Beppo-Levi space

In functional analysis, a branch of mathematics, a Beppo-Levi space, named after Beppo Levi, is a certain space of generalized functions.

In the following, D′ is the space of distributions, S′ is the space of tempered distributions in Rⁿ, D^α the differentiation operator with α a multi-index, and $\widehat{v}$ is the Fourier transform of v.

The Beppo-Levi space is

$\dot{W}^{r,p} = \left \{v \in D' \ : \ |v|_{r,p,\Omega} < \infty \right \},$

where |⋅|_r,p denotes the Sobolev semi-norm.

An alternative definition is as follows: let m ∈ N, s ∈ R such that

$-m + \tfrac{n}{2} < s < \tfrac{n}{2}$

and define:

$\begin{align} H^s &= \left \{ v \in S' \ : \ \widehat{v} \in L^1_\text{loc}(\mathbf{R}^n), \int_{\mathbf{R}^n} |\xi|^{2s}| \widehat{v} (\xi)|^2 \, d\xi < \infty \right \} \\ [6pt] X^{m,s} &= \left \{ v \in D' \ : \ \forall \alpha \in \mathbf{N}^n, |\alpha| = m, D^{\alpha} v \in H^s \right \} \\ \end{align}$

Then X^m,s is the Beppo-Levi space.

References

• Wendland, Holger (2005), Scattered Data Approximation, Cambridge University Press.
• Rémi Arcangéli; María Cruz López de Silanes; Juan José Torrens (2007), "An extension of a bound for functions in Sobolev spaces, with applications to (m,s)-spline interpolation and smoothing" Numerische Mathematik
• Rémi Arcangéli; María Cruz López de Silanes; Juan José Torrens (2009), "Estimates for functions in Sobolev spaces defined on unbounded domains" Journal of Approximation Theory