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.
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$\begingroup$ You can't expect $\lambda(p_\ell)(x) \underset{x\to\infty}{\sim} f(1) - \tfrac{f'(1)}{x} + ...$ for all $\ell$, because $p_\ell$ for fixed $\ell$ is just an arbitrary polynomial. You must have some limit as $\ell \to \infty$ in mind. $\endgroup$– Robert IsraelCommented Jan 21, 2013 at 15:23
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$\begingroup$ $p_\ell(1) \to f(1)$ by your convergence assumption, but ${p_\ell}'(1)$ can be much different from $f'(1)$. In fact the uniform limit of polynomials on $[0,1]$ need not be differentiable at all. $\endgroup$– Robert IsraelCommented Jan 21, 2013 at 15:28
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$\begingroup$ Of course, I had $l\to \infty$ in mind. For your second remark, what if I assume f smooth around 1? Don't the derivatives of $p_l$ converge to the derivatives of $f$ at 1, as $l\to \infty$? $\endgroup$– pilipilaxCommented Jan 21, 2013 at 15:38
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$\begingroup$ No they don't. For example, if $p_j(x) = T_j(x)/j$ where $T_j$ is the $j$'th Chebyshev polynomial of the first kind, $p_j \to 0$ uniformly on $[-1,1]$ but $p_j'(1) = j$. $\endgroup$– Robert IsraelCommented Jan 21, 2013 at 19:19
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$\begingroup$ Here is the reasonning that maybe makes me misunderstand: Let $a\in [0,1]$ on which every $p_l$ and $f$ are differentiable. In $\tfrac{p_l(x)-p_l(a)}{x-a}=\tfrac{p_l(x)-f(a)}{x-a} + \tfrac{f(a)-p_l(a)}{x-a}$, the first term tends to $\tfrac{f(x)-f(a)}{x-a}$ and the second to $0$ as $l\to\infty$. So when $x\to a$, we have the equality of derivatives. Am I wrong? $\endgroup$– pilipilaxCommented Jan 22, 2013 at 10:23
1 Answer
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.
