MHB Remainder converges uniformly to 0

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The discussion centers on demonstrating that the remainder \( R_n(x) \) from the Lagrange interpolation of the function \( f(x) = e^{\lambda x} \) converges uniformly to 0 as \( n \) approaches infinity. The remainder is expressed as \( R_n(x) = \frac{f^{(n+1)}(\xi_x)}{(n+1)!}\prod_{i=0}^n(x-x_i) \), and participants explore bounding this expression. They establish that the supremum of \( e^{\lambda \xi_x} \) is constant over the interval and analyze the asymptotic behavior of the factorial in the denominator compared to the polynomial terms in the numerator. Ultimately, they conclude that the remainder converges to 0, confirming the uniform convergence of the interpolation.
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Hey! :o

We have the function $f (x) = e^{\lambda x}$ on an interval $[a, b] , \ \lambda \in \mathbb{R}$.

I want to show that the remainder $R_n (x) = f (x)- p_n (x)$ at the lagrange interpolation of $f (x)$ with $n+1$ points from $[a, b]$ for $n \rightarrow \infty$ converges uniformly to $0$.

For each $n$ the points can be chosen arbitrarily in $[a, b]$.
The remainder is defined as $\displaystyle{R_n(x)=f(x)-p_n(x)=\frac{f^{(n+1)}(\xi_x)}{(n+1)!}\prod_{i=0}^n(x-x_i)=\frac{\lambda^{n+1}e^{\lambda \xi_x}}{(n+1)!}\prod_{i=0}^n(x-x_i)}$, right?

Then we have the following:

$\displaystyle{|R_n(x)|=|f(x)-p_n(x)|=\left |\frac{\lambda^{n+1}e^{\lambda \xi_x}}{(n+1)!}\prod_{i=0}^n(x-x_i)\right |=\left |\frac{\lambda^{n+1}e^{\lambda \xi_x}}{(n+1)!}\right |\prod_{i=0}^n|x-x_i|}\\ \displaystyle{\leq \frac{|\lambda|^{n+1}\sup_{\xi_x\in [a,b]}e^{\lambda \xi_x}}{(n+1)!} \cdot \sup_{x\in [a,b]}\prod_{i=0}^n|x-x_i|}$

right?

How could we continue? (Wondering)
 
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Hey mathmari!

Isn't $|x-x_i| \le b-a$? (Wondering)
 
I like Serena said:
Isn't $|x-x_i| \le b-a$? (Wondering)

Ah yes! So, we have the following:

\begin{align*}|R_n(x)|&=|f(x)-p_n(x)|=\left |\frac{\lambda^{n+1}e^{\lambda \xi_x}}{(n+1)!}\prod_{i=0}^n(x-x_i)\right |=\left |\frac{\lambda^{n+1}e^{\lambda \xi_x}}{(n+1)!}\right |\prod_{i=0}^n|x-x_i| \\ & \leq \frac{|\lambda|^{n+1}\sup_{\xi_x\in [a,b]}e^{\lambda \xi_x}}{(n+1)!} \cdot \sup_{x\in [a,b]}\prod_{i=0}^n|x-x_i|\leq \frac{|\lambda|^{n+1}\sup_{\xi_x\in [a,b]}e^{\lambda \xi_x}}{(n+1)!} \cdot \prod_{i=0}^n|b-a|=\frac{|\lambda|^{n+1}\sup_{\xi_x\in [a,b]}e^{\lambda \xi_x}}{(n+1)!} \cdot |b-a|^n\end{align*}

Can we say also something for $\sup_{\xi_x\in [a,b]}e^{\lambda \xi_x}$ ? (Wondering)
 
mathmari said:
Can we say also something for $\sup_{\xi_x\in [a,b]}e^{\lambda \xi_x}$ ?

Isn't it a constant that is independent from $x$? (Wondering)
 
I like Serena said:
Isn't it a constant that is independent from $x$? (Wondering)

Ah, so is $\xi_x$ independent from $x$ ?
 
mathmari said:
Ah, so is $\xi_x$ independent from $x$ ?

Nope. But that supremum is either in $a$ or in $b$, isn't it? (Wondering)
 
I like Serena said:
Nope. But that supremum is either in $a$ or in $b$, isn't it?
Ah ok! So, say $c=\sup_{\xi_x\in [a,b]}e^{\lambda \xi_x}$ we get $$\frac{|\lambda|^{n+1}\cdot c}{(n+1)!} \cdot |b-a|^n$$ But how do we know that this converges to $0$ ? Doesn;t this hold only when $\lambda$ and $|b-a|$ are less than $1$ ? (Wondering)
 
mathmari said:
Ah ok! So, say $c=\sup_{\xi_x\in [a,b]}e^{\lambda \xi_x}$ we get $$\frac{|\lambda|^{n+1}\cdot c}{(n+1)!} \cdot |b-a|^n$$

Shouldn't that be $\frac{|\lambda|^{n+1}\cdot c}{(n+1)!} \cdot |b-a|^{n+1}$? (Wondering)

mathmari said:
But how do we know that this converges to $0$ ? Doesn;t this hold only when $\lambda$ and $|b-a|$ are less than $1$ ? (Wondering)

Which increases asymptotically faster, $c^n$ or $n!$? (Wondering)
 
I like Serena said:
Shouldn't that be $\frac{|\lambda|^{n+1}\cdot c}{(n+1)!} \cdot |b-a|^{n+1}$? (Wondering)

Oh yes, I forgot to counter $i=0$. (Blush)

I like Serena said:
Which increases asymptotically faster, $c^n$ or $n!$? (Wondering)

The factorial? How can we see that? (Wondering)
 
  • #10
mathmari said:
The factorial? How can we see that?

A couple of years ago someone explained it here. (Clapping)
 
  • #11
I like Serena said:
A couple of years ago someone explained it here. (Clapping)

Ooh yes! (Tmi)(Blush) Thank you very much! (Yes)
 
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