Beppo-Levi space explained
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, is the space of distributions, is the space of tempered distributions in, the differentiation operator with a multi-index, and
is the
Fourier transform of .
The Beppo Levi space is
r,p=\left\{v\inD' : |v|r,p,\Omega<infty\right\},
where denotes the Sobolev semi-norm.
An alternative definition is as follows: let such that
-m+\tfrac{n}{2}<s<\tfrac{n}{2}
and define:
\begin{align}
Hs&=\left\{v\inS' : \widehat{v}\in
|\xi|2s|\widehat{v}(\xi)|2d\xi<infty\right\}\ [6pt]
Xm,s&=\left\{v\inD' : \forall\alpha\inNn,|\alpha|=m,D\alphav\inHs\right\}\
\end{align}
Then 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
External links
- L. Brasco, D. Gómez-Castro, J.L. Vázquez, Characterisation of homogeneous fractional Sobolev spaces https://link.springer.com/content/pdf/10.1007/s00526-021-01934-6.pdf
- J. Deny, J.L. Lions, Les espaces du type de Beppo-Levy https://aif.centre-mersenne.org/item/10.5802/aif.55.pdf
- R. Adams, J. Fournier, Sobolev Spaces (2003), Academic press -- Theorem 4.31