Gauss iterated map

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 $R^{n}$, $D^{\alpha }$ the differentiation operator with $\alpha$ a multi-index, and ${\hat {v}}$ is the Fourier transform of $v$.

The Beppo-Levi space is ${\dot {W}}^{r,p}(\Omega )=\{v\in D'(\Omega ):|v|_{r,p,\Omega }<\infty \}$
($|.|_{r,p,\Omega }$ denotes the Sobolev semi-norm)

An alternative definition is:

Given $m\in N,s\in R$ such that

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

Let

$H^{s}=\{v\in S'|{\hat {v}}\in L_{\text{loc}}^{1}(R^{n}),\int _{R^{n}}|\xi |^{2s}|{\hat {v}}(\xi )|^{2}\,d\xi <\infty \}$

Then $X^{m,s}$ denotes the Beppo-Levi space

$X^{m,s}=\{v\in D'|\forall \alpha \in N^{n},|\alpha |=m,D^{\alpha }v\in H^{s}\}$

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