Definition of Sobolev Space

Question

I have a definition that says that the space of functions that satisfy$$\|u\|_{H^m}^2=\sum_{k\in\mathbb{Z}}(1+|k|^2)^m|\hat{u}_k|^2<\infty$$is called Sobolev Space and when $m=1$, this is equivalent to saying, $$\int_0^{2\pi}(|u(x)|^2+|u'(x)|^2)dx<\infty$$

I have a few questions. The Sobolev Space is about the space of functions whose functions have well behaved derivatives in some sense (??). Or at least the norm incorporates some information about the derivative.

1. What is $m$ in this context?

2. I do not see any information about derivative in first statement.

3. That is obvious in second case. So the natural questions is: How does second become equivalent to first when $m=1$?

P.S. The last statement is perfectly clear and makes perfect sense. Since, it says both derivative and function are bounded in certain sense.

Answer 1

1. In this context, $m$ can be a real number.

2. We can find a link between the Fourier coefficients of a function and of its derivative. Hence in the case $m=1$ it's a consequence of Parseval's equality.

Answer 2

Notice that the second expression with $m = 1$ says that $u$ and its derivative are both square-integrable. When $m$ is a positive integer, $u \in H^m$ means that $u$ and all its derivatives up to order $m$ are square-integrable. However, this can be extended to $m$ real, by using the fact that the Fourier transform of $D^m u$ is the same as $(ik)^m \hat{u}$ and using the unitarity of the Fourier transform.