I Convergence of ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}##

[Wuberdall]

Hi Physics Forums,

I have a problem that I am unable to resolve.

The sequence $\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}$ of positive integer powers of $\mathrm{sinc}(x)$ converges pointwise to the indicator function $\mathbf{1}_{\{0\}}(x)$. This is trivial to prove, but I am struggling to decide if the convergence is uniform or not.

I hope that someone in here can help me, either by providing a reference or a sketch proof.

Thanks in regards.

 

[haruspex]

Hi Physics Forums,

I have a problem that I am unable to resolve.

The sequence $\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}$ of positive integer powers of $\mathrm{sinc}(x)$ converges pointwise to the indicator function $\mathbf{1}_{\{0\}}(x)$. This is trivial to prove, but I am struggling to decide if the convergence is uniform or not.

I hope that someone in here can help me, either by providing a reference or a sketch proof.

Thanks in regards.

What does the uniform convergence theorem say about continuous functions?

 

[Wuberdall]

What does the uniform convergence theorem say about continuous functions?

Thanks,

Each of the $\mathrm{sinc}^n(x)$ functions in the sequence are continuous. Thus IF the sequence where to converge uniformly to $\mathbf{1}_{\{0\}}$, then $\mathbf{1}_{\{0\}}$ has to be continuous (which it is not). Consequently, the sequence $\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}$ is not uniformly converging.

 

[haruspex]

Thanks,

Each of the $\mathrm{sinc}^n(x)$ functions in the sequence are continuous. Thus IF the sequence where to converge uniformly to $\mathbf{1}_{\{0\}}$, then $\mathbf{1}_{\{0\}}$ has to be continuous (which it is not). Consequently, the sequence $\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}$ is not uniformly converging.

Quite so.