Sobolev Spaces

Sobolev spaces are spaces of smooth functions i.e. functions with bounded (in a mean-squared sense) $$k$th weak derivatives.

Definition

The \(k\)th \(L^2\) Sobolev space is defined as:

\[W^{k, 2}([a, b]) := \{f: [a, b] \rightarrow \mathbb{C} : \int_a^b |f(t)|^2 dt + \int_a^b |f^{(k)}(t)| dt < \infty \}\]

where \(f^{(k)}\) is the \(k\)th weak derivative. We can define other Sobolev spaces by considering other values of \(p\), although these are

Weak Derivatives

What is a weak derivative? The weak derivative of a function \(f: \mathcal{H} \rightarrow \mathbb{C}\) is a function such that for any smooth (i.e. infinitely differentiable) function \(g\) with compact support (i.e. 0 outside some interval), the following equality holds:

\[\int f'(x) g(x) dt = - \int f(x) g'(x) dx\]

As an example, consider \(f(t) = \lvert t \lvert\). The function \(f(t) = sign(t)\) is the 1st weak derivative because for any function \(\phi(t)\), we have

TODO

\[\int_{\mathbb{R}} sign(t) \phi(t) = \int \lvert t \rvert t \phi'(t) dt\]

Because Sobolev spaces’ weak derivatives are bounded in a mean-squared sense, this is a weaker condition than being bounded in a Lipschitz sense.

Returning to Sobolev Spaces

We can define an inner product in a Sobolev space as:

\[\langle f, g \rangle_{W^{k, 2}} := \int_a^b f(t) g(t)^* dt + \int_a^b f^{(k)}(t) g^{(k)}(t)^* dt\]

Bases for Sobolev Spaces

The Fourier basis can also provide a (modified) bases for Sobolev spaces. Specifically, if we define the basis function:

\[\phi_z(t) := \frac{1}{\sqrt{T}} e^{i \frac{2 \pi z}{T} t}\]

Then the \(k\)th weak derivative will become:

\[\phi_z^{(k)}(t) := \Bigg( \frac{2 \pi i z}{T} \Bigg)^k \phi_z(t)\]

And the inner product in the Sobolev space is:

\[\langle \phi_n, \phi_m \rangle_{W^{k, 2}} := \langle \phi_n, \phi_m \rangle_{L^2} + \langle \phi_n^{(k)}, \phi_m^{(k)} \rangle_{L^2}\]

Letting \(f := \sum_z a_z \phi_z\) and \(g := \sum_z b_z \phi_z\), then

\[\langle f, g \rangle_{W^{k, 2}} = \sum_z \Bigg(1 + \big(\frac{2 \pi z}{T})^2 \Bigg) a_z b_z^*\]

Relationship to Hilbert Spaces

Relationship to Reproducing Kernel Hilbert Spaces