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Pointwise inequalities and approximation in fractional Sobolev spaces

David Swanson (2002)

Studia Mathematica

We prove that a function belonging to a fractional Sobolev space L α , p ( ) may be approximated in capacity and norm by smooth functions belonging to C m , λ ( ) , 0 < m + λ < α. Our results generalize and extend those of [12], [4], [14], and [11].

Pointwise inequalities for Sobolev functions and some applications

Bogdan Bojarski, Piotr Hajłasz (1993)

Studia Mathematica

We get a class of pointwise inequalities for Sobolev functions. As a corollary we obtain a short proof of Michael-Ziemer’s theorem which states that Sobolev functions can be approximated by C m functions both in norm and capacity.

Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

Houman Owhadi, Lei Zhang, Leonid Berlyand (2014)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough (L∞) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution H) minimizing the L2 norm of the source...

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