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 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^1_\text{loc}(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[edit]

  • 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