Sobolev Spaces on Domains

“…(Of course, the case of Ω ⊂ R 2 is included through the trivial imbedding into R 3 .) We assume the usual notion of a Hilbertian Sobolev space H s (Ω) for a positive integer s (see, e.g., [2]); Sobolev spaces with real positive index s ≥ 0 are defined through the real or K-method of function space interpolation (see, e.g., [1]), and Sobolev spaces with s < 0 are defined through duality, in a standard fashion. Since we also consider Sobolev spaces on manifolds, we assume that the pull-backs are also sufficiently smooth.…”

Section: Finite Element Spacesmentioning

confidence: 99%

“…This is a standard mollifier argument. We can choose Pt = At char(t), where At is the standard mollification operator (see, e.g., [2]), and char(ω) denotes the characteristic function of a set ω.…”

Section: Inverse-type Estimates On Hp-finite Element Spaces 207mentioning

confidence: 99%

Inverse-type estimates on $hp$-finite element spaces and applications

Georgoulis

2008

Math. Comp.

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Abstract. This work is concerned with the development of inverse-type inequalities for piecewise polynomial functions and, in particular, functions belonging to hp-finite element spaces. The cases of positive and negative Sobolev norms are considered for both continuous and discontinuous finite element functions.The inequalities are explicit both in the local polynomial degree and the local mesh size.The assumptions on the hp-finite element spaces are very weak, allowing anisotropic (shape-irregular) elements and varying polynomial degree across elements. Finally, the new inverse-type inequalities are used to derive bounds for the condition number of symmetric stiffness matrices of hpboundary element method discretisations of integral equations, with elementwise discontinuous basis functions constructed via scaled tensor products of Legendre polynomials.

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“…(2), and the original function f ρ over X is not straightforward. Some estimates for the norm of f * ρ can be found for example in [3].…”

Section: Then For Every M ≥ (mentioning

confidence: 99%

Risk Bounds for Random Regression Graphs

Caponnetto

Smale

2007

Found Comput Math

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Abstract. We consider the regression problem and describe an algorithm approximating the regression function by estimators piecewise constant on the elements of an adaptive partition. The partitions are iteratively constructed by suitable random merges and splits, using cuts of arbitrary geometry. We give a risk bound under the assumption that a "weak learning hypothesis" holds, and characterize this hypothesis in terms of a suitable RKHS.

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Sobolev Spaces on Domains

Cited by 200 publications

References 0 publications

An optimal endpoint trace embedding

An optimal endpoint trace embedding

Inverse-type estimates on $hp$-finite element spaces and applications

Risk Bounds for Random Regression Graphs

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