Even function and linear combination

Question

Why is it true that if $f(x):\mathbb{R}\longrightarrow\mathbb{R}$ is a positive, even function, decreasing for x>0, then it can be written as a convex linear combination of $\frac{1}{2h}\chi_{[-h,h]}(x)$? Thank you, I'm struggling with this a lot!

Answer 1

Suppose that $f\colon\mathbb{R}\longrightarrow\mathbb{R}$ is a positive even function, which is decreasing and left-continuous on $[0,\infty)$, and such that $f(x)\to0$ as $x\to\infty$. Then for real $t\ge0$ \begin{multline} f(t)=\int_{[t,\infty)}-df(u) =\int_{[0,\infty)}-df(u)1_{t\le u} \\ =\int_{(0,\infty)}-df(u)1_{t\le u} =\int_{(0,\infty)}-df(u)1_{|t|\le u} \\ =\int_{(0,\infty)}-df(u)\chi_{[-u,u]}(t) =\int_{(0,\infty)}\mu(du)\frac{\chi_{[-u,u]}(t)}{2u}, \tag{0} \end{multline} where the measure $\mu$ is defined by the formula $\mu(du)=-2u\,df(u)$. In display (0), the first equality holds because $f$ is left-continuous on $[0,\infty)$ and $f(x)\to0$ as $x\to\infty$; the second equality holds because $t\ge0$; the third equality holds because $f$ is left-continuous at $0$ and, being even, is also right-continuous at $0$ and hence continuous at $0$, so that $\int_{\{0\}}df(u)=0$; and the fourth equality holds because $t\ge0$; the remaining equalities in (0) are trivial.

So, \begin{equation*} f(t)=\int_{(0,\infty)}\mu(du)\frac{\chi_{[-u,u]}(t)}{2u} \end{equation*} for $t\ge0$. Since $f$ is even, the latter identity holds for all real $t$. So, $f$ is indeed a positive mixture of functions $\frac{\chi_{[-u,u]}}{2u}$ with $u>0$.

Also, \begin{multline*} \int_0^\infty \mu(du) =\int_0^\infty -2u\,df(u) =\int_0^\infty -2\,df(u)\int_{(0,u]} dt \\ =\int_0^\infty dt\int_{[t,\infty)}-2\,df(u) =\int_0^\infty dt\,2 f(t)=\int_{-\infty}^\infty dt\,f(t); \end{multline*} the third equality here is an instance of the Fubini--Tonelli theorem.

So, the positive mixture is actually a convex mixture (that is, $\int_0^\infty \mu(du)=1$) if and only if \begin{equation} \int_{-\infty}^\infty du\,f(u)=1. \tag{1} \end{equation}

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It is clear that the condition $f(x)\to0$ as $x\to\infty$ is necessary for (1), given that $f$ is positive and decreasing on $[0,\infty)$.

Also, the example of the function $f=\frac{\chi_{(-1,1)}}2$, which cannot be represented as a positive mixture of functions $\frac{\chi_{[-u,u]}}{2u}$ with $u>0$, shows that the condition that $f$ be left-continuous on $[0,\infty)$ cannot be dropped.