1
vote
1answer
116 views

Accurate bounds for derivatives of Legendre polynomials

Let $P_n(x)$ denote the $n$th Legendre polynomial. What bounds can one give for $d_{n,m}(x) = |\frac{d^m}{dt^m}P_n(t)|_{t=x}$ assuming that $|x| \le 1$? Clearly $$d_{n,m}(x) \le d_{n,m}(1) = ...
5
votes
1answer
358 views

Numerically finding a Mercer expansion for a given covariance kernel

Let $c(r)$ be a nice, continuous function with compact support. For example, $c(r) = \tfrac 1 5 (1-r)^{11} \big( 5 + 55r + 239 r^2 + 429 r^3 \big)$ for $r \in [0,1]$, and $c(r) = 0$ otherwise. On ...
14
votes
2answers
966 views

Minimal polynomial with a given maximum in the unit interval

Find the lowest degree polynomial that satisfies the following constraints: i) $F(0)=0$ ii) $F(1)=0$ iii)The maximum of $F$ on the interval $(0,1)$ occurs at point $c$ iv) $F(x)$ is positive ...