For my research, I am trying to find an integrable function $K_n$ over $[0,1]$ that satisfies following equations:
$\int_0^1 K_n (t,x_0)dt=1$
$\int_0^1 t. K_n (t,x_0)dt=x_0$
$\int_0^1 t^2 . K_n (t,x_0)dt=(x_0)^2$
I tried different functions (mostly standart and continuous ones) but none of them is useful for me. Any help or guidance be appreciated, thanks in advance.
You can use this way to get one. Let $$ K_n(t,x_0)=a(t-x_0)^2+b(t-x_0)+c $$ which means $K_n$ is quadratic. Then solving $a,b,c$ from $$ \int_0^1 K_n (t,x_0)dt=1,\int_0^1 t K_n (t,x_0)dt=x_0, \int_0^1 t^2 K_n (t,x_0)dt=(x_0)^2$$ gives $$ a= 30 \left(6 x_0^2-6 x_0+1\right),b= 36 \left(10 x_0^3-15 x_0^2+7x_0-1\right),c= 9 \left(20 x_0^4-40x_0^3+28 x_0^2-8x_0+1\right). $$ So $$ K_n(t,x_0)=\cdots. $$