Gauss iterated map

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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 v^ is the Fourier transform of v.

The Beppo-Levi space is W˙r,p(Ω)={vD(Ω):|v|r,p,Ω<}
(|.|r,p,Ω denotes the Sobolev semi-norm)

An alternative definition is:

Given mN,sR such that

m+n2<s<n2

Let

Hs={vS|v^Lloc1(Rn),Rn|ξ|2s|v^(ξ)|2dξ<}

Then Xm,s denotes the Beppo-Levi space

Xm,s={vD|αNn,|α|=m,DαvHs}

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

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