Beppo-Levi space

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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={vD : |v|r,p,Ω<},

where |⋅|r,p denotes the Sobolev semi-norm.

An alternative definition is as follows: let mN, sR such that

m+n2<s<n2

and define:

Hs={vS : v^Lloc1(𝐑n),𝐑n|ξ|2s|v^(ξ)|2dξ<}[6pt]Xm,s={vD : α𝐍n,|α|=m,DαvHs}

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