Solution to a certain moment problem I'm looking for a function $f$ that satisfies


*

*$f(x)\geq0$

*$\int f(x) \mathrm{d}x=1$

*$\int xf(x) \mathrm{d}x=0$

*$\int x^2f(x)\mathrm{d}x=1$

*$\int x^4f(x)\mathrm{d}x=\delta$

*$\int x^5f(x)\mathrm{d}x=\infty$

*It has a "nice" antiderivative, not like the density of the Student's $t$-distribution with 5 degrees of freedom and not like a series. 


So basically I'm looking for a density $f$ that solves this truncated moment problem.
My approach so far was to start with the right side of the real line:$$f_+(x)=\frac{1}{(x+\rho_1)(x+\rho_2)^2(x+\rho_3)^3}, x\geq0 $$
Then I would solve for $\rho_1,\rho_2,\rho_2$ such that $\frac{1}{2}$ of the specified moments are matched. Later I would get my $f$ by replacing $(\cdot)$ by $|\cdot|$. However, I run into ugly lengthy partial fraction decompositions and ugly integrals. Are there more accessible approaches than these rational functions?
 A: If you set $f(x)=1/x^6$ for $x\ge m>0$ then you have taken care of the tail. The remaining conditions can be met by a polynomial of degree 4 on $-m\le x\le m$ with $f(x):=0$ for $x<-m$.  However, the condition $f(x)\ge 0$ discards many solutions. In general, $m$ will depend on $\delta$. Another way of obtaining a solution is to define $f$ constant on certain intervals. For example, if $\delta=3$ you can set $m=1$ and
$$
f(x):=\begin{cases}0&\text{if }\qquad\quad\ x<-3\\
\frac{2}{75}&\text{if }\ -3\le x<-2\\
\frac{3}{25}&\text{if }\ -2\le x<-1\\
\frac{33}{100}&\text{if }\ -1\le x<0\\
\frac{97}{300}&\text{if }\ \quad\ 0\le x<1\\
\frac{1}{x^6}&\text{if }\ \ \quad 1\le x
\end{cases}
$$
Moreover, you can construct solutions that are $C^{\infty}$, approximating the locally constant solutions by such functions.
${\bf{EDIT:}}$ Playing around with the limits and locally constant functions one can set for example
$$
f(x):=\begin{cases}0&\text{if }\qquad\quad\ x<-2\delta\\
\frac{k}{6\delta^6}&\text{if }\ -2\delta\le x<-\delta\\
a&\text{if }\ 0\le |x|<1\\
b&\text{if }\ \quad\ 1\le |x|<(\delta+1)/2\\
c&\text{if }\ \quad\ (\delta+1)/2\le |x|<\delta\\
\frac{k}{x^6}&\text{if }\ \ \quad \delta\le x
\end{cases}
$$
Then $a,b,c$ can be determined by the conditions 2., 4. and 5. If one set $k:=\delta^2/800$, then $a,b,c$ are positive for 
$\delta\in[1.0026,6.9]$. 
