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, $D^{\prime }$ is the space of distributions, $S^{\prime }$ 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 ${\dot{W}}^{r,p}(\mathrm{\Omega })=\{v\in D^{\prime }(\mathrm{\Omega }):|v{|}_{r,p,\mathrm{\Omega }}<\mathrm{\infty }\}$
($|.{|}_{r,p,\mathrm{\Omega }}$ 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

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

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

$X^{m,s}=\{v\in D^{\prime }|\mathrm{\forall }\alpha \in N^n,\left|\alpha \right|=m,D^{\alpha }v\in H^s\}$

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