Question

Let us consider a sequence $(p_l)_l$ of polynomials on $[0,1]$ that converge uniformely, as $l\to \infty$, to a function $f$ defined on $[0,1]$.
I denote the polynomials $p_l(t) = \sum_{k=0}^{m(l)} a_k(l) t^k$.
I have also a certain quantity $\lambda(p_l) \in \mathbb{R}$ which has the following asymptotic behaviour :

$\lambda(p_l)(x) = \sum_{k=0}^{m(l)} a_k(l)\tfrac{1}{1+k/x} \underset{x\to\infty}{\sim} \sum_{k=0}^{m(l)} a_k(l) \sum_{j=0}^{\infty}(-1)^j (\tfrac{k}{x})^j$.
So:

$\lambda(p_l)(x) \underset{x\to\infty}{\sim} \sum_{k=0}^{m_l}a_k(l) - \sum_{k=0}^{m(l)} a_k(l)\tfrac{k}{x} + ... = p_l(1) - \tfrac{p_l'(1)}{x} + ...$

Naively, I want to apply the limit to obtain:
$\lambda(p_l)(x) \underset{x\to\infty}{\sim} f(1) - \tfrac{f'(1)}{x} + ...$
Questions : Do I have first check some hypothesis to write it? Do you have any reference about applying the limit in an asymptotic behavour?
Thank you a lot.

Answer 1

Note that the quantity $\lambda p$ depending on the parameter $x$, as you define it for polynomials $p(t):=\sum_{k=0}^m a_k t^k$, that is $$(\lambda p )(x)=\sum_{k=0}^m a_k \frac{1}{1+k/x}\ , $$ can be extended to a positive, bounded linear operator $$\lambda :C^0([0,1])\to C^0_b([0,\infty))$$ taking the function $f\in C^0(I)$ to
$$(\lambda f)(x):=\int_0^1 xt^{x-1}f(t)dt\ .$$ In particular, $(\lambda f)(x)=f(1)+o(1)$ as $x\to+\infty$. For $f\in C^1([0,1])$ we also have, integrating by parts $$(\lambda f)(x)=f(1)-\int_0^1 t^x f'(t)dt=f(1)-f'(1)/x+o(x^{-1}),\qquad (\mathrm{as}\ x\to+\infty) \ .$$

$$*$$

You can also write, changing variable in the integral, ($t=e^{-\tau}$) $$(\lambda f)(x):=x \int_0^{+\infty} e^{-\tau x}f(e^{-\tau})d\tau\ .$$
Thus your $\lambda f$ is the Laplace transform of $f(e^{-\tau})$, times $x$. Thus you can profit of the asymptotic theory for Laplace transform of functions, developed in most textbooks on the subject.