- #1

Lajka

- 68

- 0

I was always troubled by the relationships between these modes of convergence ([itex]L^1, L^2[/itex], and [itex]L^{\infty} [/itex] convergences, to be precise), so I took some books and decided to establish some relations between them. For some, I succeeded, for others I did not. Here's what I did so far:

If I is a finite interval:

[itex]L^{\infty} [/itex] implies [itex]L^{1} [/itex] (proven)

[itex]L^{\infty} [/itex] implies [itex]L^{2} [/itex] (proven)

[itex]L^{2} [/itex] implies [itex]L^{1} [/itex] (proven)

none of the converses are true (found counter-examples for each)

If I is an infinite interval:

[itex]L^{\infty} [/itex]

*doesn't*imply [itex]L^{1} [/itex] (counter-example)

[itex]L^{\infty} [/itex]

*doesn't*imply [itex]L^{2} [/itex] (counter-example)

[itex]L^{2} [/itex]

*doesn't*imply [itex]L^{1} [/itex] (counter-example)

However, I'm yet to see if, for infinite intervals:

[itex]L^{1} [/itex] implies [itex]L^{\infty} [/itex]

[itex]L^{2} [/itex] implies [itex]L^{\infty} [/itex]

[itex]L^{1} [/itex] implies [itex]L^{2} [/itex]

Intuitively, I believe that [itex]L^{1} [/itex] (or [itex]L^{2} [/itex]) cannot imply uniform convergence (I just imagine a Gaussian which shrinks (keeping his height constant)). However, I cannot think of an example to disprove if [itex]L^{1} [/itex] implies [itex]L^{2} [/itex].

Also, I should say that I don't know how to prove any of this conclusions for infinite intervals, so I'm just trying to find counter-examples (for finite intervals, I just used Cauchy-Schwarz inequality and mean value theorem, but none of that can be used here).

So, any help is appreciated, thanks.

P.S. Just one more thing that caught up in my mind whilst I was writing this: If I take a Gaussian and make new functions by letting it shrink (but increasing its height this time), that sequence would converge to what? I know that's one of the ways to define a delta function, but in a light of this discussion, I would probably say it would converge pointwise to [itex]f(x) \equiv 0[/itex].