3 problems

Legendre-function conjecture. These Legendre polynomials can be used as GKN conditions to define the $n/2$ left-definite domain, and together the $2n$ functions constitute a basis…

Increasing-density conjecture. For any given $t>0$, $f_d(t)$ is an increasing function of $d$.

Let $f(x)=\sum_{k=0}^n a_kx^k\in\mathbb{R}[x]$ have all its zeros in $(-1,1)$, and define the linear transformation $T$ by … where $P_k(x)$ is the $k$-th Legendre polynomial. Fisk'…