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Explicit Construction of Piecewise Affine Mappings with Constraints

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Pompe W.

Bulletin of the Polish Academy of Sciences. Mathematics

2010

tom 58

nr 3

209-220

We construct explicitly piecewise affine mappings u:ℝ ⁿ → ℝ ⁿ with affine boundary data satisfying the constraint div u = 0. As an application of the construction we give short and direct proofs of the main approximation lemmas with constraints in convex integration theory. Our approach provides direct proofs avoiding approximation by smooth mappings and works in all dimensions n ≥ 2. After a slight modification of our construction, the constraint div u = 0 can be turned into det Du = 1, giving new examples of piecewise affine mappings u with det Du = 1.

Unconditional biorthogonal wavelet bases in $L^{p}(ℝ^{d})$

100%

Pompe W.

Colloquium Mathematicum

2002

tom 92

nr 1

19-34

We prove that a biorthogonal wavelet basis yields an unconditional basis in all spaces $L^{p}(ℝ^{d})$ with 1 < p < ∞, provided the biorthogonal wavelet set functions satisfy weak decay conditions. The biorthogonal wavelet set is associated with an arbitrary dilation matrix in any dimension.

Ograniczanie wyników

1 Bulletin of the Polish Academy of Sciences. Mathematics

1 Colloquium Mathematicum

2 Pompe W.

1 2010

1 2002

Wyniki wyszukiwania

Explicit Construction of Piecewise Affine Mappings with Constraints

Unconditional biorthogonal wavelet bases in $L^{p}(ℝ^{d})$