# Convergence of sequence of polynomials defined by boundary conditions

I'm sorry if my question sounds trivial, but analysis is not my field.

Consider the interval $[a,b]\subset \mathbb{R}$. On $[a,b]$, for every $n\in\mathbb{N}$, $n\ge 3$, I define the polynomials $P_n:=a_{0,n}+a_{1,n}x+\dots +a_{n,n}x^n$ (i.e. $a_{i,n}$ is the coefficient of $x^i$ in $P_n$ and $a_{j,n}=0$ in $P_n$ for $j>n$), the coefficients $a_{j,n}$ are given by the conditions:

$P_n(a)=A,P_n(b)=B,P_n'(a)=C,P_n'(b)=D$, $A,B,C,D\in\mathbb{R}$ for $n=3$ and as $n\ge 3$ I gradually impose that every other derivative of $P_n$ is equal to zero in the two points $a,b$ (for $n=4$ I add the condition $P_4''(a)=0$, for $n=5$ I add $P_5''(a)=0$ and $P_5''(b)=0$, for $n=6$ I add $P_6''(a)=0$, $P_6''(b)=0$ and $P_6'''(a)=0$ and so on).

My questions are:

1) What type of convergence can I expect on the sequence $P_n$ as $n\rightarrow \infty$? Does it converge uniformly?

2) Suppose $P_n$ converges to $P$ in the better possibile way. In addition suppose that $P$ is differentiable, $C\neq D$ and $\frac{B-A}{b-a}\in[min\{C,D\},max\{C,D\}]$. Then is it true $min\{C,D\}\le P'(x)\le max\{C,D\}$ $\forall x\in(a,b)$?

Thank you!

You cannot hope for uniform convergence. Consider for instance the case C=D=0. For n=2k+1, you have $$P_n(x)=A+(B-A)\int_a^x (y-a)^k(b-y)^k\,dy/\int_a^b (y-a)^k(b-y)^k\,dy.$$ For large k, this converges to A+(B-A)H(x-(a+b)/2), where H is the Heaviside function. Since this limit is discontinuous, convergence cannot be uniform.
• yes but I excluded the case $C=D$, is it true also for $C\neq D$? – User28341 Jun 23 '15 at 19:11