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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, Dα 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 Xm,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

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

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