norm in $L^2$ and $l^2$

Question

let $f:R \rightarrow C $ , $f \in L^2 $ , also let f be continious and defined everywhere

Answer 1

$l^2$ are precisely square integrable sequences if you endow set of all integers (or natural numbers) with counting measure. In this way $l^2$ can be seen as special case of $L^2$, while summation (both finite and that of absolutely convergent series) is special case of integration.

And in case you don't know: counting measure assigns to every element of set measure $1$. So integral of sequence is its sum, provided that it converges.

Answer 2

To begin with, we have to correctly interpret what $f(n)$ means in this context. For example, if $f\in L^2(dx)$, we can change $f$ on the integers and not change the $L^2$ norm of $f$ at all. For example, let $g(x)=f(x)$ if $x$ is not an integer and let $g(n)=0$ then $||f||_{L^2}=||g||_{L^2}$ but the $\ell^2$ norms of the associated sequences are certainly not equal. You can also ``break'' it in another way by setting $g(n)=1$ and getting an infinite $\ell^2$ norm.

So, in general, the answer to your question is that there is no connection.

However, in special cases there are. One that comes to mind are the various Paley-Wiener spaces. This space is defined as the set of functions in $L^2(\mathbb{R})$ whose fourier transform is in $L^2([-\pi,\pi])$ (that is, $\hat{f}$ is supported on $[-\pi,\pi]$.) This is a special case of a Reproducing Kernel Hilbert Space - this means that for every $x\in\mathbb{R}$ there is a function $K_x$ in the Paley-Wiener space such that $f(x)=\langle f, K_x\rangle$ for all $f$ in the Paley-Wiener space. In this case, the functions $K_x$ are the sinc functions shifted by $x$ that is $K_x(y)=c\frac{\sin(x-y)}{\pi(x-y)}$ (the constant $c$ isn't important. Also, in this case, the functions $K_n$ form an orthonormal basis for the Paley-Wiener space. So, there holds: $$ ||f||_{L^2(dx)}^2=||f||_{PW}^2=\sum |\langle f,K_n\rangle|^2 \sum |f(n)|^2. $$ (This is possibly off by a multiplicative constant.) So, in this case, the answer is yes, you can recover the $L^2$ norm from the $\ell^2$ norm (again, up to a multiplicative constant.)

The point is that in Paley-Wiener space the functions are really nice in the sense that they only contain low-frequency pieces so they don't ``move around'' a lot. This means we can recover the function (almost precisely) from just having data at the integers (or really, any somewhat uniformly spaced sequence.)

There is a lot of literature on Paley-Wiener space and the connections to signal processing that explains more of this (e.g. start with the wikipedia article.)

Answer 3

Example...... contrary to the claim in a comment.

$f(t) = e^{-t^2/2}$. Then $$ \int_{-\infty}^\infty |f(t)|^2\,dt = \int_{-\infty}^\infty e^{-t^2}\,dt = \sqrt{\pi} \approx 1.772454 $$ but $$ \sum_{n=-\infty}^\infty |f(n)|^2 = \sum_{n=-\infty}^\infty e^{-n^2} =\vartheta_3(0,e^{-1}) \approx 1.772637 $$ Not equal.