Median and The Median Minimizes the Sum of Absolute Deviations (The $ {L}_{1} $ Norm)

There is a more general notion of conditional mean with respect to a $\sigma$-algebra defined in [T]. You could take the $\sigma$-algebra $\Sigma_g$ generated by $g$ and take its conditional medians, but there would be elements in $L^1(\Omega;\Sigma_g)$, which is in principle much larger that $t \cdot g$.

[T] Tomkins, R. J., On conditional medians, Ann. Probab. 3, 375-379 (1975). ZBL0307.60002.

edited May 15 '19 at 13:34

answered May 15 '19 at 10:50

Adrián González Pérez

2,346

Do you know of a similar result when $g$ is not a constant function? If you do you may also want to write an answer to this thread on MO https://mathoverflow.net/questions/331509/projection-of-a-function-f-in-l1-omega-onto-a-finite-dimensional-subspace – BigbearZzz May 15 '19 at 11:36
I don't think that characterization works, for intance for the function $f(x) = 1$ if $x<1/3$ and $f(x) = 0$ if $x>1/3$ defined in $[0,1]$, clearly $c = 0$ minimazes the integral, but it doesn't satisfies your criteria – themaker May 15 '19 at 12:52
Oh, I see. I missread the question. Clearly the solution doesn't work for nonconstant $g$. – Adrián González Pérez May 15 '19 at 13:06
Also, you are right @themaker. – Adrián González Pérez May 15 '19 at 13:35

Projection of functions in $L^1(\Omega)$ onto a $1$-dimensional subspace.

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